Antisymmetric metric fluctuations as dark matter By Tomislav Prokopec (Utrecht University) Cosmo 07, Brighton 22 Aug 2007 ˚1˚ Based on publications: T. Prokopec and Wessel Valkenburg, Phys. Lett. B636, 1-4 (2006)[astro- ph/ ]; astro-ph/ ; T. Prokopec and Tomas Janssen, gr-qc/ , Class. Quant. Grav (2006)

Nonsymmetric Gravitational Theory In 1925 Einstein proposed it as a unified theory of gravity and electromagnetism add an antisymmetric component to the metric tensor It does not work since (a) Geodesic equation does not reproduce Lorentz force (b) Equations of motion do not impose divergenceless magnetic field In 1979 Moffat proposed it as a generalised theory of gravitation: Nonsymmetric Theory of Gravitation ( NGT) change Newton´s Law on large scales -> away with DM? J. Moffat ˚2˚

Galaxy rotation curves Fits to the rotation curves for the galaxies: NGC1560, NGC 2903, NGC 4565 and NGC 5055 (Moffat 2004) ˚3˚

Instabilities of NGT Problems with Nonsymmetric Theory of Gravitation (a) When quantised, in its simplest disguise, NGT contains ghosts (b) There are instabilities (when B couples to Riemann/Ricci Tensor ) Most general problem-free quadratic action in B ˚4˚ T.Prokopec and Tomas Janssen, gr-qc/ , Class. Quant. Grav (2006) If other terms present ghosts and/or instabilities may develop in FLRW and/or Schwarzschild space-times The above NGT action cannot be obtained from a geometric theory! However when one generalises Einstein theory to complex spaces which possess a new symmetry (holomorphy), this program may be attainable [in progress with Christiaan Mantz]

Kalb-Ramond axion In the massless gauge invariant limit, one can define a pseudoscalar field (Kalb-Ramond axion)  as The action reduces to that of a massless minimally coupled scalar NB: the equivalence holds only on-shell, i.e. when equations of motion hold ˚5˚ NB2: when B field is massive then dual of B is a massive vector field; if B couples to curvature tensor or to sources, no local duality transformation exists NB3: COMMON MISCONCEPTION: since B field couples conformally during (de Sitter) inflation, it lives in conformal vacuum during inflation, and no (observable) scale invariant spectrum is generated, contrary to what is claimed in literature based on KR axion studies

Conformal space-times The (symmetric part of) the metric tensor in conformal space time is Note that in the limit a -> ∞ (late time inflation), the kinetic term drops out, and the field fluctuations can grow without a limit a = scale factor The NGT action is then This is opposite of a scalar field action (kinetic term), for which fluctuations get frozen in ˚6˚

Physical Modes Consider the electric-magnetic decomposition of the Kalb-Ramond B-field NB1: is missing may be dynamical equations of motion Lorentz “gauge” (consistency) condition implies NB2: equation is not independent (given by the transverse electric field) NB3: NB4: From is a function of Physical DOFs (massive case): pseudovector (spin = 1, parity = +) Physical DOF (massless case): longitudinal magnetic field E i = spin 1, parity - B i = spin 1, parity + ˚7˚ L

Evolution of fluctuations in Cosmological space times ˚ 8˚ antisymmetric metric (tensor) particles are produced by enhancing vacuum fluctuations produced in inflation during radiation and matter era Evolution of scales in the primordial Universe

Canonical quantisation Impose canonical commutation relation on B-field and its canonical momentum Momentum space equation of motion for the modes NB: Contrary to a scalar field (which decays with the scale factor), the Kalb-Ramond B-field grows with the scale factor a ˚9˚

Vacuum fluctuations in de Sitter inflation In De Sitter inflation the scale factor is, such that When m B ~0 the mode equation of motion reduces to conformal vacuum NB: In conformal vacuum there is very little particle production during inflation Conformal rescaling of the longitudinal B-mode Mode functions approach those of conformal vacuum ( m B <<H) ˚10˚

Radiation era In radiation era, Solutions are Whittaker functions (confluent Hypergeometric functions), which in the massless limit reduce to NB: In matter era no exact solutions to mode equations are known with the “Wronskian” condition The matching coefficients are approximately ˚11˚

Spectrum of energy density The energy density is When the only contribution to the spectrum comes from long. B-field NB1: this spectrum is relevant for coupling to (Einstein) gravity (small scale cosmological perts.) ˚12˚

Spectrum in radiation era The spectrum in radiation era for a massless Kalb-Ramond field on superhorizon scales (k  ) the energy density spectrum scales as P~k²  on subhorizon scales (k  >1), P ~ const.+ small oscillations ˚13˚

Comparison with gravitational wave spectrum massless NGT field spectrum: NB: NGT field is important at horizon crossing, gravitational waves dominate on superhorizon scales ˚14˚

Radiation era: massive B field spectrum Small B field mass Large B field mass ˚15˚

Radiation era: massive B field spectrum (2) Large B field mass Small B field mass ˚16˚

Snapshot of spectrum in radiation era Large B field mass Small B field mass ˚17˚

Spectrum in matter era The spectrum in matter era is shown in figure (log-log plot) on superhorizon scales (k  ) the energy density spectrum scales as P~k²  on subhorizon scales (k  >a/a eq ), P~const  on subhorizon scales (1<k  <a/a eq ), P~1/k² NB1: The bump in the power spectrum in matter era is caused by the modes which are superhorizon at equality, and which after equality begin scaling as nonrelativistic matter NB2: The log divergent part of the spectrum continues scaling in matter era as, such that the energy density becomes eventually dominated by the “bump” ˚18˚

Matter era: massive B field spectrum Small B field mass Large B field mass ˚19˚

Snapshots of spectra in matter era ˚20˚

Matter era: pressure oscillations These momentum space pressure oscillations may leave imprint in the relativistic Newton-like gravitational potential Φ, and thus affect CMB & structure formation ˚21˚

Summary & Discussion Is the antisymmetric metric field a good DARK MATTER CANDIDATE? ANSWER: YES, provided it has the right mass, ˚22˚ Goal 2: understand the origin of B-field mass: cosmological term (too small?); some Higgs-like mechanism, or a strong coupling regime Power is concentrated at a (comoving) scale ~ 1AU (observable consequences for structure formation?) B Below the mass scale m B, the strength of the gravitational interaction may change Goal 1: to construct covariant generalisation of Einstein’s theory that includes a dynamical torsion (mediated through antisymmetric tensor field) [work in progress with Christiaan Mantz, master’s student] Massive antisymmetric metric field is NOT equivalent to a massive axion but instead to a massive vector field (only when interactions are excluded)

Sachs-Wolfe effect NB: No tensor in 3+1 dimensions Geodesic equation ˚22˚ Perturbations Naive derivation gives a quadratic effect (unphysical!?) NB3: no dependence on  (physical dof of massless theory) with Tomas Janssen (unpublished)

Sachs-Wolfe effect (2) ˚23˚ Scalar and vector perturbations “Proper” derivation gives a linear effect in , but again no  dependence Vector SW NB: no dependence on  (physical dof of massless theory) Redefine line element such that it is conserved along geodesics Scalar SW MAIN RESULT: Perturbations induced indirectly (like isocurvature perts) (Q & G are physical dofs only in massive theory)