UV Corrections in Sgoldstino-less Inflation

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by  Emilian Dudas

ULB-TH/16-01 blank space
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CPHT-RR001.012016
UV Corrections in Sgoldstino-less Inflation
Emilian Dudas
Centre de Physique Th´eorique, Ecole Polytechnique, CNRS,
Univ. Paris-Saclay, 91128 Palaiseau Cedex, France
Lucien Heurtier
Service de Physique Th´eorique, Universit´e Libre de Bruxelles,
Boulevard du Triomphe, CP225, 1050 Brussels, Belgium
Clemens Wieck
6 Departamento de F´ısica Teo´rica UAM and Instituto de F´ısica Teo´rica UAM/CSIC,
1 Universidad Auto´noma de Madrid Cantoblanco, 28049 Madrid, Spain
0
2 Martin Wolfgang Winkler

n Bethe Center for Theoretical Physics and Physikalisches Institut der Universita¨t Bonn,
u Nussallee 12, 53115 Bonn, Germany
J
We study the embedding of inflation with nilpotent multiplets in supergravity, in particular the
6
decouplingofthesgoldstinoscalarfield. Insteadofbeingimposedbyhand,thenilpotencyconstraint

onthegoldstinomultipletarisesinthelowenergy-effectivetheorybyintegratingoutheavydegrees
]
h of freedom. We present explicit supergravity models in which a large but finite sgoldstino mass
t arises from Yukawa or gauge interactions. In both cases the inflaton potential receives two types
-
p of corrections. One is from the backreaction of the sgoldstino, the other from the heavy fields
e generating its mass. We show that these scale oppositely with the Volkov-Akulov cut-off scale,
h which makes a consistent decoupling of the sgoldstino nontrivial. Still, we identify a parameter
[ windowinwhichsgoldstino-lessinflationcantakeplace,uptocorrectionswhichflattentheinflaton

potential.
2
v
7
I. INTRODUCTION nonlinear supergravity theories are equivalent to linear
9
3 supergravities with an infinitely heavy sgoldstino scalar,
3 and that the limit relating the two is well-defined via
Constrained chiral multiplets or, equivalently, nilpo-
0 functional integration. With a few restrictions this con-
tent superfields and their application to cosmology have
. nection was previously known in the rigid limit [31].1
1 attracted a large amount of interest in recent years [1–
0 Thereforeitisdesirabletostudyfieldtheoryexamples
17]. One feature of theories with nonlinear supersymme-
6 inwhichaheavysgoldstinoexistssothatsupersymmetry
try,i.e.,withaconstrainedmultipletsatisfyingS2 =0,is
1 becomes linearly realized at a high scale. In such cases,
theabsenceofadynamicalscalardegreeoffreedom. The
v: auxiliary field of S breaks supersymmetry and the gold- the sgoldstino field cannot be infinitely heavy. Its mass,
i andhencetheVolkov-Akulovcut-offscale,mustbelower
X stino fermion is the only propagating field [18–21]. This
than the Planck scale – and favorably below the Kaluza-
makesthemappealingincosmologicalmodelbuildingfor
r Kleinandstringscales. Astrongerconstraintarisesfrom
a various reasons.
unitarity which signals a perturbative breakdown of the
The connection of such theories to string theory has nonlinear theory at a scale ∼√m in Planck units. A
3/2
recentlybeenstudiedin[15,22–27]. Effectivesupergrav- UV complete theory which can describe both the linear
ity theories with a constrained goldstino multiplet can andnonlinearregimesisboundtoyieldcorrectionswhich
be shown to arise from D3-branes in certain geometries aremissedbysimplyimposinganilpotencyconstrainton
[23–27]. The emergence of nonlinear supersymmetry in the goldstino multiplet in supergravity. In this letter we
string models with anti-branes was proven in [28], in the compute these corrections and evaluate their effects in
context of global string theory vacua [29]. In such UV simple inflation models previously studied in the litera-
embeddings it is difficult to extract the behavior of the ture. It is our aim to prove that in a well-defined regime
supergravityabovethecut-offscaleoftheVolkov-Akulov of the theory, corrections are under control – though in
action. Usually there is no scale at which linear super-
symmetryisrestoredandthereforethescalarcomponent
of S does not exist. A step towards understanding the
connectionbetweenthelinearandnonlinearregimeswas 1 Forarecentstudyregardingtheapplicabilityofnilpotencycon-
recently made in [30], where it was shown explicitly that ditionscf.[32].
