1. Introductory remarks
a) Cosmological considerations
b) Vacuum energy
2. Motivation for SUSY
4. The model
5. Calculations of the vacuum energy
density and pressure
3. 1.Introductory remarks
a) Cosmological considerations
In a homogeneous, isotropic and spatially flat spacetime
(FRW), i.e., with metric
ds dt a(t ) dx
2 2 2 2
Einstein’s equations take the form
a 8 G 0
a 4 G 0
(T0 Ti )
a 3 i
where Tµν is the energy momentum tensor
The Hubble “constant” H describes the rate of the expansion.
4. Owing to the isotropy we can set
T00 ; T11 T22 T33 p
In a hydrodynamical description in which Tµν
represents a perfect fluid. i.e.,
T ( p )u u pg
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
u ( g00 ,0,0,0); u ( ,0,0,0)
5. We obtain the Friedmann equations
a 8 G
a 4 G
( 3 p)
6. b) vacuum energy
If we assume T g
and we reproduce Einsten’s equations with a cosmological
constant equal to
7. In this case the metric takes the form
ds dt e
2 2 2 Ht
We have a universe with an accelerating expansion.
This metric describes the so called flat patch of de Sitter
(dS) spacetime with the de Sitter symmetry group.
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
8. 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)
9. Consider a real scalar field. Assuming the so called
minimal interaction, the Lagrangian is
L g V ( )
with the correspoding energy-momentum tensor
T g L
10. We define
vac = T H >
pvac Ti i
Where < A> denotes the vacuum expectation value of
an operator A. In FRW spacetime
2 ( )2 V ( )
2 ( )2 V ( )
11. For a free massive field in flat spacetime one finds
1 d 3k
vac k 2 m2
2 (2 )3
1 d k k
6 (2 )3 k 2 m2
and with a 3-dim momentum cutoff K we obtain
K4 m2 K 2 1 K2
vac ln 2 ....
1 K 4 1 m2 K 2 1 K2
pvac ln 2 ....
3 16 2
3 16 2
12. Assuming that the ordinary field theory is valid up to
the scale of quantum gravity, i.e. the Planck scale, we find
vac ... 1073 GeV 4
16 2 16 2
compared with the observed value
vac 1047 GeV4
13. Fine tuning problem
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.
14. Adding gravity
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
15. 2. Motivation for SUSY
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!
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
16. Question No 2
How does the SUSY vacuum behave in curved spacetime,
e.g., in de Sitter spacetime?
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.
17. 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.
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)
18. 3. Summary
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.
N cut a 2
1 ( cut ln cut )
N cut a 2
p 2 2
2 1 ( cut ln cut )
24 a a
NB, Phys Rev D 2011
λ an arbitrary positive parameter 0 1
N number of chiral species
19. 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
peff weff eff
eff weff w
1 3 1
Friedman equations take the standard form
a 2 8
G(1 3weff ) eff
20. 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
2. The contribution of the vacuum fluctuations to the effective
equation of state is always positive and, hence, it goes against
A similar conclusion was drawn by M. Maggiore, PRD (2011)
who considers massless scalar fields only and removes the
flat-spacetime contribution by hand.
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.
21. 4. The model
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)
22. 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
where ϕi are the complex scalar and Ψi the Majorana
are the curved space time gamma matrices
The symbol ea denotes inverse of the vierbein.
23. In the chiral (m→0) limit, this Lagrangian becomes
invariant under the chiral U(1) transformation:
This symmetry reflects the R-invariance of the
24. 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
25. The potential for the scalar fields then reads
Variation of the action with respect to Ψ yields the Dirac
equation of the form
26. Effective action
We introduce the background fields and and
redefine the fields
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.
28. Effective pottential
V( , )
at 2 , 0 m m mF m
0, 0 m m mF m
29. 4.Calculations of the vacuum
energy density and pressure
We need the vacuum expectation value of the energy-
momentum tensor. The energy-momentutensor is derived
from S(2) as
30. It is convenient to work in the conformal frame with
ds a( ) (d dx )
2 2 2
where the proper time t of the isotropic observers is
related to the conformal time η as
dt a( )d
In particular, we will be interested in de Sitter space-
32. • Scalar fields
As in the flat space time, each real scalar field operator
is decomposed as
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]
33. 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
L.E. Parker and D.J. Toms, Quantum Field Theory in Curved Spacetime
To second adiabatic order we find
34. The vacuum expectation value of the components of
the energy-momentum tensor for each scalar is then
35. • Spinor fields
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 .
The Majorana field may be decomposed as
36. The spinor uks is given by
with the helicity eigenstates
vks is related to uks by charge conjugation
37. 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 m 2 k 2 and solving a recursive set
of differential equations Baacke and Patzold, Phys. Rev. D 62 (2000)
b) using a WKB ansatz similar to the boson case and the
adiabatic expansion Cherkas and Kalashnikov, JCAP 0701(2007)
38. The divergent contributions to these expressions were
calculated for a general spatially flat FRW metric.
Baacke and Patzold, Phys. Rev. D 62 (2000)
39. From T00 we find the boson and fermion contributions to
the vacuum energy density
41. 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.
42. 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
N a 2
1 ( cut ln cut )
8 a 2
N cut a
p 2 2
2 1 ( cut ln cut )
24 a a
Clearly, we do not reproduce the usual vacuum equation of state.
E.g., in the de Sitter background
43. It is convenient to introduce a free dimensionless cutoff
parameter of order such that
Then, the vacuum energy
8 G a 2
is of the order H2 mPl2
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.
Dvali and Solodukhin, arXiv:0806.3976
Dvali and Gomez PLB (2009)
44. Concluding remark
We do not reproduce the vacuum energy-momentum tensor
in the form Tvac vac g required by Lorentz invariance.
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
Tvac Rg R
Where ∙∙∙ denote higher order terms in Riemann tensor,
involving its contractions and covariant derivatives.