Natural Inflation from 5D SUGRA and Low Reheat Temperature

0.28 MB·English

by  Filipe Paccetti Correia

January 15, 2015
Natural Inflation from 5D SUGRA
5
1
and Low Reheat Temperature
0
2

r
p
A
Filipe Paccetti Correiaa1, Michael G. Schmidtb2, and Zurab Tavartkiladzec3
9

] aDeloitte Consultores, S.A., Pra¸ca Duque de Saldanha, 1 - 6o, 1050-094 Lisboa, Portugal4
h
p bInstitut fu¨r Theoretische Physik, Universit¨at Heidelberg, Philosophenweg 16,
-
p 69120 Heidelberg, Germany
e cCenter for Elementary Particle Physics, ITP, Ilia State University, 0162 Tbilisi, Georgia
h
[

2
v
0
Abstract
2
5
Motivatedbyrecentcosmologicalobservationsofapossiblyunsuppressedprimordialtensor
3
0 component r of inflationary perturbations, we reanalyse in detail the 5D conformal SUGRA
.
1 originated natural inflation model of Ref. [1]. The model is a supersymmetric variant of 5D
0 extra natural inflation, also based on a shift symmetry, and leads to the potential of natural
5
inflation. Coupling the bulk fields generating the inflaton potential via a gauge coupling to
1
: the inflaton with brane SM states we necessarily obtain a very slow gauge inflaton decay rate
v
anda very low reheating temperatureT < (100) GeV. Analysis of the requirednumberof e-
i r ∼
X O
foldings(fromtheCMBobservations)leadstovaluesofn inthelowerrangeofpresentPlanck
s
r
a 2015 results. Some related theoretical issues of the construction, along with phenomenological
and cosmological implications, are also discussed.
1E-mail: [email protected]
2E-mail: [email protected]
3E-mail: [email protected]
4Disclaimer: This address is used by F.P.C. only for the purpose of indicating his professional affiliation. The
contents of the paper are limited to Physics and in no ways represent views of Deloitte Consultores, S.A.
1
1 Introduction
Inflation solves the problems of early cosmology in a natural way [2] and besides that produces
a primordial fluctuation spectrum [3] which allows to discuss structure formation successfully. In
detailed models (i) a sufficient number of e-folds for the inflationary phase has to be produced,
(ii) guided by bounds presented recently by the Planck Collaboration [4], the cosmic background
radiationand a spectral index n = 0.968 0.006should be generated.5 And (iii), the normalization
s
±
of fluctuations has to be reproduced. Rather flat potentials for the inflaton field lead to the “slow
roll” needed for (i). Such potentials appear naturally in (tree level) global supersymmetric models;
higher loop corrections can be controlled, but the inclusion of supergravity easily produces an
inflaton mass of the order of the Hubble scale.
Inmodelswithanextra dimension thefifthcomponent ofaU(1)gaugefieldentering inaWilson
loop operator can act as an inflaton field of pseudo Nambu-Goldstone type which is protected
against gravity corrections and avoids a transplanckian scale [6], [7], present in the original model
of “natural inflation” [8]. We have presented such a model [1] based on5D conformal SUGRAon an
orbifold S1/Z with a predecessor based on global supersymmetry with a chiral “radion” multiplet
2
on a circle in the fifth dimension [9]. We also made the interesting observation that a spectral
index n 0.96 as observed recently [different from a value very close to one usually obtained
s
∼
in straightforward SUSY hybrid inflation [11]], is obtained rather generically in gauge inflation.
Actually, in the supersymmetric formulation we have a complex scalar field which besides the
gauge inflaton A1 contains a further “modulus” field M1 which also might allow for successful
5
inflation [1]. The main difference between the two inflation types is that gauge inflation leads to a
large tensor to scalar ratio r( 0.12 in [1]) whereas modulus inflation leads to very small r(< 10−4
∼
∼
in [1]).6 Recently the BICEP2 data [12] gave strong indication of a large ratio r = 0.2+0.07 though
−0.05
recent joint analysis of BICEP2/Keck and Planck [13] gave a reduced upper bound r < 0.12,7 with
∼
the likelihood curve for r having a maximum for r 0.05. Because of this, we here consider the
≃
gauge inflation of ref. [1] again with particular emphasis on the required length of inflation. The
well known 62 e-folds solving the horizon problem will turn out to require a substantial expansion
during the reheating period within the natural inflation scenario emerged from 5D SUGRA.
Let us present the organization of the paper and summarize some of the results. In Sec. 2 we
perform a detailed analysis of natural inflation with cos-type potential. For the calculation of the
spectral index n and the tensor to scalar ratio r, we use a second order approximation with respect
s
to slow roll (SR) parameters. Since these quantities (n ,r) are determined at the point where the
s
SR parameters are tiny, this approximation is sufficient for all practical purposes. However, near
