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A MANIFESTLY LOCAL THEORY OF VACUUM ENERGY SEQUESTERINGGeorge Zahariade
UC Davis
KICP Chicago
2015
Exploring Theories of
Modified Gravity
OUTLINE
Cosmological constant problem Prelim: CC problem in Unimodular Gravity Original vacuum energy sequestering
proposal (Kaloper, Padilla 2013-2015) arXiv:1309.6562 arXiv:1406.0711 arXiv:1409.7073
Localized sequestering model (Kaloper, Padilla, Stefanyszyn, Zahariade 2015) arXiv:1505.01492
THE COSMOLOGICAL CONSTANT PROBLEMOf vacuum energy, phase transitions, bubbles and radiative corrections
THE COSMOLOGICAL CONSTANT
Einstein-Hilbert action
Cosmological observations: accelerated expansion of the universe
Main energy component of the universe
What is it?
S = d4x −gMP
2
2R−Λ
⎡
⎣⎢
⎤
⎦⎥− d4∫∫ x −gLm(g
μν ,φ)
Λ : (meV )4
: 68%
VACUUM ENERGY
Equivalence principle: ALL energy gravitates
Two contributions Classical minimum of the potential Zero-point energy of quantum fluctuations
Vacuum energy and the cosmological constant
Problem: high accuracy cancellation!!!
⟨Ω |Tμν | Ω⟩=Vvac gμν
Λ =Λbare +Vvac
COSMOLOGICAL CONSTANT PROBLEM(S)
Classical cosmological constant problem Phase transitions: classical component of Λ
cancellation before XOR after Quantum mechanical cosmological constant
problemQuantum corrections: zero-point energy
component computed in QFT in the locally flat frame (Zel’dovich) Quantum matter Classical gravity
e.g. scalar λφ4 theory
Radiative instability / non-naturalness
PROBLEMS TO SOLVING THE PROBLEM
Radiative instability = knwoledge of the UV details of the theory
Supersymmetry: not enough supersymmetry in the world… Technically natural value of Λ~ (MSUSY)4 too big
No-Go theorem (Weinberg 1989) No local self adjustment mechanisms for
Poincaré invariant vacua
Would imposing global constraints help?
A PEAK AT UNIMODULAR GRAVITYOr the Cosmological Constant problem in GR revisited...
THE MODEL
Henneaux-Teitelboim (1989): unimodular gravity
Diffeomorphism invariance recovered
Key point: New volume form New gauge symmetry
Sg=−1 = d4xMP
2
2R+ Λ 1− −g( )−Lm(g
μν ,φ)⎡
⎣⎢
⎤
⎦⎥∫
c
SHT = d4xMP
2
2R+Λ Úμνρσ∂μAνρσ − −g( )−Lm(gμν ,φ)
⎡
⎣⎢
⎤
⎦⎥∫
Fμνρσ =4∂[ μAνρσ ]
EQUATIONS OF MOTION
Einstein equations
Λ variation
A variationGlobal variable Λ Cosmological constant: constant of motion set by
the boundary conditions and potentially small
Do matter sector quantum corrections spoil this small cosmological constant configuration?
M P2Gμν =−Λgμν +Tμν∗F = 1
∂μΛ =0
RADIATIVE CORRECTIONS
Background solution of the EOMs
QFT in the locally flat frame Einstein box
EOM decomposition
Scalar curvature sector unstable...
M P2 Rμ ν −
14
Rδ μ ν⎛⎝⎜
⎞⎠⎟=T μ
ν −14
T μμ δ
μν
MP2 R=4Λ−T μ
μ
: R−1/2
ORIGINAL SEQUESTERING MECHANISMGR with global constraints (at the price of a not so local action)
(GLOBAL) SEQUESTERING ACTION
Idea: couple matter sector scales and cosmological constant
Action
Matter couples to Λ, λ: global variables σ smooth odd function: determined
phenomenologically μ mass scale ~ MP
S = d4x −gMP
2
2R−Λ−λ 4Lm(λ
−2gμν ,φ)⎡
⎣⎢
⎤
⎦⎥∫ +σ
Λλ 4μ 4
⎛⎝⎜
⎞⎠⎟
%gμν =λ2gμν
FIELD EQUATIONS AND VACUUM ENERGY SEQUESTERING
Einstein equations
Global equations
Key equation:
M P2Gμ
ν =−Λδμν +T μ
ν
′σλ 4μ 4
= d 4x −g∫
4Λ′σ
λ 4μ 4= d 4x −g T μ μ∫
Λ =1
4
d 4x −gT μμ∫
d 4x −g∫≡ ⟨T μ μ ⟩
M P2Gμ
ν =T μν −14⟨T μ
μ ⟩ δμν
DISCUSSION Classical and (matter sector) quantum contributions
to Λ cancel to all loop orders!
