Derivation of the Friedmann Equations• The universe is homogenous and isotropic • ds2 = -dt2 + a2(t) [ dr2/(1-kr2) + r2 (dθ2 + sinθ dɸ2)]

where k = 1, 0, -1• The energy momentum tensor is that of a perfect fluid and is

given by Tμν = (p + ρ ) Uμ Uν + p gμν , and becomes

Tνμ = diag (- ρ, p , p, p ) at the rest frame.

The conservation of energy; μ Tμ0 = 0 0ρ – 3(å/a)(ρ + p) = 0

For a perfect fluid, ρ and p are related by p = w ρ where w = 0 (matter ) dust , w = 1 radiation ,

w = -1 vacuum energy The energy conservation, [(dρ/dt)/ρ] = -3 (1+w) [(da/dt)/a]

The Einstein Equations, Rμν – (1/2) gμν R = 8π GN Tμν or Rμν = 8π GN ( Tμν - (1/2) gμν T ) results in

Friedmann Equations:

μν = 00 (ä/a) = -(4π GN /3)(ρ + 3p)

μν = ij (å/a)2 = (8π GN /3)ρ – (k/a2)

(å/a) : Hubble parameter

The observation of the acceleration of the expansion of

the universe implies (ä/a) > 0 , that is, most of the

dark energy is in the form of vacuum energy i.e.

cosmological constant

A symmetry for vanishing cosmological constant

Recai Erdem, 2006

The Theoretical Source of the Cosmological Constant

• Einstein field equations : Rμν – (1/2) gμν R = 8GNTμν - gμν Λ( i.e gravitational action: IG = (1/ 8GN)-g d4x (R - Λ) + -g d4x ℒ )

Λ: cosmological constant energy density with the energy-momentum tensor

Tμν = - gμνρ , ρ = Λ /(8GN)

(with positive energy density and negative pressure for Λ>0)ρ can be considered as a vacuum energy densityρ = <ℒ>.

Motivation for studying Cosmological Constant

• A conservative candidate for a dark energy (component) with a repulsive force to explain the accelerated expansion rate of the universe a positive Cosmological Constant

• Any contribution to the vacuum energy (e.g. vacuum polarization in quantum field theory) contributes to the cosmological constant

Cosmological Constant Problems• 1- Why so huge discrepancy between the observed and the

theoretical values of Λ?

(Λtheoretical /Λexperimetal ) ~ 1041 – 10118

Due to zero-point energies in chromodynamics ~ 1041 - Due to zero-point energies in

gravitation at Planck scalle ~ 10118 • 2- Why is Λ so small? ( in other words, is there a

mechanism which sets Λ to zero or almost to zero?) the subject of this talk

• 3- Why is Λ not exactly equal to zero?• 4- Why is the vacuum energy density so close to the

matter density today?• 5- Does Λ vary with time?

Basic schemes to explain “Why is the cosmological constant so small?”

• 1- Symmetries ( supersymmetry, supergravity, superstrings, conformal symmetry, Signature Reversal Symmetry the subject of this talk )

• 2- Anthropic Considerations• 3- Adjustment Mechanisms• 4- Changing Gravity• 5- Quantum Cosmology

A symmetry for vanishing cosmological constant ( Signature Reversal Symmetry)

The First Realization• Require action functional be invariant under

xA i xA , gAB gAB

( A = 0, 1, 2,.....D-1 where D is the dimension of the spacetime )

ds2 = gAB dxAdxB - ds2

( R.Erdem, “A symmetry for vanishing cosmological constant “, Phys.Lett. 621 (2005) 11-17, ArXiv hep-th/0410063

S. Nobbenhuis, “Categorizing different approaches to cosmological constant problems”, ArXiv gr-qc/0411093

G. ‘t Hooft, S. Nobbenhuis, “Invariance under complex transformations and its relevance to the cosmological constant problem”, Clas. Quant. Grav. 23 (2006) 3819-3832, ArXiv qr-qc/0602076 )

.

