Supersymmetric Dark Energy

Neven Bilić

Ruđer Bošković Institute

Zagreb, Croatia

BW2011, Donji Milanovac, 31 Aug 2011

Outline

1. Introductory remarks

a) Cosmological considerations

b) Vacuum energy

2. Motivation for SUSY

3. Summary

4. The model

5. Calculations of the vacuum energy

density and pressure

Einstein’s equations take the form

2

2 0

0

0

0

8

3

4( )

3

i

i

i

a GH T

a

a GT T

a

where Tµν is the energy momentum tensor

The Hubble “constant” H describes the rate of the expansion.

a) Cosmological considerations

In a homogeneous, isotropic and spatially flat spacetime

(FRW), i.e., with metric2222 )( xdtadtds

1.Introductory remarks

Owing to the isotropy we can set

0 1 2 3

0 1 2 3;T T T T p

In a hydrodynamical description in which Tµν

represents a perfect fluid. i.e.,

pguupT )(

p and ρ may be identified with pressure and density.

This identification is correct only in a comoving frame,

i.e., when the cosmic fluid velocity takes the form

00

00

1( ,0,0,0); ( ,0,0,0)u g u

g

We obtain the Friedmann equations

2

2 8

3

4( 3 )

3

a GH

a

a Gp

a

b) vacuum energy

p

8 G

and we reproduce Einsten’s equations with a cosmological

constant equal to

If we assume T g

then

Thus, a nonzero cosmological constant implies the dS

symmetry group of space-time rather than the Poincar´e

group which is the space time symmetry group of Minkowski

space.

This metric describes the so called flat patch of de Sitter

(dS) spacetime with the de Sitter symmetry group.

In this case the metric takes the form

2222 xdedtds Ht

We have a universe with an accelerating expansion.

It is generally accepted that the cosmological constant

term which was introduced ad-hoc in the Einstein-Hilbert

action is related to the vacuum energy density of

matter fields. It is often stated that the vacuum energy

density estimated in a quantum field theory is by about 120

orders of magnitude larger than the value required by

astrophysical and cosmological observations.

e.g., S.Weinberg, Rev. Mod. Phys., 61 (2000)

1( )

2g V

L

T g L

Consider a real scalar field. Assuming the so called

minimal interaction, the Lagrangian is

with the correspoding energy-momentum tensor

2 2

vac 2

2 2

vac 2

1 1( ) ( )

2 2

1 1( ) ( )

2 6

Va

p Va

0

vac 0

vac

1

3

i

i

i

T

p T

= >H

We define

Where < A> denotes the vacuum expectation value of

an operator A. In FRW spacetime

32 2

vac 3

3 2

vac 3 2 2

1

2 (2 )

1

6 (2 )

d kk m

d k kp

k m

4 2 2 2

vac 2 2 2 2

4 2 2 2

vac 2 2 2 2

1ln ....

16 16 64

1 1 1ln ....

3 16 3 16 64

K m K K

m

K m K Kp

m

For a free massive field in flat spacetime one finds

and with a 3-dim momentum cutoff K we obtain

4 473 4Pl

vac 2 2... 10 GeV

16 16

K m

compared with the observed value

Assuming that the ordinary field theory is valid up to

the scale of quantum gravity, i.e. the Planck scale, we find

47 4

vac 10 GeV

eff vac

In addition to the vacuum fluctuations of the field there may

exist an independent cosmological term Λ equivalent to

so that one would find an effective vacuum energy

In order to reproduce the observed value one needs

a cancelation of these two terms up to 120 decimal places!

The problem is actually much more severe as we have many

other contributions to the vacuum energy from different fields

with different interactions and all these contributions must

somehow cancel to give the observed vacuum energy.

8 G

Fine tuning problem

In fact, there would be no problem if there were no gravity!

In flat space one can renormalize the vacuum energy

by subtracting the divergent contributions since the energy

is defined up to an arbitrary additive constant.

However , in curved space this cannot be easily done because

the energy is a source of the gravitational field and

adding (even constant) energy changes the spacetime

geometry .

Adding gravity

Question No 1

Can supersymmetry cure the mentioned problems?

