Antisymmetric metric fluctuations as dark matter By Tomislav Prokopec (Utrecht University) Cosmo 07,...
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Transcript of Antisymmetric metric fluctuations as dark matter By Tomislav Prokopec (Utrecht University) Cosmo 07,...
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)