2
a quite constrained parameter space. The limit c → ∞ in (1) then corresponds to taking the
Forthispurposetheclassofmodelsdevelopedin[7]is couplingλtoinfinityorthemassm tozero. Sinceboth
X
particularly instructive.2 They feature the coupling of a must be finite and m must be large for the effective
X
nilpotent stabilizer multiplet to a holomorphic function field theory (EFT) to make sense, we must consider the
oftheinflatonmultiplet,givingrisetoaplethoraofpossi- regime where the sgoldstino has a finite mass, i.e., finite
ble potential shapes for the inflaton scalar. During infla- c. In the remainder of this letter we strive to determine
tion and in the vacuum supersymmetry is broken by the whetherinflationisstillpossibleinthiscase. Specifically,
auxiliary field of the nilpotent multiplet. The setup can we determine whether the inflaton potential obtained in
accommodate low-energy supersymmetry which is non- the nilpotent limit still holds and corrections are under
trivial given the high scale of inflation. We extend this control.
setuptoasupergravitywithheavyfieldsinwhichalarge We will find that such corrections are of two differ-
mass for the sgoldstino scalar is generated dynamically. ent natures. Additional heavy fields at the energy scale
We expect our results to be relevant in many other su- Λ backreact on the inflaton potential, introducing cor-
pergravity theories with nilpotent goldstino multiplets. rections which vanish as Λ → ∞.3 On the other hand,
Thus, we hope that this work is another step towards thefinitemassofthesgoldstinofieldleadstocorrections
understanding nilpotent multiplets and their role in cos- which vanish in the limit where the latter is infinitely
mology. heavy. This corresponds to Λ → 0. Therefore, it is far
from obvious that both types of corrections can be sup-
pressed simultaneously.
II. SGOLDSTINO DECOUPLING
Note that while we study this in the class of inflation
modelsproposedin[7],ourfindingscanstraightforwardly
The success of nilpotent fields in cosmology has trig- be applied to alternative scenarios with nilpotent multi-
gered growing interest in their field-theoretical origin. It plets.
is well-known that in spontaneously broken linear su-
persymmetry, the sgoldstino field acquires a large mass
through the operator
III. SGOLDSTINO-LESS MODELS OF
|S|4 INFLATION
K ⊃c (1)
Λ2
Let us briefly review the inflation models of [7]. They
in the K¨ahler potential. In the limit c → ∞, the sgold-
feature the K¨ahler and superpotential
stino becomes infinitely heavy and the resulting theory
isequivalenttononlinearlyrealizedsupersymmetrywith
1
a nilpotent goldstino multiplet S [30, 31]. Clearly this K = (Φ+Φ)2+|S|2, (4a)
2
theory is only a low-energy effective theory. With the
(cid:16) √ (cid:17)
sgoldstinodecoupled,itviolatesperturbativeunitarityat W =f(Φ) 1+ 3S , (4b)
√
the intermediate energy scale m [34]. Requiring in-
3/2
flationintheperturbativeregimeoneobtainsthegeneric
where Φ denotes the inflaton superfield and S contains
constraint [7]
the stabilizer field. This setup is a generalization of the
m >H2, (2) models developed in [36, 37] with built-in supersymme-
3/2
try breaking by the auxiliary field of S. The function f
whereH denotestheHubblescale. However,thescaleof satisfies f(0) (cid:54)= 0, f(cid:48)(0) = 0 and f(x) = f(−x¯). In [7] it
supersymmetrybreakingmaybedifferentduringandaf- is assumed that S fulfills the boundary condition S2 =0
terinflation. Hence,nilpotentinflationmodelsconsistent of a nilpotent chiral multiplet. This implies (cid:104)s(cid:105) = 0 for
with low-energy supersymmetry can be constructed [7]. its scalar component.4
√
A different concern is the limit c → ∞: in a UV- The factor 3 ensures the cancellation of the cosmo-
complete model the operator (1) arises from couplings logical constant in the vacuum at (cid:104)φ(cid:105) = 0. Along the
of S to heavy degrees of freedom. As an example, we inflationary trajectory the potential reads
may consider the superpotential coupling W ⊃λSX2 of
Sgenteorattheeshaeoanvye-lfioeolpdcXorrwecittihonm[a3s5s],mX. This coupling V =(cid:12)(cid:12)(cid:12)f(cid:48)(cid:18)i√ϕ (cid:19)(cid:12)(cid:12)(cid:12)2 , (5)
(cid:12) 2 (cid:12)
λ4 |S|4
K ⊃− . (3)
16π2 m2
X
3 Thesewecall“UVcorrections”becausetheyarisefromembed-
dingthenilpotentmultipletinacompletetheoryofsupergravity.