At least we know that in a field theory with exact SUSY the

vacuum energy, and hence the cosmological constant, is

equal to zero as the contributions of fermions and bosons to

the vacuum energy precisely cancel!

2. Motivation for SUSY

Unfortunately, in the real world SUSY is broken at small

energy scales. The scale of SUSY breaking required by

particle physics phenomenology must be of the order of 1 TeV

or larger implying Λ still by about 60 orders of magnitude too

large.

Our aim is to investigate the fate of vacuum energy when

an unbroken supersymmetric model is embedded in

spatially flat, homogeneous and isotropic spacetime.

In addition, we assume the presence of a dark energy

type of substance obeying the equation of state

pDE =wρDE, with w<0.

Question No 2

How does the SUSY vacuum behave in curved spacetime,

e.g., in de Sitter spacetime?

This type of “soft” supersymmetry breaking is known

in supersymmetric field theory at finite temperature

where the Fermi-Bose degeneracy is lifted by statistics.

Das and Kaku, Phys. Rev. D 18 (1978)

Girardello, Grisaru and Salomonson, Nucl. Phys. B 178 (1981)

The space time symmetry group of an exact SUSY is

the Poincar´e group.. The lack of Poincar´e symmetry will lift

the Fermi-Bose degeneracy and the energy density of

vacuum fluctuations will be nonzero.

The final expressions for the vacuum energy density and

pressure are free from all divergent and finite flat-spacetime

terms.The dominant contributions come from the leading terms

which diverge quadratically.

2 2

cut Pl

3m

N

2 2

2cutcut cut2 2

1 ( ln )8

N a

a

2 2

2cutcut cut2 2

2 1 ( ln )24

N a ap

a a

3. Summary

λ an arbitrary positive parameter

N number of chiral species

0 1

NB, Phys Rev D 2011

Combining effects of dark energy with the equation of state

pDE = wρDE and vacuum fluctuations of the supersymmetric

field we find the effective equation of state

eff eff effp w

Friedman equations take the standard form

DEeff eff

2

1 3 1w w

2

eff2

8

3

aG

a

eff eff

4(1 3 )

3

aG w

a

2. The contribution of the vacuum fluctuations to the effective

equation of state is always positive and, hence, it goes against

acceleration!

A similar conclusion was drawn by

who considers massless scalar fields only and removes the

flat-spacetime contribution by hand.

M. Maggiore, PRD (2011)

1. Imposing a short distance cutoff of the order mPl we have

found that the leading term in the energy density of vacuum

fluctuations is of the same order as dark energy (H2 mPl2)

and no fine tuning is needed

3. If we require accelerating expansion, i.e., that the effective

equation of state satisfies weff < −1/3, the range −1 < w < −1/3

is compatible with 0 < λ < 1/2, whereas w < −1 (phantom)

would imply λ> 1/2.

We consider the Wess-Zumino model with N species and

calculate the energy momentum tensorof vacuum fluctuations

in a general FRW space time. The supersymmetric Lagrangian

for N chiral superfields Φi has the form

W(Φ) denotes the superpotential for which we take

Bailin and Love, Supersymmetric Gauge Field Theory and String Theory( 1999)

4. The model

From now on, for simplicity, we suppress the dependence

on the species index i. Eliminating auxiliary fields by

equations of motion the Lagrangian may be recast in the

form

where ϕi are the complex scalar and Ψi the Majorana

spinor fields.

are the curved space time gamma matrices

The symbol denotes inverse of the vierbein.ae

In the chiral (m→0) limit, this Lagrangian becomes

invariant under the chiral U(1) transformation:

This symmetry reflects the R-invariance of the

cubic superpotential

The action may be written as

where LB and LF are the boson and fermion Lagrangians

The Lagrangian for a complex scalar field ϕ may be

expressed as a Lagrangians for two real fields, σ and π

The potential for the scalar fields then reads

Variation of the action with respect to Ψ yields the Dirac

equation of the form

We introduce the background fields and and

redefine the fields

Effective action

;

The effective action at one loop order is is given by

S0 is the classical part of the action and S(2) is the part of

the action which is quadratic in quantum fields.

