Antisymmetric metric fluctuations as dark matter By Tomislav Prokopec (Utrecht University) Cosmo 07, Brighton 22 Aug 2007 ˚1˚ Based on publications: T.

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/0503289]; astro-ph/0606315; T. Prokopec and Tomas Janssen, gr-qc/0604094, Class. Quant. Grav. 23 1-15 (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

μν μνμνg =g +B

μν (μν)g =g , [ ]

1B g g g

2

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 curvesFits to the rotation curves for the galaxies: NGC1560, NGC 2903, NGC 4565 and NGC 5055 (Moffat 2004)

0

120 Sun

r 14kpc,

M 10 M

˚3˚

Instabilities of NGTProblems 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/0604094, Class. Quant. Grav. 23 1-15 (2006)

matNGTEH SSSS

Λ2 RgxdG

SN

EH4

16

1

,4

1

12

1 24

BBmHHgxdS BNGT R BBBH

If other terms present ghosts and/or instabilities may develop in FLRW and/or Schwarzschild space-times

BBBB RR ,

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]

gBBBBggmB2,8212

Kalb-Ramond axionIn 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˚

eμναβαμH

))((2

1][ 4

gxdS

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

HHgxdSNGT 12

14

Conformal space-timesThe (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

4 2BNGT 2

1 1S d x H H m B B

12a 4

a = scale factor

The NGT action is then

2μνg =a diag(1,-1,-1,-1)

4 2scalar

1S d x a

2

This is opposite of a scalar field action (kinetic term), for which fluctuations get frozen in

˚6˚

Physical ModesConsider the electric-magnetic decomposition of the Kalb-Ramond B-

field

NB1: is missing may be dynamical

2 2Ba m E 0

IIIIIIIIIIIIII

equations of motion

1 2 3

1 3 2

2 3 1

2 13

0 E E E

B E 0 B B

E B 0 B

B B 0E

2 2B

2a'a m B B xE 0

a

IIIIIIIIIIIIII IIIIIIIIIIIIII

·E 0 IIIIIIIIIIIIIIIIIIIIIIIIIIII

E B 0, IIIIIIIIIIIIII

x

Lorentz “gauge” (consistency) condition implies

B 0

T·B 0 B

NB2: equation is not independent (given by the transverse electric field)

TB

NB3:

LE 0IIIIIIIIIIIIII

NB4: From is a function of

T TT TE B 0 B E IIIIIIIIIIIIIIIIIIIIIIIIIIII

x Physical DOFs (massive case): pseudovector (spin = 1,

parity = +) B

Physical DOF (massless case): longitudinal magnetic field

LB

Ei = spin 1, parity -Bi = 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 quantisationImpose canonical commutation relation on B-field and its canonical

momentum

3

ik.x *k kk k

d kB x a( ) e (k) B ( )b B ( )b

(2

L LIIIIIIIIIIIIII IIIIIIIIIIIIII

IIIIIIIIIIIIIIIIIIIIIIIIIIII

†

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

3 3k k'b ,b (2 ) (k k')

IIIIIIIIIIIIII

†

k (k) 0L

˚9˚

0)(222

LkBma

aa

aa

k 2B

22

Vacuum fluctuations in de Sitter inflation

In De Sitter inflation the scale factor is, such that

When mB~0 the mode equation of motion reduces to conformal vacuum

NB: In conformal vacuum there is very little particle production during inflation

II

1a ( H )

H

2

a' ' a'2 0

a a

Conformal rescaling of the longitudinal B-mode

LL L

cB (x )

B (x ) B (x )a

IIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIII

Mode functions approach those of conformal vacuum (mB<<H)

˚10˚

0)(22

222

Lk

B BHm

k

2

2)2(

/412

12

1)(

4)(

22 Hm

Ok

kHB B

Hm

Lk

ikeB

Radiation eraIn 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

I Ia H ( H )

with the “Wronskian” condition

The matching coefficients are approximately

I

2 22ik/ HI

2 2kI I

1H k k1 2i 2 e ,

2 k H H

2k k

| 1

2I2k

1H2 k

ik ikk k k

1 i iB ( ) 1 e 1 e

k k2k

˚11˚

Spectrum of energy densityThe 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.)