the end of the inflation, when SR breaks down, we perform an accurate numerical determination of
the point via the condition ǫ = 1 on the Hubble slow roll parameter (see [15]- [17] for definitions).
H
This is needed to compute, with desired precision, the number of e-foldings (Ninf) before the end of
e
the inflation. We carry out our analysis by using recent fresh data [4] for n , N , the amplitude of
s e
5In the originalversionofthis paper (see v1of arXiv:1501.03520)we hadthe 2013value [5] n =0.9603 0.0073
s
±
which being about a standard deviation below this value makes quite a difference for our analysis.
6Genuinetwofieldinflationwasdiscussedinref.[10]. Thetwobasicinflationtypesdependingoninitialconditions
turn out to be still like in [1]. Since inclusion of the M1 modulus into the inflation process is fully legitimate, one
can reserve this scenario as an alternative with a tiny tensor perturbations, if it should be.
7Earlier,Planck’sintermediate results [14]noted about a possible ordinarydust contributioninstead of the light
polarization effect really due to gravitationalwaves.
2
curvature perturbations and bounds for r [13]. Our results are in agreement with a recent analysis
of natural inflation by Freese and Kinney (see 2nd citation in [8]). For various cases we have also
calculated the reheat temperature, which we use later on for contrasting with our 5D SUGRA
emerged inflation scenario requiring a very low reheat temperature.
In Sec. 3 and Appendix A we shortly review our model of ref. [1] in a more self-contained way
and discuss how natural inflation emerges from 5D SUGRA. Using a superfield formulation, we do
not need to go into the details of the component expressions in conformal 5D SUGRA of Fujita,
Kugo and Ohashi (FKO) [18]. Indeed this emerged from our discussion [19,20] (see also Ref. [23])
bringing the 5D conformal SUGRA formulation closer to the 4D global SUSY language [20]. We
concentrate here on gauge inflation, i.e. on the case M1 = 0 (stabilized moduli in the origin or a
choice of initial conditions8). In section 3, discussing the realization of natural inflation within 5D
SUGRA, we present a new mechanism for inflaton decay, which eventually leads to the reheat of the
Universe. Notethat,besidesaspecificstringtheoryrealization[25],theinflatondecayandreheating
has never been discussed before in the context of natural inflation. We show that the inflaton’s
slow decay is a natural consequence of the 5D construction (with consistent UV completion), being
realized by couplings of the heavy bulk supermultiplets generating the inflaton potential through
their gauge coupling with brane SM states. Since the inflaton decay proceeds by 4-body decay
and the decay width is strongly suppressed by the 2-nd power of the tiny U(1) gauge coupling
constant9 (of the gauge inflaton-charged fields) and a relatively small inflaton mass coupled to the
intermediate bulk fields, a strong suppression of the reheat temperature T comes out naturally.
r
Our 5D SUGRA construction allows us to make an estimate T 0.34ρ1/4 λ 2 100 GeV (where
r ∼ reh ∼ | | ×
λ < 1 is a brane Yukawa coupling). At the end of Sec. 3 we show that, by the parameters we are
∼
dealing with, preheating is excluded within the considered scenario.
Appendix A discusses the Kaluza-Klein spectrum of the fields involved, as well as the SUSY
breaking effects for brane fields. We also perform a derivation of higher dimensional operators
involving the inflaton φ and light (MSSM) states relevant for the inflaton decay. As it turns out,
Θ
the dominant decay channel is φ llhh (with l and h denoting SM lepton and Higgs doublets
Θ
→
respectively). Sec. 4 includes a discussion and concluding remarks about some related issues.
2 Natural inflation
In this section we analyse inflation with the potential of natural inflation [8] given by:
= (1+cos(αφ )) , (1)
0 Θ
V V
whereφ isacanonicallynormalizedreal scalarfieldofinflation. Intheconcretescenario ofRef.[1],
Θ
we focus later on, theinflaton originatesfroma 5D gaugesuperfield, while the parameters/variables
of (1) are derived through the underlying 5D SUGRA. See Eqs. (24), (25), (A.17) and also the
comment underneath Eq. (A.17).
8For a discussion of moduli stabilization in the superfield formalism within 5D SUGRA see [24]. For a choice of
initial conditions leading approximately to M1 =0 see Ref. [10].
9From a very recent paper [26] we learned that the ‘weak gravity conjecture’ (going back to Ref. [27]) based on
magnetically charged black hole considerations and the dangerous neighborhood to a global symmetry, applies in
disfavor of gauge (extranatural) inflation and might explain difficulties to embed the model in string theory.
3
The slow roll parameters (”VSR” - derived through the inflaton potential) are given by
M2 ′ 2 (M α)2 αφ
ǫ = Pl V = Pl tan2 Θ
2 2 2
(cid:18)V (cid:19)
′′ (M α)2 αφ 1
η = M2 V = Pl tan2 Θ 1 = ǫ (M α)2 ,
Pl 2 2 − − 2 Pl
V (cid:18) (cid:19)
′ ′′′ αφ