2 Werecommend[33]asareviewoftheseandotherinflationmod- 4 We use capital letters for superfields and small letters for their
elsinvolvingnonlinearsupersymmetry. scalarcomponents.
3
√
where ϕ = 2Imφ denotes the canonically normalized denoting the inflaton potential in the limit where S is
inflaton. Two examples for f are discussed in [7]. One is nilpotent and hence s is infinitely heavy. The gravitino
mass along the inflationary trajectory can be approxi-
m
f(Φ)=f − Φ2, (6) mated as
0 2
leadingtothepotentialofchaoticinflation, V = 12m2ϕ2. m2 =eK|W|2 (cid:39)(cid:12)(cid:12)(cid:12)f(cid:18)i√ϕ (cid:19)(cid:12)(cid:12)(cid:12)2 . (11)
The other is 3/2 (cid:12) 2 (cid:12)
(cid:32) √ (cid:33)
(cid:112) 3 √
f(Φ)=f −i V Φ+i e2iΦ/ 3 , (7) As only the real part of the stabilizer field is displaced
0 0 2
duringinflation, wesets¯=sinthefollowing. Atsecond
√ order in s the scalar potential reads
(cid:16) (cid:17)
producingtheplateaupotentialV =V 1−e− 2/3ϕ .5
0 √ (cid:16) (cid:17)
InthefollowingwecallS thegoldstinomultipletands V =V +m2 Λ2+ 3 2V −4m2 s+m2s2, (12)
0 3/2 0 3/2 s
thesgoldstino, itsscalarcomponent. Thisisbecausesis
theheavyscalarthatissupposedtodecouple,anddespite
including only terms up to O(Λ2).7 The sgoldstino mass
the fact that the inflaton multiplet has a sub-dominant
is given by
but nonvanishing auxiliary field during inflation.
m2
m2 =12 3/2 , (13)
IV. CORRECTIONS FROM THE SGOLDSTINO s Λ2
which,throughm ,dependsonϕduringinflation. The
3/2
inflaton-dependent minimum of s lies at
Let us discuss corrections to the inflaton potential
which arise if the sgoldstino has a finite mass. To this
end we consider (cid:104)s(cid:105)= 2m23/2−V0 Λ√2 . (14)
m2 4 3
3/2
W =f(Φ)(1+δS), (8a)
1 |S|4 The scalar potential after integrating out s reads
K = (Φ+Φ)2+|S|2− . (8b)
2 Λ2
(cid:34) (cid:32) (cid:33) (cid:35)
V
The difference compared to the previous section is that V =V 1+ 1− 0 Λ2+O(Λ4) . (15)
we do not impose the nilpotency constraint S2 = 0. 0 4m2
3/2
Instead we introduce the term |S|4/Λ2 in the K¨ahler
potential which generates a large – but finite – mass As mentioned above, the corrections from the sgoldstino
for the sgoldstino s and dynamically keeps s close to sector appear in powers of m2 /m2 and H2/m2, where
the origin.6 Supersymmetry breaking introduces an √ 3/2 s s
H ∼ V again denotes the Hubble parameter. Correc-
inflaton-dependent linear term for the stabilizer field 0
tionsareundercontrolaslongasm >HΛwhichisthe
which slightly shifts it away from the origin [4]. As this 3/2
case during inflation in the two examples of Section III.8
effect scales inversely with the mass of s, it is absent in
Note that even when the corrections are small the
the nilpotent limit. Notice that we introduced the pa-
sgoldstino can affect post-inflationary cosmology. If the
rameter δ which allows us to tune the vacuum energy to
above constraint is violated after inflation, s may no
zero at the minimum of the potential. Due to the shift
√
longer trace its minimum. If it gets trapped the associ-
of s, δ is close to but not exactly 3. We find
atedpotentialenergycanalterlate-timecosmology. This
√ Λ2 isnotnecessarilyproblematicandmayeveninduceinter-
δ = 3+ √ +O(Λ4). (9)
esting signatures. We merely point out that decoupling
2 3
s from all dynamics in the universe requires the bound
For a compact notation we introduce m >HΛtobesatisfiedduringtheentirecosmological
3/2
V0 =(cid:12)(cid:12)(cid:12)f(cid:48)(cid:18)i√ϕ (cid:19)(cid:12)(cid:12)(cid:12)2 , (10) esivsotleunttioEnF.TWΛecwainllnoshtobweainrbtihtrearfiollyloswminagll,tmhaatkiinngathciosna-
(cid:12) 2 (cid:12) very severe constraint.
