Superfluids and the Cosmological Constant Problem

Transcription

1 Superfluids and the Cosmological Constant Problem Adam R. Solomon Carnegie Mellon University With Justin Khoury & Jeremy Sakstein (UPenn) arxiv: (JCAP)

2 Outline 1. Intro to the cosmological constant problem(s) 2. Proposed solutions and obstructions 3. Our proposal: finite-temperature superfluid/ Lorentz-violating massive gravity

3 The cosmological constant problem is the unwanted child of two pillars of twentieth century physics: quantum field theory and general relativity. Tony Padilla

4 What is the cosmological constant? Einstein s equations allow a cosmological constant: G μν + Λg μν = 1 T M 2 μν Pl Most pronounced effect is on long distances/cosmology: Λ > 0: late-time acceleration Λ < 0: collapse Observations indicate the Universe is accelerating, consistent with a cosmological constant of size Λ O ( mev 4 M 2 Pl )

5 What could produce Λ? One known theoretical source of Λ: vacuum energy Equivalence principle: the vacuum gravitates Lorentz invariance: T μν ρ vac M 2 Pl Massive fields contribute ρvac ~ m 4 g μν Observed Λ is about the vacuum energy due to neutrino masses, tiny compared to any other particles in Standard Model (and beyond) Electron vacuum energy alone would lead to de Sitter horizon ~ 10 6 km Pauli: the radius of the world nicht einmal bis zum Mond reichen würde (would not even reach the Moon!)

6 Where is the vacuum energy? The problem is even worse with more particle species: ρ vac,electron ρ vac,obs ρ vac,sm ρ vac,obs ρ vac,planck ρ vac,obs The worst theoretical prediction in the history of physics!

7 Cosmological constant problems old and new Cosmic acceleration poses two (logically distinct?) problems: Old problem : why does an enormous vacuum energy not gravitate? New problem : why is there some residual acceleration anyway? Often treated separately! Solve one while ignoring the other This talk: focus on the old problem

8 Approaches to the old problem There are many proposed solutions. I will discuss a small sample. For comprehensive lists, see reviews by Weinberg, Rev. Mod. Phys. 61 (1989) 1-23 Nobbenhuis, gr-qc/ Martin, Burgess, Padilla,

9 Some approaches Anthropics: if Λ were bigger, we wouldn t be around to remark on it \_( ツ )_/ Could follow from string landscape + eternal inflation Dimopoulos: danger of premature application Modifications of gravity: leave Λ alone, but change how it gravitates Degravitation: weaken gravitational response to long-wavelength sources Self-tuning: introduce new field(s) which dynamically counteract Λ

10 Self-tuning and our modest goal We will set a modest goal: field equations solved by Minkowski for arbitrary Λ Necessary but not sufficient condition for solving old CC problem Other criteria: UV insensitivity, radiative stability, no pathologies, agreement with experiments, reproduce observed cosmological history, etc.! Self-tuning runs into a famous obstruction due to Weinberg

11 Weinberg s famous no-go theorem Weinberg (1988); see also Padilla review Assume some fields φ A eat up vacuum energy, T ϕ μν = TΛ μν Assume Poincaré-invariant vacua, ϕ A = const, g μν = η μν

12 Weinberg s famous no-go theorem: two paths Add a potential: L = g(v(ϕ A ) + 2Λ) + derivatives Fine-tuning: only cancels out one specific value of Λ Scaling symmetry: L = gv 0 e 4 ϕ + derivatives Particle masses also vanish; not physical

13 Evading Weinberg s theorem Any no-go theorem has assumptions, pointing the way forward Our (and others ) approach: break Poincaré invariance Φ A = Φ A (x μ ), g μν = η μν e.g., in cosmology, we might have fields with time dependence This implies, at a minimum, that the fields must be accompanied by derivatives To leading order in EFT, one derivative per field