ξ = M4 V V = (M α)4tan2 Θ = 2(M α)2ǫ , (2)
Pl 2 − Pl 2 − Pl
V
where M = 2.4 1018 GeVisthe reduced Planck mass. Inorder to make notationscompact, forthe
Pl
·
VSR parameters we do not use the subscript ‘V’ (denoting them by ǫ,η,ξ, ). However, for HSR
···
parameters (derived through the Hubble parameter) we use subscript ‘H’ (e.g. ǫ ,η ,ξ , ),
H H H
···
as adopted in literature [15], [17], [16]. The number of e-foldings during inflation, i.e. during
exponential expansion, denoted further by Ninf, is calculated as
e
1 φiΘ 1
Ninf = dφ . (3)
e √2M √ǫ Θ
Pl ZφeΘ H
In this exact expression the HSR parameter ǫ (defined below), participates. The point φe , at
H Θ
which inflation ends, is determined by the condition ǫ = 1. The point φi corresponds to the
H Θ
begin of the inflation. Also further, symbols with superscript or subscript ’i’ will correspond to
values at the beginning of the inflation, while superscript/subscript ’e’ will indicate end of the
inflation.
The observables n andr dependonthevalueofφi (thepoint atwhich scales crossthehorizon).
s Θ
This allows to determine φi as follows. Via HSR parameters, the expressions for n and r are given
Θ s
by [15], [17], [16]:
1 1
n = 1 4ǫ +2η 2(1+C)ǫ2 (3 5C)ǫ η + (3 C)ξ ,
s − Hi Hi − Hi − 2 − Hi Hi 2 − Hi
r = 16ǫ (1+2C(ǫ η )) , with C = 4(ln2+γ) 5 0.0815 , (4)
Hi Hi Hi
− − ≃
where we have limited ourself with second order corrections. The HSR parameters ǫ ,η ,ξ are
H H H
given by:
H′ 2 H′′ H′H′′′
ǫ = 2M2 , η = 2M2 , ξ = 4M4 , (5)
H Pl H H PlH H Pl H2
(cid:18) (cid:19)
with the Hubble parameter H and it’s derivative with respect to the inflaton field. The subscript
′i′ in (4) indicates that the parameter is defined at the point at which scales cross the horizon. As
it turns out, at this scale the slow roll parameters are small and second order corrections in n and
s
r are small and the approximations made in (4) are pretty accurate. Exact relations between VSR
(ǫ,η,ξ, ) and HSR parameters (ǫ ,η ,ξ , ) are given by [15], [17], [16]:
H H H
··· ···
3 η 2 3(ǫ +η ) η2 ξ
ǫ = ǫ − H , η = H H − H − H,
H
3 ǫ 3 ǫ
(cid:18) − H(cid:19) − H
3 η 1 H(iv)
ξ = 3 − H 3ǫ η +ξ (1 η ) σ , with σ = 4M4 ǫ . (6)
(3 ǫ )2 H H H − H − 6 H H Pl H H
− H (cid:18) (cid:19)
4
When the slow roll parameters are small, from (6), the HSR parameters to a good approximation
can be expressed in terms of VSR parameters as
4 2 8 8 1 1
ǫ ǫ ǫ2 + ǫη , η η ǫ+ ǫ2 ǫη + η2 + ξ,
H H
≃ − 3 3 ≃ − 3 − 3 3 3
ξ 3ǫ2 3ǫη +ξ. (7)
H
≃ −
Using these approximations in (4), we can write n and r in terms of VSR parameters:
s
2 2 1
n = 1 6ǫ +2η + (22 9C)ǫ2 (14 4C)ǫ η + η2 + (13 3C)ξ
s − i i 3 − i − − i i 3 i 6 − i
2
r = 16ǫ 1 ( 2C)(2ǫ η ) , (8)
i i i
− 3 − −
(cid:18) (cid:19)
where we have still restricted the approximations up to the second order. Applying these expres-
sions, for the model (determining ǫ,η and ξ as given in Eq. (2)), we arrive at:
αφi 1 1 αφi 1 1 αφi
n = 1 1+2tan2 Θ (M α)2+ +(1 C)tan2 Θ +( C)tan4 Θ (M α)4 , (9)
s Pl Pl
− 2 6 −2 2 3 − 2 2
(cid:18) (cid:19) (cid:18) (cid:19)
and
1 αφi αφi
r = 8(M α)2 1 ( C)(1+tan2 Θ)(M α)2 tan2 Θ . (10)
Pl Pl
− 3 − 2 2
(cid:18) (cid:19)
αφi
From Eq. (10) we can express tan Θ in terms of r and M α. As will turn out, the latter’s value
2 Pl
is small, so to a good approximation we find:
αφi r 1 r 1 1
tan2 Θ 1+( C)( +(M α)2)+ ( C)2(M α)4 . (11)
2 ≃ 8(M α)2 3− 8 Pl 8 3 − Pl
Pl (cid:18) (cid:19)
Plugging this into Eq. (9) for the spectral index we get:
r 1 r2 1 3 r 1 3 r2 10 13
n 1 = (M α)2+ (M α)4 ( C)+ ( + C)(M α)2+ ( C( 3C))(M α)2.
s Pl Pl Pl Pl
− −4− 6 −64 3−2 8 3 2 128 9 − 3 −
(12)
Using the recent value n = 0.968 0.006from Planck [4]10,11 relation (12) provides anupper bound
s
±
for the value of M α:
Pl
M α < 0.19 (obtained via 2σ variations of n ) . (13)
Pl ∼ s
This will be used as orientation for further analysis and various predictions.
So far, we have performed calculations in a regime of small slow roll parameters, determining
the value of φi via Eq. (11). As was mentioned, the value of φe is determined from the condition
Θ Θ
ǫ = 1. Near this point both ǫ and η parameters turn out to be large and instead of an expansion
H
we need toperformnumerical calculations. This will berelevant uponthecalculation ofthenumber
of e-foldings Ninf.
e
10The central value of n , is larger, though the range (within 1σ) is consistent with Planck’s old result n =
s s
0.9603 0.0073 [5]. This required modification of our first version appeared before the new results.
11Rec±entjointanalysisofBICEP2/KeckandPlanck[13]gaveanupperboundr∼< 0.12,whilethe likelihoodcurve
for r has a maximum for r 0.05. Note that the value of r reported before by BICEP2 collaboraton [12] was
r =0.2+0.07 although, later o≃n, Planck’s intermediate results [14] warnedabout possible ordinarydust contribution
−0.05
instead of the light polarization effect really due to the gravitational waves.
5
=0.04
(cid:2169)(cid:2172)(cid:2194)(cid:2235)
(cid:2035)(cid:3009)