5 AspointedoutinSection5of[7],thefunctionf canbeextended
to include matter fields like an MSSM sector. Tachyonic direc- 7 Noticethats=O(Λ2)andm2=O(Λ−2).
s
tions are avoided automatically for matter fields which appear 8 ForthesgoldstinotobeheavierthanRe(Φ)onewouldaddition-
atleastquadraticallyinf. ally have to require |f(cid:48)(cid:48)(Φ)|,|f(cid:48)(cid:48)(cid:48)(Φ)|<|f(cid:48)(Φ)| on the inflation-
6 ComparedtoSectionIIweabsorbedtheparametercinthedef- ary trajectory. These conditions are, however, already fulfilled
initionofΛ. byrequiringslow-rollinflation.
4
V. CORRECTIONS FROM UV COMPLETIONS m m H Λ
3/2 s
inflation 8·1014GeV 9·1015GeV 9·1013GeV 7·1017GeV
In the previous section we have included corrections
to the inflaton potential which arise from the sgoldstino vacuum 105GeV 106GeV ∼0 7·1017GeV
sector. The corrections disappear in the limit Λ → 0 in
whichthesgoldstinobecomesinfinitelyheavy. Butthere TABLE I.Representativeexampleofthedifferentscalesap-
are more corrections related to the heavy fields living pearing in sgoldstino-less inflation. The values during infla-
at the scale Λ. Contrary to the sgoldstino corrections, tion refer to the beginning of observable inflation, 50-60 e-
thesescalewithΛ−1 andpreventusfromtakingthelimit folds in the past.
Λ→0. In the following we discuss these UV corrections
intwoexamples. Inthefirstexamplethesgoldstinomass
Notethatitisthesameasthemasstermarisingfromthe
isgeneratedbyYukawainteractionswithheavyfields, in
equivalent quantum correction to the K¨ahler potential
the second example by gauge interactions. Despite their ∆K =−|S|4/Λ2 with
simplicityweexpectthatourexamplesarerepresentative √
of more sophisticated UV embeddings. 2 3π
Λ= M. (22)
λ2
WeconcludethatweobtainthemodelofSectionIVasa
A. Example 1
low-energy effective theory and the small shift of s does
not affect inflation for sufficiently small Λ.
Consider two additional chiral multiplets X,Y with a
However,wehaveyettoconsidertheeffectofinflation
vector-like mass M. We consider M to be large com-
onthesectorofheavyfieldsX andY. Inflationdoesnot
pared to the Hubble scale and the gravitino mass during
induce linear terms for the scalar components x and y.
inflation. We define the model as follows,
However, it generates a bilinear mass term for x. The
W =f(Φ)(1+δS)+λSX2+MXY , (16) mass of Imx is given by
√
K = 1(Φ+Φ)2+|S|2+|X|2+|Y|2. (17) m2Imx (cid:39)M2−2 3λm3/2. (23)
2
Thus, taking the limit M → 0 to make S nilpotent in-
It bears resemblance to the O’Raifeartaigh model [38]. troduces a tachyonic direction in the full theory, which
ThesgoldstinosuperfieldS obtainsamasstermthrough makesinflationimpossible.9 Toobtainapositivesquared
its coupling to X. The parameter δ is chosen such that mass, we obtain the constraint
the vacuum energy vanishes at the minimum of the po- √
tential M2 >2 3λm3/2. (24)
√ (cid:18) 2π2M2(cid:19) Taking the example of chaotic inflation (6) and using
δ = 3 1+ +O(M4). (18) √
λ4 m (cid:39) 6·10−6, ϕ ∼ 15, this translates into M > 0.03 λ.