14 Warmup: one scalar Can we degravitate with a single scalar? No. Why? At leading order in derivatives, most general action is L g = M2 Pl 2 [ R 2Λ + m 2 P(X)], X ( Φ) 2 NB: this is the EFT of a zero-temperature superfluid

15 The total stress tensor is 1 M 2 Pl In order to have flat solutions for arbitrary Λ, this must vanish: 1. PX = 0 2. m 2 P = 2Λ T μν = 1 2 m2 ( Pg μν 2P X μ Φ ν Φ ) Λg μν, The latter requires fine-tuning. e.g., ghost condensate, P(X) = X + λx 2 /2 P X P X Solution: Φ = λ -1/2 t. But m 2 P = 2Λ only if λ is carefully tuned against Λ! λ = m2 4Λ

16 Four scalars Fine-tuning can be alleviated with four scalars Φ A Can eliminate problem of inhomogeneous term in stress tensor Consider a simple (and trivially wrong) model: L g = M2 Pl 2 [R 2Λ m2 η AB μ Φ A μ Φ B ]

17 The stress tensor no longer has an inhomogeneous term: 1 M 2 Pl T μν = 1 2 m2 ( η AB α ΦA α Φ B g μν + 2η AB μ Φ A ν Φ B ) Λg μν This admits flat solutions Φ A = αx A, with α = Λ m Degravitates any negative Λ without fine-tuning! Key: instead of tuning theory parameters against Λ, tune integration constant α to Λ Tuning is achieved dynamically

18 Fatal problem: Φ 0 is a ghost L g = M2 Pl 2 [R 2Λ m2 η AB μ Φ A μ Φ B ] = M2 Pl 2 [ R 2Λ + m2 ( Φ 0 ) 2 m 2 ( Φ i ) 2 ] i=1 3 Direct result of internal Lorentz symmetry, which is what we used to remove inhomogeneous term! Blessing and a curse Can we self-tune without a ghost?

19 Ghosts and massive gravity The ghost is easy to understand if we recognize this is a theory of massive gravity Why? Consider adding to GR a mass term of the form L g = M2 Pl 2 [R 2Λ m2 g μν η μν ] This breaks diff invariance due to ημν. Can restore diffs by introducing Stückelberg fields Φ A, η μν η AB μ Φ A ν Φ B And we recover the action discussed in the previous slides

20 Massive gravity and degravitation This connects to an old and venerable story: Massive graviton finite range of gravity gravity acts as high-pass filter screening out sources with wavelengths >> m -1 Λ is infinite-wavelength source! Massive gravity at linear level: Fierz-Pauli (1939) h μν h μν (h μ μ )2 Other linear mass terms have ghosts, just like the example we discussed

21 Massive gravity and degravitation Non-linear: another ghost! (Boulware-Deser, 1972) Unique non-linear, ghost-free, Lorentz-invariant massive gravity: de Rham-Gabadadze-Tolley (drgt, 2010) drgt cannot degravitate large Λ without violating solar system tests of GR ( ) This means we cannot use a theory of the form L g = M2 Pl 2 [R 2Λ + m2 U( μ Φ A, η AB, ε ABCD ) ]

22 Is Lorentz invariance too strong a requirement? For cosmology, we only need SO(3), not SO(3,1) Idea: break internal boosts L g = M2 Pl 2 [R 2Λ + m2 U( μ Φ 0, μ Φ i, δ ij, ε ijk ) ] Aim: use newfound freedom to avoid ghosts (and other pathologies) while retaining degravitation We will analyze this theory for physically sensible degravitating models with Φ 0 = αt, Φ i = βx i such that T Φ μν g=η = M2 Pl Λη μν

23 Interpreting our theory Two physical interpretations of this type of theory: 1. Lorentz-violating massive gravity 2. Low-energy EFT of self-gravitating fluid Difference hinges only on coordinate choice

24 Lorentz-violating massive gravity as a fluid EFT Consider the fields Φ A = Φ A (t,x) to be comoving (Lagrangian) coordinates of a fluid Fluid rest frame is a coordinate system in which Φ A = αx A EFT describing excitations of fluid is a derivative expansion in Φ A obeying any relevant symmetries