Residual cosmological constant (radiatively stable) BUT given by a 4-volume average
Non-zero mass gap: finite volume universe
Weinberg No-Go evaded: global constraints decouple cosmological constant from matter mass scales
BUT non-local action (although GR recovered locally)
LOCALIZED SEQUESTERING MECHANISMOr how to enforce global constraints with local degrees of freedom?
(LOCAL) SEQUESTERING ACTION
Idea: work in Jordan frame and couple the two “eventually global” variables using HT trick
Action
Λ,κ: local fields F = dA , H = dB : 4-forms σ,τ: smooth functions determined
phenomenologically μ mass scale ~ MP
S = d4x −gκ 2
2R−Λ−Lm(g
μν ,φ)⎡
⎣⎢
⎤
⎦⎥∫ + σ
Λμ4
⎛⎝⎜
⎞⎠⎟F∫ + τ
κ 2
MP2
⎛
⎝⎜⎞
⎠⎟H∫
FIELD EQUATIONS
Einstein equations
Variation of F and H
Variation of Λ and κ2
κ 2Gμν = (∇μ∇ν − δ μ ν∇
2 )κ 2 + T μ ν − Λδ μ ν
∂μΛ =0 = ∂μκ2
′σμ4F = ∗1
−′τ
M P2H = ∗1
R
2
VACUUM ENERGY SEQUESTERING
Cosmological constant equation
Key equation
New residual cosmological constant component
Λ =1
4⟨T μ μ ⟩−
1
2
κ 2μ 4 ′τ
M P2 ′σ
H∫F∫
κ 2G νμ = T μ ν −
1
4⟨Tμ μ ⟩δ
μν +
1
2
κ 2μ 4 ′τ
M P2 ′σ
H∫F∫δ νμ
RADIATIVE CORRECTIONS
Background solution of the EOMs
QFT in the locally flat frame Einstein box
Quantum corrections renormalize κ2
Radiatively stable if MUV ~ κ2 ~ μ
: R−1/2
κ ren2 ; κ 2 +O(N)MUV
2 +L
RADIATIVE CORRECTIONS
Vacuum energy loops? Decomposition of the EOMs
Form sector
Cosmological constant sector
Curvature sector
∗F − ∗F = 0 , ∗F =μ 4
′σ, ∗H − ∗H =
M P2
2κ 2 ′τT μ
μ − T μμ( )
T μ ν ≡−Vvacδ νμ +T
μν
4Λ + 4Vvac = T μμ +κ
2 R ,14κ 2 R =−
κ 2 ′τ2MP
2 ∗H
κ 2 Rμ ν −1
4Rδ ν
μ⎛⎝⎜
⎞⎠⎟
= T νμ −
1
4T ρ ρ δ
μν , κ 2 R − R( ) = − T μ
μ − T μμ( )
DISCUSSION
σ,τ smooth: quantum corrections give at most O(1) corrections as long as κ ~ MP
Form sector insensitive to UV details Volume integrals: IR quantities
New residual cosmological constant component also radiatively stable
GR recovered locally (globally, different theories)
Weinberg No-Go evaded: equivalence principle violated, vacuum energy sector non-gravitating
CONCLUSION
Vacuum energy sequestering: mechanism for cancelling matter loop corrections exhaustively via global constraints
Residual cosmological constant radiatively stable
Original scenario: non-local action
Localization via the unimodular HT trick
NB: graviton loops not included
THANK YOU FOR YOUR ATTENTION