1001.0010.txt

Minimal In
ation
Luis Alvarez-Gaum ea, C esar G omezb,a, Raul Jimenezc,a
aTheory Group, Physics Department, CERN, CH-1211, Geneva 23, Switzerland.
bInstituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, E-28049 Madrid, Spain.
cICREA & Institute of Sciences of the Cosmos (ICC), University of Barcelona, 08028 Barcelona, Spain.
Abstract
Using the universal Xsupereld that measures in the UV the violation of conformal invariance we build up a model
of multield in
ation. The underlying dynamics is the one controlling the natural 
ow of this eld in the IR to the
Goldstino supereld once SUSY is broken. We show that 
at directions satisfying the slow roll conditions exist only if
R-symmetry is broken. Naturalness of our model leads to scales of SUSY breaking of the order of 101113Gev, a nearly
scale-invariant spectrum of the initial perturbations and negligible gravitational waves. We obtain that the in
aton eld
is lighter than the gravitino by an amount determined by the slow roll parameter . The existence of slow-roll conditions
is directly linked to the values of supersymmetry and R-symmetry breaking scales. We make cosmological predictions
of our model and compare them to current data.
Key words: SUSY; cosmology; in
ation
1. Introduction
In spite of the enormous success of in
ationary cosmol-
ogy [1, 2, 3, 4, 5, 6, 7, 8, 9] at describing the observed
properties of the Universe, we are still missing a deriva-
tion from rst principles where the in
aton eld is iden-
tied with one, or several, fundamental elds in particle
physics. This manifests itself the in the fact that we still
do not count with a natural way of identifying the in
aton
eld and the properties of its potential required to satisfy
experimental constraints [10, 11].
It was quickly realized after the in
ationary scenario
was proposed more than 30 years ago, that supersymmetry
could provide a natural scenario with plenty of 
at direc-
tions which could lead to in
ation [18, 19, 20, 21, 22, 23].
When the theory couples to supergravity, there are a num-
Email addresses: Luis.Alvarez-Gaume@cern.ch (Luis
Alvarez-Gaum e), cesar.gomez@uam.es (C esar G omez),
jimenez@icc.ub.edu (Raul Jimenez)ber of new problems that appear [24], and we will discuss
some of them later on.
Current observational constraints from CMB tempera-
ture and polarization experiments and large-scale struc-
ture limit the amount the in
aton eld has moved to ap-
proximately <2Mpl[14], where Mplis the reduced Planck
mass. Therefore, in
ationary models that search for the
in
aton at very large energies, like for example chaotic
in
ation, are severely constrained already by current ob-
servations. With the current new generation of CMB ex-
periments (Planck, EBEX, Spider, SPUDS etc...) it will
be possible to further constraint how much the in
aton
eld has displaced during the in
ationary period that gave
rise to our current casual horizon. It is therefore useful to
revisit again the problem of steep directions in SUGRA
models to understand if a 
at direction can be obtained at
all.
In this paper we will suggest a natural embedding of in-

ationary dynamics in the eective low-energy Lagrangian
Preprint submitted to Physics Letters B October 26, 2018arXiv:1001.0010v1  [hep-th]  30 Dec 2009describing supersymmetry breaking. Our approach will
be quite independent of the microphysics underlying su-
persymmetry breaking, and will only rely on universal
properties of this symmetry. Since we are not commit-
ting ourselves to any particular microscopic realization of
supersymmetry breaking, some of our comments about re-
heating for instance will be rather sketchy. A more de-
tailed and precise presentations of our ideas will appear
elsewhere [25]. Like most in
ationary theories containing
supersymmetry, we present a simple model of multield
in
ation (sometimes called hybrid) [26], identify naturally
the in
aton eld and its potential, and then t a few obser-
vational data to estimate the few parameters of our model.
We compute, in particular, the number of e-folding and the
amplitude of density 
uctuations at horizon crossing. It is
surprising to nd that the scale of supersymmetry break-
ing indicated by this analysis is between 10111014GeV.
An interesting spin-o of our model is that the in
aton is
lighter than the gravitino by an amountp, whereis
one of the slow roll parameters (see below).
We would like to stress that in this paper we are always
assuming F-breaking of supersymmetry. In D-breaking
scenarios our arguments do not apply, at least as presented
here1.
2. General framework
Supersymmetry is a natural framework to dene in-