Under this symmetry transformation

• RAB -RAB , R = gABRAB -R , g g , dDx (i)DdDx

• So IG = ∫ g dDx R remains invariant only if

• D = 2(2n+1) , i.e. D = 2, 6, 10, 14, ....

while

IG = ∫ g dDx Λ is forbidden.

If the symmetry is imposed on the Lagrangian then

• We require ℒ - ℒthen through the kinetic terms we obtain

Φ +/- Φ , Φμ +/- Φμ for scalar and gauge

fields

Fermion kinetic term is not allowed on the bulk it may be allowed only on a lower dimenion with D=2n+1, n=1,2,..

In D=2n+1

Ψ eiΨ constant

So the n-point functions < Φ1 Φ2..... Φn> are invariant • and• D=2n+1 for fermions for any n, and for D=4n+2 for scalars and

gauge fields (if one takes + sign) and for even number of n if one adopts – sign

• Moreover two-point functions (propagators), in any case, respect the symmetry

• So this symmetry is more promising for extension into quantum field theory unlike the usual scale symmetry.

• The straightforward application of the symmetry in its present realization only forbids bulk cosmological constant. In order to forbid brane cosmological constant one must impose an additional symmetry.

• The straightforward application of the symmetry in its this realization only forbids bulk cosmological constant in order to forbid brane cosmological constant (that may be induced by the extra dimensional part of the curvature scalar) one must impose an additional symmetry.

• In order to make the brane cosmological constant vanish one may let, in D=6,

• 1- abondon the singnature reversal so that the extra dimensional piece of the metric is invariant under the symmetry

gabdxadxb gabdxadxb

• 2- let gab - gab as x4(5) x5(4) • Then g44 = - g55 which makes the extra dimensional piece of the

curvature scalar zero for metrics with Poincaré invariance

• In the next realization of the symmetry we make both the bulk and the brane cosmological constants zero by signature reversal symmetry alone.

Second Realization• Require the action be invariant under

gAB - gAB , xA xA that is, ds2 = gAB dxadxB - ds2 ( as in the first realization )

(R. Erdem, “A symmetry for vanishing cosmological constant: Another realization”, Phys. Lett. B ( to be published)M.J. Duff and J. Kalkkinen, “Signature reversal invariance”, ArXiv, hep-th/0605273M.J. Duff and J. Kalkkinen, “Metric and coupling reversal in string theory”, ArXiv, hep-th/0605274 )

thenRAB RAB , R = gABRAB -R , g dDx (-1)D/2 g dDx

• So IG = ∫ g dDx R remains invariant if

D = 2(2n+1) , i.e. D = 2, 6, 10, 14, ....

• This realization of the symmetry as well forbids a bulk cosmological constant provided one assumes that the Einstein-Hilbert action is non-zero.

• This symmetry may be used to forbid a possible contribution of the extra dimensional curvature scalar to the 4-dimensional cosmological constant

by taking the 4-dimensional space-time be in the intersection of a 2(2n+1) and a 2(2m+1) dimensional space ( n,m = 1,2,3,.........)

• The 4-Dimensional Space as the Intersection of two Spaces

2(2n+1)dimensionalspace

2(2m+1) dimensional space

The usual 4-dimensional space

• In this case the symmetry transformation becomes

gAB - gAB ; A,B = 0,1,2,3,4’,5’....D’-1; D’ = 2(2n+1)

gCD - gCD ; C,D = 0,1,2,3,4’’,5’’....D’’-1; D’’ = 2(2m+1)

• Under this transformation the metric (with 4-dimensional Poincaré invariance), the curvature scalars, and the volume element transform as

ds2 = gμν dxμdxν + gab dxadxb gμν dxμdxν - gab dxadxb

R4 R4 , Re -Re , g dDx g dDx

where R4, Re are the 4-dimensional, and the extra dimensional pieces

of the curvature scalar.

