Instabilities during Einstein-Aether Inflation

Transcription

1 Instabilities during Einstein-Aether Inflation Adam R Solomon DAMTP, University of Cambridge, UK Based on work to appear soon: AS & John D Barrow arxiv:1309.xxxx September 3 rd, 2013

2 Why consider Lorentz violation? Lorentz invariance is a key ingredient of the most successful modern theories. What do we gain from theories in which it breaks down?

3 Why consider Lorentz violation? Completeness Anything as crucial to physics as Lorentz symmetry deserves testing. More to the point: LI is tested extremely well in the standard model Mattingly 2005 gr-qc/ , but significant room for improvement remains in gravity, dark matter, and inflation. Lorentz-violating theories (Examples) Gravity: Einstein-aether theory, Hořava-Lifschitz gravity Dark Matter: Blas, Ivanov, & Sibiryakov 2012 arxiv: Inflation: This talk

4 Why consider Lorentz violation? Beyond the Standard Model/Quantum Gravity Beyond the Standard Model/Quantum Gravity: Lorentz symmetry is not guaranteed to survive all the way to the Planck scale. Example: Hořava-Lifschitz gravity Hořava 2009 arxiv: Lorentz-violating gravity theory power-counting renormalizable, UV completion of GR? Note that the low-energy limit of HL is closely related to our model.

5 Why consider Lorentz violation? Modified Gravity Ties to Modified Gravity: We will consider LV in the gravitational sector. This is part of the ongoing investigation into modified gravity, with motivations including completeness and dark energy. LV gravity may have something to say about dark energy e.g., Blas & Sibiryakov 2011 arxiv: closely related to this work.

6 LV in the gravitational sector: Einstein-aether theory Einstein-aether theory is a modification of GR which spontaneously violates Lorentz invariance. Ingredients: Metric g µν with usual Einstein-Hilbert term Aether u µ : Fixed norm and timelike preferred frame at every point, violating LI. Lagrange multiplier λ: Enforces fixed norm condition.

7 LV in the gravitational sector: Einstein-aether theory Einstein-aether theory is a modification of GR which spontaneously violates Lorentz invariance. Ingredients: Metric g µν with usual Einstein-Hilbert term Aether u µ : Fixed norm and timelike preferred frame at every point, violating LI. Lagrange multiplier λ: Enforces fixed norm condition. The action - most general up to second derivatives S = d 4 x 1 g 16πG R +L kin +λ ( u µ u µ +m 2) }{{}}{{} E-H Imposes u 2 = m 2 L kin = c 1 µ u ν µ u ν c 2 ( µ u µ ) 2 c 3 µ u ν ν u µ

8 LV in the gravitational sector: Einstein-aether theory Major restriction: Couplings between u µ and SM matter fields are strongly constrained. But what about a scalar field?

9 Coupling the aether to a scalar field Donnelly & Jacobson 2010 arxiv: Add canonical scalar field φ allow it to couple to the divergence of the aether: θ µ u µ The coupled action S = d 4 x [ 1 g 16πG R +L kin +λ ( u µ u µ +m 2) 1 2 ( φ)2 V(φ,θ) }{{} coupling

10 Coupling the aether to a scalar field Donnelly & Jacobson 2010 arxiv: Add canonical scalar field φ allow it to couple to the divergence of the aether: θ µ u µ The coupled action S = d 4 x [ 1 g 16πG R +L kin +λ ( u µ u µ +m 2) 1 2 ( φ)2 V(φ,θ) }{{} coupling NB: In FRW background, θ = 3mH measures the Hubble rate. Hence this is sometimes considered expansion-coupled inflation.

11 Aether-scalar coupling introduces tachyonic instability AS & John D Barrow 2013 arxiv:0913.xxxx Now consider cosmological perturbations around slow-roll background for general V(φ, θ). In background, aether aligns with cosmic rest frame ū µ = m a(τ) δµ 0. Define aether perturbations V i by 1 u i = m a Vi. 1 u 0 fixed by u µu µ = m 2

12 Aether-scalar coupling introduces tachyonic instability Spin-1 Perturbations During slow-roll inflation, the spin-1 aether perturbations V (±1) obey the wave equation Note the effective mass is 1 a (av) +cs 2 k2 V + M2 eff H 2 τ2v = 0. M 2 eff = V φ V θφ 6mc 1 H (1+O(ε)). V (±1) becomes unstable for M 2 eff < 2H2.