ationary scenarios for two main reasons. First of all,
SUSY naturally leads to the existence of 
at, or nearly

at directions (pseudomoduli), allowing for slow roll sce-
narios. Second, and more important, the order parameter
of supersymmetry breaking is the vacuum energy density.
Hence, naturally associated with its breaking, supersym-
metry contains two main ingredients necessary in in
a-
tionary scenarios: vacuum energy and reasonably 
at di-
rections.
1We thank Gia Dvali for raising this point. See for instance the
last entry in [21]In a remarkable recent work, Komargodski and Seiberg
[27] have presented a new formalism to understand super-
symmetry breaking, its general properties, its non-linear
realizations [28], and a systematic way to understand the
low-energy couplings of goldstinos to other elds. Al-
though many things were known before (see references in
[27]) this work, the presentation is quite insightful, and it
played a major part in the inspiration of this work.
The basic starting point in [27] is the Ferrara-Zumino
multiplets of currents [29]. A vector supereld composed
of the R-symmetry current, the supercurrent, and the en-
ergy momentum tensor. This vector supereld satises the
general relation:
D_J;_=DX: (1)
The chiral supereld Xis essentially dened uniquely2
in the ultraviolet. Following [27] the supereld Xhas the
following properties:
In the UV description of the theory, it appears in the
right hand side of 1, where it represents a measure of
the violation of conformal invariance.
The expectation value of its 2component is the or-
der parameter of supersymmetry breaking. In this
work we are only considering F-breaking of super-
symmetry. We denote by fthe expectation value of
theF-component of X. It will sometimes be useful
to writef=2, whereis the microscopic scale of
supersymmetry breaking.
When supersymmetry is spontaneously broken, we
can follow the 
ow of Xto the infrared (IR). In the IR
this eld satises a non-linear constraint and becomes
2The ambiguities in the supercurrent multiplet and Xare related
to improvement terms in the various currents.
2the \goldstino" supereld3.
X2
NL= 0; (2)
XNL=G2
2F+p
2G+2F: (3)
The scalar component xofXbecomes a goldstino
bilinear. Its fermionic component is the goldstino
fermionG, andFis the auxiliary eld that gets the
vacuum expectation value. A major part in the anal-
ysis in [27] is based on this novel nonlinear constraint
satised by the supereld Xin the IR. As shown
there, the correct normalization of the goldstino su-
pereld to derive all relevant low-energy theorems of
broken supersymmetry is XNL=3
8fX.
Finally,Xgeneralizes the usual spurion couplings ap-
pearing in the description of low-energy supersymmet-
ric lagrangians. If msoftdescribes the soft supersym-
metry breaking masses at low energies, the standard
spurion in the lagrangian is replaced bymsoft
fXNL.
This allows one to write the leading low-energy cou-
plings of the goldtino to other matter elds.
Since we are going to consider goldstino couplings, we will
work with a eld whose expectation values are well below
the Planck scale.
Our proposal is to identify in the UV the in
aton eld
with the scalar component of the supereld X. SinceXis
dened uniquely (up to the ambiguity mentioned in foot-
note one) in the UV, this provides a well dened prescrip-
tion. Furthermore, we will identify the in
ationary period
precisely with the 
ow of Xfrom the UV to the IR i.e.
X!XNL. Note that by making this assumption we do
not need to think of the in
aton as any extra fundamental
eld. In fact, independently of how SUSY is broken, and
3A modied version of the nonlinear constraint (2) appears when
one considers spontaneous R-symmetry breaking. In that case, the
goldstino and the corresponding axion will be part of the same mul-
tiplet.what is the underlying fundamental theory we can always
identify the Xsupereld as well as its scalar component
x. More importantly, by making this assumption we are
identifying the vacuum energy driven in
ation with the
actual SUSY breaking order parameter.
In the supergravity context, once we have the K ahler
potentialK(X;X) and the superpotential W(X), the full
scalar potential is given by [30]:
V=eK
M2(K1
X;XDW DW3
M2jWj2) (4)
with
DW =@XW+1
M2@XKW: (5)
Mis the high energy scale below which we can write the ef-
fective action describing the dynamics of the X-supereld.