2 20NGT 2 2 20 B NGT6

1T B E ( B) a m E B

2a

IIIIIIIIIIIIIIIIIIIIIIIIIIII

2 x +

2Bm 0

0NGT0 NGT

dk0| T | 0 P (k )

k

3

0NGT0NGT

kP (k, ) T (k, )

2

IIIIIIIIIIIIIIIIIIIIIIIIIIII

24

L L 2 2 2 L 2IBNGT 4 k k k

H a'P (k ) B ( ) B ( ) k a m | B ( )|

a a

˚12˚

Spectrum in radiation eraThe spectrum in radiation era for a massless Kalb-

Ramond field

4I

NGT 4 2 2

H 1 1 sin(2k ) 1 cos(2k )P (k ) 1

a 2 (k ) k 2 (k )

4

NGT4I

8 aP

H

k

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 spectrummassless NGT field

spectrum:

2 4

INGT 2 2 2

a H 1 2 1P' (k ) 1 sin(2k ) 1 cos(2k )

k (k ) k (k )

4

NGT4I

8 aP'

H

k

L 2

NGT

dk0| [B (x, )] | 0 P' (k )

k

IIIIIIIIIIIIII

NB: NGT field is important at horizon crossing, gravitational waves dominate on superhorizon scales

2

GW 2

sin (k )P' (k )

k

˚14˚

Radiation era: massive B field spectrum

eraradiationfunction WhittakerLBk

inflationSitterdefunction HankelLBk

Small B field Small B field massmass

Large B field Large B field massmass

˚15˚

4

NGT4I

8 aP

H

Radiation era: massive B field spectrum (2)

Large B field mass

Small B field mass

˚16˚

4

NGT4I

8 aP

H

Snapshot of spectrum in radiation era

Large B field mass

Small B field mass

˚17˚

4

NGT4I

8 aP

H

k

Spectrum in matter eraThe spectrum in matter era is shown in figure (log-log

plot)

on superhorizon scales (k) the energy density spectrum scales as P~k²

4

NGT4I

8 aP

H

k

on subhorizon scales (k>a/aeq), P~const

on subhorizon scales (1<k<a/aeq), 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

3NGTP a

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”

NGT 4Pa

4I

NGT 3eq

Ha a

˚18˚

Matter era: massive B field spectrum

Small B field Small B field massmass

Large B field Large B field massmass

˚19˚

Snapshots of spectra in matter era ˚20˚

)10(10 eBeIB HmHm 22 2

10B I em H 2

1B I em H 2

Matter era: pressure oscillations

2 2" 3H ' (2H' H ) 4 Ga P

These momentum These momentum space space pressure pressure oscillationsoscillations may leave may leave imprint in the imprint in the relativistic Newton-relativistic Newton-like gravitational like gravitational potential potential ΦΦ, and thus , and thus affect affect CMBCMB & & structure formationstructure formation

˚21˚

2 2 23H ' 3H 4 Ga

Summary & DiscussionIs 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?)

eVHGeVmB413 /1003.0

4/1)1/(0 eqzHmk B

Below the mass scale mBB, the strength of the gravitational interaction may change 41 1310/1.0 GeVHmB μm

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˚

B scalar

a20 i

j ijll

B vector

a20 Qi

Q j ijlGl

du

d

uu 0, u dx

dPerturbations

Naive derivation gives a quadratic effect (unphysical!?)

iiji

jiirir

scalar

nnndTT f

i

))((

3

2

16

)(

4

)( 2

NB3: no dependence on (physical dof of massless theory)

with Tomas Janssen (unpublished)

Sachs-Wolfe effect (2) ˚23˚

B scalar

a20 i

j ijll

B vector

a20 Qi

Q j ijlGl

Scalar and vector perturbations

2

3

2)(

3

4

f

i

dTT

iscalar

“Proper” derivation gives a linear effect in , but again no dependence

ljiijl

ii

vector

nGnQdTT f

i

)(3

4

Vector SW

ds2 D dxdx ,

NB: no dependence on (physical dof of massless theory)

E D uP

D 0,

Redefine line element such that it is conserved along geodesics

Scalar SW

MAIN RESULT: Perturbations induced indirectly (like isocurvature perts)

T g T /T photons

(Q & G are physical dofs only in massive

theory)