The contribution of Re vanishes after integration

so the contribution of Re to the cosmological constant is zero

Realization of the symmetry through reflections

For example one may take

gμν = Ω4ημν(x) , gAB = Ω1eηAB(x) , gCD = Ω2eηCD(x)

where Ω4= Ω1(xA) Ω2(xC) , Ωe1= Ω1(xA) , Ωe2 = Ω2(xC)

Ω1(xA) = cos k1x5’ ; Ω2(xC) = cos k2x6’

Then the reflections in extra dimensions given by

k1x5’ π - k1x5’ ; k2x6’ π - k2x6’

induce the symmetry transformation

gAB - gAB ; A,B = 0,1,2,3,4’,5’....D’-1; D’ = 2(2n+1)

gCD - gCD ; C,D = 0,1,2,3,4’’,5’’....D’’-1; D’’ = 2(2m+1)

If the symmetry is identified as a reflection in extra dimensions then the forbiddance of a term materializes as the vanishing of that term after integration. For example

considerds2 = Ω4gμν(x) dxμdxν + Ωe1gAB dxAdxB + Ωe2gCD dxCdxD

where Ω4= Ω1(xA) Ω2(xC) , Ωe1= Ω1(xA) , Ωe2 = Ω2(xC)

and the metric transforms in accordance with the symmetryThe corresponding volume element and the curvature scalar are

g dDx = Ω12(2n+1) Ω2

2(2m+1) g’ dDx , where g’FG = Ω1-1 Ω2

-1 g FG

R = Ω1-1 Ω2

-1 {R’4 + R’e – (D-1)[g’4’(d2 ln Ω1/(dx4’)2)

+ g’4’’(d2 ln Ω’/(dx4’’)2)] – [(D-1)(D-2)/4][g’4’(d ln Ω1/dx4’)2

+ g’4’’(d ln Ω’/dx4’’)2]

Then only the term, R’4 survives in the action. If we simply consider the signature reversal we say that the other terms are forbidden. However if one identifies it as a reflection then it manifest itself as; the other terms vanish after integration.

Transformation rules for fields

• The invariance of the action under the symmetry requires

• ℒ - ℒThen, as in the first realization,

Φ +/- Φ for scalars

while the gauge fields are allowed in D = 4n

Like the first realization fermions live in

D=2n+1. However an ambiguity is introduced by

the veilbeins. A veilbein may roughly be considered as

the square root of the corresponding metric tensor. So

+/- ambiguity due to the signature reversal

transformation is introduced. To circumvent this

problem we double the dimension of the spinor space.

So if we take gAB = cos u ηAB

then the corresponding vielbein may be taken as

EAK’ = [cos (u/2) τ3 + i sin (u/2) τ1] IA

K’

Where τ3(1) denote sigma matrices and denotes tensor product.

Some similarities with and differences from Linde’s model

• The action for Linde’s model is• S = Sψ - Sφ where

Sψ = N ∫ g(x) g’(y) d4x d4y [(Mpl /16π)2 R(x) + ℒ(ψ(x)]

Sφ = N ∫ g(x) g’(y) d4x d4y [(Mpl /16π)2 R(y) + ℒ(φ(y)]• Under the symmetry

gμν (x) g’μν (y) , ψ(x) φ(y)

If one imposes

S - S

then this symmetry forbids a cosmological constant

This conclusion is somewhat similar to what we

have found in the sense that S - S corresponds to

a vanishing cosmological constant. On the other

hand our scheme does not contain ghosts.

In SummaryBoth realizations of the symmetry are quite similar:• Both impose zero bulk cosmological constant• In the first realization one needs an extra symmetry to

make the possible contribution of the curvature scalar to cosmological constant zero while in the second realization this may be accomplished through the same symmetry provided we put the usual 4-dimensional space to the intersection of two spaces.

• Scalars live in D = 2(2n+1) and fermions in D = 2n+1 for both realizations while the gauge fields live in D =2(2n+1), D = 4n for the first and the second realizations, respectively. The n-point functions are invariant in both.

• The symmetry seems to be promising.