It could be the Planck scale, or a GUT scale depending on
the microscopic theory. We will work well below the scale
M, and for simplicity take M=MplIn equation (4) we
can see one of the basic problems in supergravity in
a-
tion [24]. As we will see later on, to satisfy the slow roll
conditions, a necessary condition is that the -parameter,
dened by:
=M2
plV00
V; (6)
be much smaller than one. If we choose a K ahler potential
K(X;X) with R-symmetry, for instance the canonical one
K(X;X) =XX+:::, where the :::represents a function
ofXX, it is easy to see that from the exponent of (4) we
always get a contribution to equal to 1: = 1 +:::, no
matter which component of Xis taken as the in
aton eld.
This of course violates the slow roll conditions. Since we
are considering a situation with supersymmetry breaking
and gravity (early universe), we cannot exclude supergrav-
ity from the picture, and this leads to the -problem in
these theories.
The simplest way out of this problem without unreason-
able ne tuning, is to have explicit R-symmetry breaking
3in the K ahler potential4. If we have explicit R-breaking,
the expansion of Vfor small elds takes the form:
X=M(+i) (7)
V=f2(1 +A1(2+2) +B1(22) +:::)(8)
fis the supersymmetry breaking parameter representing
the expectation value of an F-term, and hence with square
mass dimensions. We assume that Vis locally stable at
least during in
ation. Hence A1B1>0. We express the
potential in terms of the dimensionless elds ;. Their
masses can be read o from (8):
m2
=2f2
M2(A1+B1); m2
=2f2
M2(A1B1):(9)
The numbers A1;B1are taken to be O(1).
One could be more explicit, and choose some super-
symmetry breaking superpotential, like W=fX, and
K ahler potential explicitly breaking R-symmetry, like:
K=XX+ (c=M2)(X3X+XX3) +:::as in [27] lead-
ing to an eective action description of Xfor scales well
belowM. At this stage, we prefer not to consider explicit
examples of UV-completions of the theory.
We consider the beginning of in
ation well below M,
hence the initial conditions are such that ; << 1. In
fact, sinceis the lighter eld, we take this one to be the
in
aton, and consider that initially ;pf=M . For us
the in
ationary period goes from this scale until the value
of the eld is close to the typical soft breaking scale of the
problemmsoft, where the eld X!XNL(2), at this scale
XNLbehaves like a spurion [27] and as shown in Ref. [27],
the leading couplings to low-energy supersymmetric mat-
ter can be computed as spurion couplings, for instance5,
ifQ;V represent respectively low energy chiral and vector
4R-symmetry is a well-known problem in phenological applica-
tions of supersymmetry. R-symmetry does not allow soft breaking
masses for the gauginos; and spontaneous breaking of the symmetry
may lead to axions with unacceptable couplings. Often one wants to
preserve R-parity to avoid other possible phenomenological disasters.
5The details can be found in[27] section 4, in particular around
equations (4.3,4).superelds, we can have the couplings:
L=Z
d4XNL
f2
m2QeVQ (10)
+Z
d2XNL
f1
2BijQiQj+:::+h:c:
plus gauge couplings.
Once we reach the end of in
ation, the eld Xbecomes
nonlinear, its scalar component is a goldstino bilinear and
the period of reheating begins. The details of reheating de-
pend very much on the microscopic model. At this stage
one should provide details of the \waterfall" that turns the
huge amount of energy f2into low energy particles. Part of
this energy will be depleted and converted into low energy
particles through the soft couplings in (10), and hence we
can in principle compute a lower bound on the reheating
temperature. Before making some comments on the re-
heating period, we analyze the cosmological consequences
of a potential as simple as (8), as well as the assumptions
we have made earlier about the in
aton and its range as
in
ation takes place.
3. The In
aton Potential and Slow Roll Conditions
To study the conditions under which our potential pro-
vides in
ation consistent with the latest cosmological con-
straints, we examine the slow-roll parameters, dened as
[13]:
=M2
pl
2V0
V2
; (11)
=M2
plV00
V; (12)
whereMplis the reduced Planck mass and ' denotes deriva-
tive with respect to the in
aton eld. The observables are
then expressed in terms of the above slow roll parameters
as:
nS= 16+ 2; (13)
r= 16 (14)
4nt=2; (15)

