Unstable Klein-Gordon Modes in an

Accelerating Universe

Unstable Klein-Gordon modes in an accelerating universe Dark Energy

-does not behave like particles or radiation Quantised unstable modes

-no particle or radiation interpretation Accelerating universe

-produces unstable Klein-Gordon modes

Plan Solve K-G coupled to exponentially accelerating space

background Canonical quantisation ->Hamiltonian partitioned into stable and unstable components Fundamental units of unstable component have no Fock

representation Finite no. of unstable modes + Stone von Neumann theorem -> Theory makes sense

BASICS

CM

QM QFT

-Qm Harmonic -Fock Space Oscillator

Classical Mechanics Lagrangian

Euler-Lagrange equations

Conjugate momentum

Hamiltonian (energy)

Quantum Mechanics Dynamical variables → non-commuting operators

Most commonly used

Expectation value

Quantum Harmonic Oscillator Hamiltonian – energy operator Eigenstates with eigenvalue

Creation and annihilation operators=

Number operator

Quantum Field Theory Euler-Lagrange equations

→ Klein-Gordon equation Conjugate field

Commutation relations

Hamiltonian density

0)( 2 Rmg

Fock Space Basis where are e’vectors with energy e’value Vectors Vacuum state Creation and annihilation operators Number operator Commutation relations

Klein-Gordon

222 1)16(23

m

Unstable when requires

02

te 1 Change to time coordinate

K-G becomes

0)( 2 Rmg

Canonical Quantisation)}()({),(

)(),,,( *†

0

21

kklmkk lm

l

lmlm

l

lSk

fafaYkr

krJr

'''†

'''

†'''

†'''

],[

0],[

0],[

mmllkkmlkklm

mlkklm

mlkklm

iaa

aa

aa

𝜋=𝜕ℒ𝜕 ��

=𝜕𝜙𝜕𝜂=∑𝑘∈𝑆∑𝑙=0

∞

∑𝑚=−𝑙

𝑙 𝐽𝑙+1

2

(𝑘𝑟 )

√𝑘𝑟𝑌 𝑙𝑚(𝜃 ,𝜑) {𝑎𝑘𝑙𝑚 𝑓 ′

𝑘(𝜂)+𝑎𝑘𝑙𝑚† 𝑓 𝑘′ ∗(𝜂) }

Commutation relations for creation and annihilation operators

Hamiltonian density

Hamiltonian Sum of quadratic terms

Bogoliubov transformation

†

††† ][

21

mkl

klm

mkl

klm

klmmklk lmmklklmklm

aa

aa

DaaaaH

),'(00),'()1(0),'(),'()1(00),'()1(),'(0

),'()1(00),'(

*

*

***

***

kkkkm

kkkkm

kkm

kk

kkm

kk

klklm

ffWffWffWffW

ffWffWffWffW

AD

Bogoliubov transformation preserves Canonical Commutation Relations

†

††† ][

21

mkl

k lm

mkl

k lm

klmmklk lmmklk lmklm

bb

bb

DbbbbH

TDTD klmklm†

†

†

†

†

mkl

klm

mkl

klm

mkl

klm

mkl

klm

bb

bb

T

aa

aa

Bogoliubov Transformation Preserves eigenvalues of

Real when

Purely imaginary when

k lmDI

2

2

00ˆI

II

22 k

klmklm i 22 k

Energy Partitioning

}{2

)( ††††

1022

mklmklk lmklmklmklmmklmklk

l

mlk

SkL bbbbbbbbH

DL HHH

}{2

)( ††††

1022

mklmklklmklmklmklmmklmklk

l

mlk

SkD bbbbbbbbiH

𝑆= {𝑘∈ℝ : (∃ ℓ∈ℕ∪0 ) 𝑗ℓ′ (𝑘 )=0 }

𝜕𝜙𝜕𝑟 =0 ,𝑟=1

𝑆= {𝑘∈ℝ : (∃ ℓ∈ℕ∪0 ) 𝑗ℓ′ (𝑘 )=0 }

𝜕𝜙𝜕𝑟 =0 ,𝑟=1

22 k

Existence of Preferred Physical Representation Stone-von Neumann Theorem guarantees

a preferred representation for HD

HL has usual Fock representation There is a preferred representation for the

whole system

Cosmological Consequences Modes become unstable when

First mode k=2.2 t ≈ now

Modes of wavelength 1.07μm t ≈ 100×current age of universe

Current/Future work This theory is semi-classical Dark energy at really long wavelengths

A quantum gravity theory Dark energy at short wavelengths (we hope!)

Horava Gravity (Horava Phys. Rev. D 2009)

𝜔2=𝑚2+𝑘2+𝑎1𝑘4+𝑎2𝑘6

Candidate for a UV completion General Relativity Higher derivative corrections to the Lagrangian Dispersion relation for scalar fields (Visser Phys. Rev. D 2009)

Development of unstable modes