At the same time, to make the sgoldstino sufficiently
The tree-level scalar potential along the direction
heavy we have to require that Λ (cid:46) 1 which is equiv-
x=y =0 reads
alent to M (cid:46) 0.09λ2, cf. (22). After combining these
12π2M2 √ (cid:16) (cid:17) two constraints there is a small window at λ (cid:38) 1 and
V = V + m2 +2 3 V −2m2 s
0 λ4 3/2 0 3/2 M ∼0.05,wheresgoldstino-lessinflationcanconsistently
(cid:16) (cid:17) takeplace. Inthisregimetheheavyfieldsremainattheir
+ 4V2−2m2 s2+O(s3), (19)
0 3/2 minimaandinflationdoesnotreceivecorrectionsbesides
those of Section IV. Notice, however, that the sgoldstino
where V = |f(cid:48)|2 and m2 (cid:39) |f|2 as before and s¯ = s
0 3/2 mass can at most be enhanced by an O(10) factor com-
is assumed. The imaginary part of s is stabilized at the
paredtothegravitinomass. ThisisillustratedinTableI,
origin and does not play a role in our discussion. The
where we show a possible choice of scales which leads to
tree-level mass of s is negligible compared to the one-
successful sgoldstino-less inflation. In the vacuum, the
loop contribution due to the interaction with X. We use
gravitino mass is much smaller than in the inflationary
the Coleman-Weinberg formula
epoch and low-energy supersymmetry can be obtained.
1 M2
V = StrM4log , (20)
CW 64π2 Q2
B. Example 2
where StrM4 =(cid:80)i(−1)2Ji(2Ji+1)m4i is the trace over
the field-dependent mass eigenvalues of states with spin
Second, we consider an example where the sgoldstino
J . The Coleman-Weinberg potential gives rise to an ad-
i receives its mass from gauge interactions. We introduce
ditional mass term
m2
V =λ4 3/2 s2+... . (21)
CW π2M2 9 Weassumeλ>0. IntheoppositecaseRexisthetachyon.
5
three new chiral multiplets X, Y, Ψ which carry the 108V(φ)
charges q(X) = −1, q(Y) = 1 and q(Ψ) = 0 under a
U(1) symmetry.10 We further assume that q(S)=1 and
2
define the model by
W =f(Φ)(1+δXS)+λΨ(XY −v2), (25)
K = 1(Φ+Φ)2+|S|2+|X|2+|Y|2+|Ψ|2, (26) 1
2
where δ is again chosen to adjust the cosmological con-
stant. We find
φ
√ 10 20 30 40 50
3(cid:18) 2v2 (cid:19)
δ = 1− +O(v4) . (27)
v 9
Figure1. Effectiveinflatonpotentialforthechaoticinflation
The second term in the superpotential is introduced to
model with v = 0.03 (blue) v = 0.1 (orange) and v = 0.3
break the U(1) symmetry at a high scale. For the same (green). The backreaction of the heavy fields flattens the
reason as before we set x¯ = x, y¯ = y, ψ¯ = ψ in the inflaton potential. For v (cid:38) 0.1 corrections from the heavy
following. The imaginary parts do not play a role in our fields are under control, while sgoldstino decoupling requires
discussion. Giventhatv (cid:29)Max(m ,H)theU(1)sym- v (cid:46)1. This leaves a small window of viable parameter space
3/2
metryisbrokenalongthealmostD-andF-flatdirection in which sgoldstino-less inflation consistently proceeds.
xy =v2, s2−x2+y2 =0, ψ =0. (28)
Using these three conditions to eliminate x, y, and ψ Noticethedifferencetoourfirstexample. Inthiscasethe
yields the scalar potential shift of the heavy fields during inflation causes a back-
reaction on the potential. Expression (31) only includes
2 √ (cid:16) (cid:17)
V =V + m2 v2+ 3 2V −2m2 s the corrections due to the heavy fields. In addition, the
0 3 3/2 0 3/2 sgoldstino-induced corrections of Section IV arise.