25 Building blocks and symmetry At leading order in derivatives, action is built out of C AB g μν μ Φ A ν Φ B L = U(C 00, C 0i, C ij ) Choice of operators determines symmetry-breaking pattern and hence fluid, e.g., Solids: L = U(C ij ) Zero-temperature superfluids: L = U(C 00 ) Finite-temperature superfluids: L = U(C 00, det C ij, det C AB )

26 The importance of coordinates If we move to the fluid rest frame, Φ A = αx A, then we recover the Lorentz-violating massive gravity picture: U(C 00, C 0i, C ij ) Φ A =αx A U(h 00, h 0i, h ij ) We refer to this as unitary gauge e.g., X = C 00 = g μν μ Φ 0 ν Φ 0 Φ A =αx A C 00 = α 2 g 00 Potential for g 00

27 Criteria for degravitation Existence of a Minkowski (degravitating) solution: equations of motion must be solved by g = η for arbitrary Λ No fine-tuning: Tune integration constants, not parameters, against Λ Massless tensors: Tensor mass generically is huge, m ~ O(Λ 1/2 ), unless they are exactly massless. (LIGO: m < ev) No pathologies: No ghosts, tachyons, gradient instabilities, infinite strong coupling, instantaneous modes UV insensitivity: Higher-derivative EFT corrections should not introduce new modes at low energy

28 Strategy 1. Identify parameter space of Lorentz-violating massive gravity which satisfies these criteria 2. Look for symmetries that protect our parameter choice 3. Determine fluid building blocks 4. Solve cosmological constant problem

29 Analysis Work in unitary gauge at linear level, Φ A = (αt, βx i ), g μν = η μν + h μν Most general SO(3)-invariant mass term Dubovsky hep-th/ L mass = M2 Pl 2 (m2 0 h m2 1 h2 0i m2 2 h2 ij + m2 3 h2 ii 2m2 4 h 00 h ii) Massless tensors: m2 = 0 Stability: m1 = 0 (see our paper for gory details! )

30 Searching for a symmetry With our parameter choice in hand, m1 = m2 = 0, we need to find a symmetry to protect it Otherwise we re just fine-tuning and not solving anything! Several candidate symmetries Most either don t degravitate or are UV-sensitive Only symmetry that works: time-dependent, volumepreserving spatial diffeomorphism

31 Time-dependent, volume-preserving spatial diffeomorphisms Fluid language: Φ i Ψ i (Φ 0, Φ i Ψ i ) with det ( Φ j ) = 1 Massive gravity language: break diffs while leaving x i x i + ξ i (t, x j ) with i ξ i = 0 Closely related to time-independent volume-preserving spatial diffs, which set m2 = 0 and forbid massive tensors Adding time dependence further restricts m1 = 0, as necessary for stability

32 Building blocks of degravitation The building blocks invariant under our symmetry, X = ( Φ 0 ) 2 Yb = det( Φ) g Our degravitating theory is therefore L g = M2 Pl 2 [ R 2Λ + m 2 U(X, Yb)] Unique theory that satisfies our criteria This describes a finite-temperature superfluid!

33 Particle content Two massless tensors (ω 2 = p 2 ) No vectors One global scalar (ω 2 = 0) Propagates due to higher-derivative EFT corrections Sounds weird, but well-known from ghost condensation

34 Hints from the UV Including next-to-leading operators in derivative expansion, scalar dispersion relation becomes ( m2 0 m4 4 m 2 3 ) ω2 = ap 4 a: dimensionless constant determined by UV physics Ghost-free condition: m 2 0 m4 4 m 2 3 > 0 No gradient instability: a > 0 UV insensitive: ω 2 p4 M 2, M cutoff