For the quadratic part we find

Effective masses

V( , )

2m

Effective pottential

F0, 0 m m m m

atF2 , 0

mm m m m

We need the vacuum expectation value of the energy-

momentum tensor. The energy-momentutensor is derived

from S(2) as

4.Calculations of the vacuum energy density and pressure

It is convenient to work in the conformal frame with

metric

where the proper time t of the isotropic observers is

related to the conformal time η as

In particular, we will be interested in de Sitter space-

time with

2 2 2 2( ) ( )ds a d dx

( )dt a d

1Hta eH

Specifically for the FRW metric

The function χk(η) satisfies the field equation

Where ’ denotes a derivative with respect to the

conformal time η .

[N.D. Birell, P.C.W. Davies, Quantum Fields in Curved Space]

As in the flat space time, each real scalar field operator

is decomposed as

• Scalar fields

If m ≠ 0, the solutions may be constructed by making

use of the WKB ansatz

where the function Wk (η)may be found by solving the field

equation iteratively up to an arbitrary order in adiabatic

expansion.

L.E. Parker and D.J. Toms, Quantum Field Theory in Curved Spacetime

To second adiabatic order we find

where

The vacuum expectation value of the components of

the energy-momentum tensor for each scalar is then

calculated from

Rescaling the Majorana field as

we obtain the usual flat space-time Dirac eq. with time

dependent effective mass am.

The quantization of is now straightforward .

• Spinor fields

The Majorana field may be decomposed as

The spinor uks is given by

with the helicity eigenstates

vks is related to uks by charge conjugation

The mode functions ςk satisfy the equation

In massless case the solutions to (a) are plane waves.

For m≠ 0 two methods have been used to solve (a) for

a general spatially flat FRW space-time: 1) expanding

in negative powers of and solving a recursive set

of differential equations

b) using a WKB ansatz similar to the boson case and the

adiabatic expansion

Baacke and Patzold, Phys. Rev. D 62 (2000)

Cherkas and Kalashnikov, JCAP 0701(2007)

22 km

The divergent contributions to these expressions were

calculated for a general spatially flat FRW metric.

Baacke and Patzold, Phys. Rev. D 62 (2000)

From we find the boson and fermion contributions to

the vacuum energy density

0

0T

and from and we obtain the pressure0

0T T

To make the results finite we need to regularize the integrals.

We use a simple 3-dim momentum cutoff regularization

for the following reasons

1. It is the only regularization scheme with a clear physical

meaning: one discards the part of the momentum

integral over those momenta where a different, yet

unknown physics should appear.

2. We apply this in a cosmological context where we have

a preferred reference frame: the frame fixed by the CMB

background or large scale matter distribution.

3. As we have an unbroken SUSY, the cancelation of the

flat-spacetime contributions takes place irrespective

what regularization method we use.

We change the integration variable to the physical momentum

p = k/a and introduce a cutoff of the order Λcut ~mPl. The

leading terms yield

2 2

2cutcut cut2 2

1 ( ln )8

N a

a

2 2

2cutcut cut2 2

2 1 ( ln )24

N a ap

a a

Clearly, we do not reproduce the usual vacuum equation of state.

E.g., in the de Sitter background

vac vac

1

3p

It is convenient to introduce a free dimensionless cutoff

parameter of order such that

The factor 1/N is introduced to make the result independent

of the number of species. A similar natural cutoff has been

recently proposed in order to resolve the so called species

problem of black-hole entropy.

2 2

cut Pl

3m

N

Dvali and Solodukhin, arXiv:0806.3976

Dvali and Gomez PLB (2009)

2

2

3

8

a

G a

Then, the vacuum energy

is of the order H2 mPl2

One may argue that our result is an artifact of the 3-dim

regularization which is not Lorentz covariant. However,

even a Lorentz covariant approach (e.g., Schwinger -

de Witt expansion) would give something like

vac vacT g We do not reproduce the vacuum energy-momentum tensor

in the form required by Lorentz invariance.

vacT Rg R

Where ∙∙∙ denote higher order terms in Riemann tensor,

involving its contractions and covariant derivatives.

Concluding remark

Thank you