Unstable Klein-Gordon Modes in an Accelerating Universe

1. Unstable Klein-Gordon Modes in an Accelerating Universe

2. 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

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

4. BASICS • CM • QMQFT • -Qm Harmonic -FockSpace Oscillator

5. Classical Mechanics • Lagrangian • Euler-Lagrange equations • Conjugate momentum • Hamiltonian (energy)

6. Quantum Mechanics • Dynamical variables → non-commuting operators • Most commonly used • Expectation value

7. Quantum Harmonic Oscillator • Hamiltonian – energy operator • Eigenstates with eigenvalue • Creation and annihilation operators • = • Number operator

8. Quantum Field Theory • Euler-Lagrange equations • → Klein-Gordon equation • Conjugate field • Commutation relations • Hamiltonian density

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

10. Klein-Gordon • Change to time coordinate • K-G becomes • Unstable when requires

11. Canonical Quantisation • Commutation relations for creation and annihilation operators • Hamiltonian density

12. Hamiltonian • Sum of quadratic terms • Bogoliubov transformation

13. Bogoliubovtransformation preserves Canonical Commutation Relations

14. Bogoliubov Transformation • Preserves eigenvalues of • Real when • Purely imaginary when

15. Energy Partitioning

18. 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

19. Cosmological Consequences • Modes become unstable when • First mode k=2.2 t ≈ now • Modes of wavelength 1.07μm t ≈ 100×current age of universe

20. 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!)

21. Horava Gravity (HoravaPhys. Rev. D 2009) • Candidate for a UV completion General Relativity • Higher derivative corrections to the Lagrangian • Dispersion relation for scalar fields (VisserPhys. Rev. D 2009)

22. Development of unstable modes