+m2s2+O(s3), (29) Requiring the correction to be suppressed compared
s
to the leading-order inflaton potential leads to the con-
with
straint
9m2 m3/2
m2 = 3/2 . (30) v (cid:29) , (32)
s 2v2 V1/4
0
This resembles (12) if we identify Λ = (cid:112)4/3v.11 The for λ,g ∼O(1). In the model of chaotic inflation defined
large mass m decouples the sgoldstino and the small by (6), with m ∼ 6·10−6 and ϕ ∼ 15, the constraint
s
shift of s does virtually not affect inflation. translates into
Unfortunately, this is not the end of the story. So far
v (cid:29)0.03. (33)
we have worked in the regime m ,H (cid:28) v. We expect
3/2
additionalcorrectionsifeitherm orH areclosetothe Even for larger v there are substantial corrections. We
3/2
scale v. To find these corrections we must treat x, y and depict the effective inflaton potential of the example (6)
ψ as dynamical fields. We perform a Taylor expansion in Figure 1 for f = 10−14, m = 6·10−6, λ = g = 1,
0
around s=0, ψ =0, x=v, y =v up to second order in and different values of v. Again there is a small window
theshiftofthefourfields. Settingthefourfieldstotheir at v (cid:38) 0.1 where the backreaction is under control and
new minima, we arrive at the following effective inflaton sgoldstino-less inflation can take place. As in the previ-
potential ous example, choosing v too large decreases the mass of
the sgoldstino scalar beyond the point where it can be
9(cid:18) 1 1 (cid:19)m43/2 consistently decoupled. The corrections from the heavy
V =V − + . (31)
0 4 2g2 λ2 v4 fields of the UV completion cause a flattening of the in-
flaton potential.
10 In order to avoid anomalies we have to introduce another field VI. DISCUSSION
Z withchargeq(Z)=−1. ThefieldZ canbecoupledtoanew
singletΘviaatermYZΘinthesuperpotential. WhenY breaks
Wehaveemphasizedthatsgoldstinodecouplingincos-
the U(1) symmetry this becomes a large vector-like mass term
mology is nontrivial. Working in spontaneously broken
for Z. In this case Z and Θ do not affect our analysis, and we
neglecttheminthefollowingdiscussion. linearsupergravity,insteadofimposinganilpotencycon-
11 To recove√r the exact form of (12) we would have to substitute straint by hand we assumed that the mass of the sgold-
m3/2→ 2m3/2. stino field is produced dynamically. This required the
6
inclusion of heavy degrees of freedom which couple to we calculated the corrections to the inflaton potential
the sgoldstino. We discussed two possible UV embed- which typically appear in the form of flattening effects.
dings of nilpotent goldstino multiplets. Both scenarios The constraints on Λ imply that the sgoldstino mass
result in the sgoldstino-less inflation models of [7] as a canatmostbeenhancedbyoneorderofmagnitudecom-
low-energy effective theory. The sgoldstino has a large pared to the gravitino mass. Requiring the sgoldstino to
but finite mass ms ∼ m3/2/Λ during inflation, where Λ decouple in the post-inflationary cosmology as well puts
is the mass scale of the heavy fields which couple to the strong additional constraints on the form of the scalar
sgoldstino. As m3/2 > H the sgoldstino decouples from potential.
the inflationary dynamics. Duetothestructureofthedangeroustermsthatarise,
ThescaleΛsetsanewcut-offatwhichthelow-energy we expect these results to be relevant for many other
effective theory breaks down and the heavy fields be- applications of constrained multiplets in cosmology.
comedynamicaldegreesoffreedom. Forinflationtotake
place in a controlled regime, where the heavy fields can
be integrated out, one has to require that the Hubble
scale does not exceed Λ. However, an even more severe ACKNOWLEDGEMENTS
constraint arises in the class of models [7] which feature
m3/2 (cid:29)H duringinflation. Thereinflationinduceslarge The authors thank G. Dall’Agata and A. Uranga for
“soft terms” which may destabilize the heavy fields. De- discussions. The work of C.W. is supported by the ERC
pendingonthespecificUVembedding,wefindthattad- AdvancedGrantSPLEundercontractERC-2012-ADG-
pole terms ∝ m23/2MP2/Λ and bilinear terms ∝ m3/2MP 20120216-320421, by the grant FPA2012-32828 from the
are particularly dangerous. In all examples we find that MINECO,andbythegrantSEV-2012-0249ofthe“Cen-
inflation is generically spoiled if Λ (cid:46) 0.1. This does not tro de Excelencia Severo Ochoa” Programme. The work
leavemuchroomforacompletetheorybelowthePlanck of L.H. is supported by the IISN and the Belgian Fed-
scale with a decoupled sgoldstino. Still, a window of vi- eral Science Policy through the Interuniversity Attrac-
able parameter choices survives in which sgoldstino-less tion Pole P7/37 “Fundamental Interactions”. The work
inflationcansuccessfullytakeplaceandthebackreaction ofM.W.issupportedbytheSFB-TransregioTR33“The
ontheheavyfieldsinundercontrol. Withinthiswindow Dark Universe” (Deutsche Forschungsgemeinschaft).
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