35 Aside: relation to unimodular gravity Well-known attempt to solve cosmological constant problem: unimodular gravity Traceless version of Einstein equations: Λ reintroduced as integration constant This is one of a class of theories invariant under transverse diffeomorphisms (TDiff), corresponding to U(Yb) R μν 1 4 Rg μν = 1 M 2 Pl (T μν 1 4 Tg μν )

36 Aside: relation to unimodular gravity GR: L g = M2 Pl Unimodular: 2 R L g = M2 Pl 2 [ R + m 2 U(Yb)] Ghost condensate: L g = M2 Pl 2 [ R + m 2 U(X)] Our theory: L g = M2 Pl 2 [ R + m 2 U(X, Yb)] Our theory is to unimodular as ghost condensate is to GR! Same symmetry-breaking pattern

37 UV sensitivity of unimodular/tdiff gravity Aside: we find that TDiff theories are UV-sensitive Not a good solution to the CC problem! TDiff: m0 = m3 = m4 Lowest order, no mode: ( m2 0 m4 4 m 2 3 ) ω2 = 0 = 0 ω 2 Next order in EFT: Lorentz invariance implies a massless mode, ap 4 + bω 2 p 2 + cω 4 M 2 = 0 (ω2 p 2 ) 2 M 2

38 Degravitating solutions in practice Finally, we can see how this all works! Example: U(X, Yb) = K 1 2 (X + 1)2 + K 2 2 (Yb)2 Ghost condensate plus term quadratic in Yb Has degravitating solutions! g μν = η μν, Φ 0 = t, Φ i = ( 4Λ K 2 m 2 ) 1/6 x i Can degravitate any positive Λ for K2 < 0 and vice versa No ghost: K1 > 0

39 Degravitating solutions in practice Another simple example: U(X, Yb) = X + γxyb λ 2 (Yb)2 Degravitating solutions: g μν = η μν, Φ 0 = 2Λ m λ 2 2γ t, 2 2γ 2 Λ Φi = λ 1/6 ( m 2 2 ) x i Can degravitate any Λ > λm 2 /4γ 2 Ghost-free: λ > 0

40 Future directions Cosmology! Embed this model in a realistic scenario Is the Minkowski solution an attractor? Hints we may need to (softly) break shift symmetry

41 Summary New method for self-tuning Λ by breaking Lorentz Circumvent Weinberg Unique theory: finite-temperature superfluid Next step: see whether this cancellation can occur dynamically

42 Bonus slides

43 Weinberg s famous no-go theorem Weinberg (1988); see also Padilla review Assume some field φ eats up vacuum energy density for arbitrary Λ, i.e., We ll consider a single scalar field, but proof extends to arbitrary spins/ number of fields Assume Poincaré-invariant vacua, T ϕ μν = TΛ μν ϕ = const, g μν = η μν For constant φ and gμν, the field equations are L ϕ g,ϕ=const. = 0, L g μν g,ϕ=const. = 0

44 Scenario 1: equations of motion hold independently. Translation invariance: δ M x μ = M μ νx ν δ M g μν = 2M (μν), δ M L = L Tr M General variation of L, assuming φ eom, is δl = L g μν δg μν + L ϕ δϕ Lgμν M μν = 2 L g μν M μν This implies (on-shell) L g μν = 1 2 Lgμν L = g(v(ϕ) + 2Λ) + derivatives The metric eom requires V(φ) = -2Λ at min; fine-tuning

45 Scenario 2: equations of motion imply each other, L 2g μν = f(ϕ) L g μν ϕ L ϕ, ϕ dϕ f(ϕ) For constant fields, this implies scaling symmetry, δ ϵ g μν = 2ϵg μν, δ ϵ ϕ = ϵ δ ϵ L = 0 This requires L = V 0 g + derivatives, g μν e 2 ϕ g μν implying (by gμν equation) V 0 e 4 ϕ = 0

46 Our equation of motion (dropping tildes), V 0 e 4ϕ = 0, has two solutions: V0 = 0: fine-tuning again e φ = 0: all particles massless QED