2
R=VM4
pl
242: (16)
nSis the slope of the scalar primordial power spectrum,
ntis the corresponding tensor one, ris the scalar to tensor
ratio and 
2
Ris the amplitude of the initial perturbations.
All these numbers are constrained by current cosmological
observations [10, 11, 12]. We will use their constraints to
explore the naturalness of our in
ationary trajectories. In-

ation takes place when the slow-roll parameters are much
smaller than 1.
We will use the amplitude of initial perturbations and
the number of efoldings to t some of the paramenters
of the toy model in the previous section. Recall that the
potential in the range of interest is:
V=f2(1 +A1(2+2) +B1(22) +:::);(17)
which appears in gure 1. We can compute ;while
rolling in the direction:
= 2 ( (A1B1))2+::: (18)
= 2 (A1B1) +:::; (19)
since << 1,is naturally small. We can make small
by a slight ne tuning of the dierence A1B1. We will
writelater as a ratio of the in
aton and gravitino masses.
Once the slow roll conditions are satised, we can compute
the number of efoldings (see for instance [16, 17]):
N=1
MZdxp
2=Zf
id
2p(20)
From (19) we get:
N=1p
2jA1B1jlogf
i: (21)
In most models of supersymmetry breaking, the gravitino
mass is given by:
m3=2=f
M; (22)
hence, we can rewrite the parameters and masses in (9)
as:
jA1B1j=1
2m2

m2
3=2;jA1+B1j=1
2m2

m2
3=2;(23)thus:
N=p
2m3=2
m2logf
i(24)
The number of efoldings is considered normally to be be-
tween 50100. Finally we will use the amplitude of initial
perturbations to get one extra condition in the parameters
of our potential. Using [11] (16) can be written as:
V
1=4
=f1=2
21=4(jA1B1j)1=2=:027M; (25)
whereis taken atN-efoldings before the end of in
ation.
Summarizing, the two cosmological constraints we get on
the parameters of our potential can be written as:
N=p
2m3=2
m2logf
i; (26)
21=4m3=2
mpf
M1=2
= 0:027; (27)
and theparameter can be written as:
=m
m3=22
: (28)
We takeiabove the supersymmetry breaking scale
pf=M ==M , andfclose tomsoft=M, therefore we
can easily get values for Nbetween 50100 for moderate
values of, which is expressed here as the square of the
ratio of the in
aton to the gravitino mass. It is interesting
to notice that from (27), we can write the supersymmetry
breaking scale in terms of the -parameter:

M5:2 104: (29)
Hence for a value of :1 we can get 1013GeV.
Lower values of the supersymmetry breaking scale can be
obtained by reducing . However, since the in
aton mass
is
m=m3=2p; (30)
we may end up with an in
aton whose mass is substantially
lighter than the gravitino. For these values of ;, we
have thati1013=M;f103=M, and the number of
efoldings is110.
5We conclude then that with moderate values of be-
tween:1:01 we can get supersymmetry breaking scales
between 10111013without major ne tunings. We eas-
ily get enough efoldings, and furthermore, the in
aton is
lighter than the gravitino by an amount given byp.
For the above range of parameters we can compare the
predicted value of nSin our model with observational con-
straints. This is shown in the right panel of Fig. 1. The
yellow region is the current cosmological constraints from
WMAP5 [11] and the other colored areas are the predic-
tions for our model with minimal ne tuning for an stable
(unstable)Xpotential, i.e. the eld is concave (convex) re-
spectively. The constraints will improve greatly when the
Planck satellite releases its results next year, and therefore
our model can be tested much more accurately.
Reheating can proceed in many ways, since we have not
provided a detailed microscopic model. Once in the non-
linear regime, the XNLeld (whose scalar component is
made of a goldstino bilinear) could eciently convert the
f2-energy density into radiation. We can calculate the
amount of entropy and particle density by using the Boltz-
man equation and assuming that the pair of Goldstinos
will have an out-of-equilibrium decay[16]. Using that
TRH= 1010p
f=GeV3=2
GeV (31)
we obtain a range 107< TRH<109. This produces a
particle abundance of n107090which are standard
values. We can also compute the amount of entropy gen-
erated by the out-of-equilibrium decay as
Sf=Si= 107(p
f=GeV )1=2(32)
which yields values in the range 10 to 1, and assures that
there is no entropy overproduction. We could also compute
the depletion of this energy through the soft couplings (10)
yielding very similar values as above. In both cases, we
can get sucient reheating with temperatures betweenpf
and a fraction of m3=2. The true value depends very much
on the details of the microscopic model. However, thereseems to be no obstruction to reheating the universe to
and acceptable value of temperature, particle abundances
and entropy. We are currently working in a more detailed
theory incorporating our scenario [25].
4. Conclusions
In this short note we have studied the possibility of hav-
ing supersymmetry breaking as the driving force of in
a-
tion. We have used the unique chiral supereld Xwhich
represents the breaking of conformal invariance in the UV,
and whose fermionic component becomes the goldstino at
low energies. Its auxiliary eld is the F-term which gets
the vacuum expectation value breaking supersymmetry.
It is crucial in our analysis to have explicit R-symmetry
breaking along with supersymmetry breaking. This allows
us to avoid the problem in supergravity and to take the
supersymmetric limit. The simplest model we obtain de-
scribes the components of Xwell below the Planck scale.
It is written in terms of three parameters: the supersym-
metry breaking parameter fand the masses of the real and
imaginary components of the eld x(the scalar component
of X). In our analysis the imaginary part of xplays the role
of the in
aton, and its mass was shown to be smaller than
the gravitino mass by an amount given byp. This imag-
inary component represents a pseudo-goldstone boson, or
rather, a pseudomoduli. In supersymmetric theories such
elds abound, and any of them could be used to construct
some form of hybrid in
ation. In our case, however, we
want to use the minimal choice that is naturally provided
by the universal supereld Xthat must exist in any su-
persymmetric theory.
Since we have not presented any detailed model, the cos-
mological consequences are a bit rudimentary, especially
concerning reheating at the end of in
ation. However, the
comparison of the simplest model with present data, yields
very interesting values for the supersymmetry breaking
scale, and the ratio of the in
aton and gravitino masses.
6Figure 1: Left panel: The potential as a function of ( ) and () components of the eld X. Note the nearly 
at direction ( ) that we use for our
in
ationary trajectories. Graceful exit and particle creation occurs in the non-linear part of the Xeld. Right panel: WMAP5 cosmological
constraints (yellow region) in the rnSplane. For no-ne-tuned minimal in
ation models the green and red area show our predictions for
both cases of a stable (concave) potential and unstable (convex) potential. The Planck satellite will be able to provide signicantly tigther
constraints on rand especially nS(at the<0:5% level) thus further constraining our model. The dashed line is the limit in rthat can be
achieved with an ideal CMB polarization experiment [14]
These are bonuses which come directly from the observa-
tions of the initial density perturbations from WMAP data
[11]. The fact that the in
aton is lighter than the gravitino
may have interesting low-energy phenomenological impli-
cations. Furthermore in this simple model it is easy to
obtain sucient number of efoldings with moderate values
of theparameter.
To explore our proposal in more detail, it is important
to construct an explicit model, even if not very realistic,
in order to understand in more detail the end of in
ation,
the reheating mechanisms, and also the ne structure of
the in
aton potential. We hope to report on this in the
near future [25].
Acknowledgements
We would like to thank G. Dvali, G. Giudice, J. Les-
gourgues, S. Matarrese, G. Ross, Nathan Seiberg, M.A.
V azquez Mozo, and L. Verde for useful discussion. C.G.
and R.J. would like to thank the CERN Theory Group for
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8