arXiv:hep-th/0411011v5 29 Jan 2005
Landscape, the Scale of SUSY Breaking, and Inflation
Renata Kallosh and Andrei Linde
Department of Physics, Stanford University, Stanford, CA 94305-4060, USA
We argue that in the simplest version of the KKLT model, the maximal value of the Hubble
constant during inflation cannot exceed the present value of the gravitino mass, H<
may have important implications for string cosmology and for the scale of the SUSY breaking in
this model. If one wants to have inflation on high energy scale, one must develop phenomenological
models with an extremely large gravitino mass. On the other hand, if one insists that the gravitino
mass should be O(1 TeV), one will need to develop models with a very low scale of inflation. We
show, however, that one can avoid these restrictions in a more general class of KKLT models based
on the racetrack superpotential with more than one exponent. In this case one can combine a small
gravitino mass and low scale of SUSY breaking with the high energy scale of inflation.
PACS numbers: 11.25.-w, 98.80.-k hep-th/0411011
Soon after the invention of inflationary cosmology, it
became clear that our universe may consist of many expo-
nentially large locally homogeneous regions correspond-
ing to different stable or metastable vacuum states, which
could be used for justification of anthropic principle .
It was argued that the total number of such states ob-
tained as a result of compactification of 10D or 11D uni-
verse can be exponentially large . Early estimates of
the total number of different vacua in heterotic string
theory gave astonishingly large numbers such as 101500
. A more recent investigation, based on the idea of flux
compactification, gave similarly large number of possible
However, none of the stable or metastable 4D vacua
which were known in string theory at that time described
dS space or accelerating/inflationary universe.
progress in finding metastable dS vacua with a positive
cosmological constant was achieved only recently  (see
also , where this problem was addressed for noncriti-
cal string theory). The main idea of KKLT was to find a
supersymmetric AdS minimum taking into account non-
perturbative effects, and then uplift this minimum to dS
state by adding the positive energy density contribution
of D3 branes or D7 branes. The position of the dS mini-
mum and the value of the cosmological constant there
depend on the quantized values of fluxes in the bulk
and on the branes. This provided a starting point for
a systematic investigation of the landscape of all possible
metastable dS string theory vacua [7, 8].
The idea of string theory landscape may be useful, in
particular, for understanding the scale of supersymme-
try breaking. Different points of view on this issue have
been expressed in the literature, with the emphasis on
statistics of string theory flux vacua [9, 10, 11, 12, 13].
Depending on various assumptions, one can either con-
clude that most of the vacua in the string theory land-
scape correspond to strongly broken supersymmetry, or
find a large set of vacua with a small-scale SUSY break-
ing. The methods used in [9, 10, 11, 12, 13] study the
distribution of flux vacua and do not involve the study of
the shape of the effective potential after the uplifting to
dS state. The relation between the parameters of SUSY
breaking, the height of the barrier stabilizing dS vacua
and the Hubble constant during inflation was not studied
so far. This will be the main subject of our investigation.
First of all, we will show that the gravitino mass in
the KKLT scenario with the superpotential used in  is
extremely large, m3/2∼ 6 × 1010GeV. This means that
in the context of this model one should either consider
particle phenomenology with superheavy gravitino, see
e.g. [14, 15], or modify the scenario in such a way as to
make it possible to reduce the gravitino mass by many
orders of magnitude.
We will show also that the Hubble parameter during
inflation in the simplest models based on the KKLT sce-
nario cannot exceed the present value of the gravitino
∼m3/2. This means that even if we succeed
to find the models with small gravitino mass, following
[11, 12, 13], we may face an additional problem of finding
successful inflationary models with extremely small H.
We will then suggest a possible resolution of these
problems in the context of a volume modulus stabiliza-
tion model where the gravitino mass is not related to the
scale of inflation and can be made arbitrarily small.
II.GRAVITINO-HUBBLE RELATION IN THE
SIMPLEST KKLT MODEL
Recent ideas on string cosmology rely on a possi-
bility to stabilize string theory moduli, in particular
the dilaton and the total volume modulus.The sim-
and E. Silverstein, “Beauty is attractive: Moduli trap-
ping at enhanced symmetry points,” JHEP 0405, 030
 L. McAllister and I. Mitra, “Relativistic D-brane scatter-
ing is extremely inelastic,” arXiv:hep-th/0408085.
 S. Watson, “Stabilizing moduli with string cosmology,”
 S. Weinberg, “Does Gravitation Resolve The Ambiguity
Among Supersymmetry Vacua?,” Phys. Rev. Lett. 48
 S. R. Coleman and F. De Luccia, “Gravitational Ef-
fects On And Of Vacuum Decay,” Phys. Rev. D 21,
3305 (1980); T. Banks, “Heretics of the false vac-
uum: Gravitational effects on and of vacuum decay. II,”
 A. H. Guth, “The Inflationary Universe: A Possible Solu-
tion To The Horizon And Flatness Problems,” Phys. Rev.
D 23, 347 (1981); A. H. Guth and E. J. Weinberg, “Could
The Universe Have Recovered From A Slow First Order
Phase Transition?,” Nucl. Phys. B 212, 321 (1983).
 A. Vilenkin, “The Birth Of Inflationary Universes,”
Phys. Rev. D 27, 2848 (1983).
 A. D.Linde,“Eternally
Chaotic Inflationary Universe,” Phys. Lett. B 175, 395
 S. W. Hawking and I. G. Moss, “Supercooled Phase
Transitions In The Very Early Universe,” Phys. Lett.
B 110, 35 (1982); A. A. Starobinsky, “Stochastic De
Sitter (Inflationary) Stage In The Early Universe,” in:
Current Topics in Field Theory, Quantum Gravity and
Strings, Lecture Notes in Physics, eds. H.J. de Vega
and N. Sanchez (Springer, Heidelberg 1986) 206, p. 107;
A. S. Goncharov and A. D. Linde, “Tunneling In Expand-
ing Universe: Euclidean And Hamiltonian Approaches,”
Fiz. Elem. Chast. Atom. Yadra 17, 837 (1986) (Sov. J.
Part. Nucl. 17, 369 (1986)); K. M. Lee and E. J. Wein-
berg, “Decay Of The True Vacuum In Curved Space-
Time,” Phys. Rev. D 36, 1088 (1987); A. D. Linde,
“Hard art of the universe creation (stochastic approach
to tunneling and baby universe formation),” Nucl. Phys.
B 372, 421 (1992) [arXiv:hep-th/9110037]; A. D. Linde,
D. A. Linde and A. Mezhlumian, “From the Big Bang
theory to the theory of a stationary universe,” Phys.
Rev. D 49, 1783 (1994) [arXiv:gr-qc/9306035]. J. Gar-
riga and A. Vilenkin, “Recycling universe,” Phys. Rev.
D 57, 2230 (1998) [arXiv:astro-ph/9707292]; L. Dyson,
M. Kleban and L. Susskind, “Disturbing implications
of a cosmological constant,” JHEP 0210, 011 (2002)
[arXiv:hep-th/0208013]; B. Freivogel and L. Susskind,
“A framework for the landscape,” Phys. Rev. D 70,
126007 (2004) [arXiv:hep-th/0408133]; S. M. Carroll and
J. Chen, “Spontaneous Inflation and the Origin of the
Arrow of Time,” arXiv:hep-th/0410270.