13 Aether-scalar coupling introduces tachyonic instability Spin-1 Perturbations The metric shift perturbation B (1) (spin-1 piece of g 0i ) is related to the aether perturbation by where B (±1) = γv (±1) γ 16πGm 2 c 13. This means that the off-diagonal metric component blows up exponentially, destroying FRW background. (Slightly more complicated to show: this instability persists to the spin-0 modes as well.)

14 Aether-scalar coupling introduces tachyonic instability Parameter constraints For the quadratic Donnelly-Jacobson potential DJ 2010 arxiv: V(φ,θ) = 1 2 M2 φ 2 +µθφ the previous constraint (flat space stability) was DJ 2010 µ M < 2(c 1 +c 2 +c 3 ) O(1). The absence of the inflationary instability µ M 2 6c 1 }{{} O(1) m M Pl }{{} O(10 7 ) This is many orders of magnitude stronger. ( ) M 2. H }{{} O(10 2 )

15 Aether-scalar coupling effects are negligible in the CMB

16 Aether-scalar coupling effects are negligible in the CMB The spin-0 perturbations (which affect CMB) are difficult to solve: 4π G c ( φ δφ a 2 V φ δφ ) = ( 3H 2 A ) Φ 3HΨ G c G k2 Ψ 8π G c c 1 m 2 k 2 Φ +8π G c c 1 m 2 k(v +HV) 8π G c αhkv +4π G c ma V θφ ( φ Φ 3Hδφ) 1 8πG (khφ kψ ) = k 2 φ δφ αav +c 1 m 2 a 1 (akφ) c 1 m 2ξ a ma V θφ φ V 4π G c ( φ δφ a 2 V φ δφ ) = ( 3H 2 A ) Φ+HΦ 2HΨ Ψ 8π G c m 2 c 123 k 2 (Φ+Ψ) γ 3m 3 +4π G c a A V ( θθθ 3Ψ 3HΦ+kV ). 4π G c ma [ Vθφ (3Hδφ+δφ )+ V θφφ φ δφ ] k 2 (Φ+Ψ) = γa 2 (a 2 kv), +4π G c m 2 Vθθφ [ 3Aδφ φ (3Ψ 3HΦ+kV) ], 1 a (av) + c 123m 2 1 ( α(1 γ)a c 1 m 2 + αγ k2 2 V+ ma V ) θφ φ c 1 m 2 V = c 1m 2 + α + αγ c 1 m 2 + αγ k(aφ) 1 ma V θφ a 2c 1 m 2 + αγ kδφ.

17 Aether-scalar coupling effects are negligible in the CMB Can make progress by using the smallness of m/m Pl Strategy: Expand equations of motion in m/m Pl and solve order-by-order.

18 Aether-scalar coupling effects are negligible in the CMB Φ evolution At O(m/M Pl ) 0, metric perturbation Φ on superhorizon scales obeys the usual equation Mukhanov, Feldman, and Brandenberger 1992 ) ) ( ) Φ + (2H A Φ + (2H 2 2A A m A A H Φ+O = 0 where A H 2 H (vanishes in the de Sitter limit). M Pl

19 Aether-scalar coupling effects are negligible in the CMB Φ evolution At O(m/M Pl ), Φ obeys...

20 Aether-scalar coupling effects are negligible in the CMB Φ evolution At O(m/M Pl ), Φ obeys... Φ + ) ) ( ) (2H A Φ + (2H 2 2A A m 2 A A H Φ+O = 0 M Pl the exact same equation.

21 Aether-scalar coupling effects are negligible in the CMB Φ evolution At O(m/M Pl ), Φ obeys... Φ + ) ) ( ) (2H A Φ + (2H 2 2A A m 2 A A H Φ+O = 0 M Pl the exact same equation. This is unusual! This tells us: No superhorizon isocurvature modes (to O(m/M Pl )) Coupling effects on CMB are O(m/M Pl ) corrections. Unexpected cancellations deeper physical mechanism?

22 Summary The model: Einstein-aether theory coupled to scalar inflaton (fairly general) Coupling strongly constrained by instabilities about inflationary background much stronger (5 6 OOM?) than previous constraints No isocurvature modes (to O(m/M Pl )) surprising for a coupled theory! Coupling effects on CMB are O(10 15 ) corrections. Unexpected cancellations deeper physical mechanism? work ongoing