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  • Horndeskis Vector-Tensor Theory

    Mikjel Thorsrud,University of Oslo

    March 30, 2015

  • The Maxwell vector

    Standard kinetic term conformally invariant:LM = 14

    gFF = 14

    gggFF

    FLRW conformally flat

    Thus: (LM)FLRW = 14FF

    Field energy decays adiabatically (a4)

    Needs to break the conformal invariance...

  • History

    Vectors in cosmology:

    Varying fine-structure constant.Beckenstein 1982:Lint = (1/4)I 2()FF

    Production of primordial magnetic fieldsTurner and Widrow (1988) considered:RFF

    , RFF , RF

    F .RAA

    , RAA

    Acceleration driven by vectors.Ford (1989)

    Statistical anisotropiesAckerman, Carroll and Wise (2008)

  • Instability of vector theories

  • Horndeskis theory

    Horndeskis Vector-Tensor Theory:

    Second order field equations, gauge invariant, reduces toMaxwell theory in flat spacetime:

    L = 12R 1

    4FF

    +1

    4M2FF R .

    Remarkably simple containing only one free parameter M2.

    Totally neglected in the literature!

  • Example: inhomogeneous electromagnetic field in FLRW

    Equations of motion:

    Energy-density and pressure components:

  • JHEP02(2013)146

    We considered a non-linear dynamics

    Homogenous vector

    Axisymmetric Bianchi I

    Perfect fluid p = w

    Hamiltonian constraint:1 = 2 + H + M + m

    Dynamical system approach, but H does not couple off...

  • JHEP02(2013)146

    For M2 < 0 a finite time singularity inevitable if |H | > 3M .

    (w = 0.9)

  • JHEP02(2013)146

    Phase space portrait:

  • JHEP02(2013)146

    Similar conclusions for magnetic case:

  • JCAP 1310 (2013) 064

    Flat FLRW:

    S =1

    2

    d3kd

    [(1 +

    4H2

    a2M2

    )( ~A,~k)

    2 +

    (1 4qH

    2

    a2M2

    )k2(~A,~k

    )2],

  • JCAP 1310 (2013) 064

    Hamiltonian stability in flat FLRW:

  • JCAP 1310 (2013) 064

    Hamiltonian stability in Schwarzschild:

    Free of ghosts and Laplacian instabilities if

    14R2s