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