(cid:2022)(cid:3009)

Figure 1: Dependence of ǫ ,3η and ξ on t(cid:2261)he value of ǫ, for M α = 0.04.
H H H Pl
Since, within our model, via Eq. (2) VSR parameters are related to each other as
1
η = ǫ (M α)2 , ξ = 2(M α)2ǫ, (14)
Pl Pl
− 2 −
the three equation in (6) can be rewritten as
2
3 η
H
ǫ − = ǫ
H
3 ǫ
(cid:18) − H(cid:19)
3(ǫ +η ) η2 ξ 1
H H − H − H = ǫ (M α)2
Pl
3 ǫ − 2
H
−
3 η
3 − H (3ǫ η +ξ (1 η )) = 2(M α)2ǫ , (15)
(3 ǫ )2 H H H − H − Pl
H
−
where σ has been dropped because of it’s smallness. From the system of (15), for a fixed value
H
of M α, the parameters ǫ ,η and ξ can be found in terms of the single parameter ǫ. The
Pl H H H
dependance of these parameters on the value of ǫ, for M α = 0.04 are shown in Fig. 1 (for different
Pl
values of M α shapes of the curves are similar). We see that ǫ = 1 is achieved when ǫ = ǫ 2
Pl H e
≈
and thus, the expansion with respect to ǫ,η within this stage of inflation is invalid. On the other
hand, the values of η and ξ remain relatively small. From the relation 2ǫ = (M α)2tan2 αφΘ
H H Pl 2
one derives:
M √2
Pl
dφ = dǫ. (16)
Θ
√ǫ(2ǫ+(M α)2)
Pl
Using this, the integral in (3) can be rewritten as
ǫi 1 dǫ
Ninf = . (17)
e 2ǫ+(M α)2√ǫǫ
Zǫe Pl H
Having the numerical dependence ǫ = ǫ (ǫ) (depicted in Fig. 1), we can evaluate the integral in
H H
(17) and find Ninf for various values of M α. The results are given in Fig. 2. While BICEP2/Keck
e Pl
6
6655 ø
ø
6600
ø ò
ø
5555 ø ø ø ò
ì
NNeieinnff 5500 r=0.15 r=ò0.13 òr=0.12ò òr=0.1 ò r=0. ìr=0.06ì5r=0.05
0
4455 r=0.18 ì 8
ì
ì
ì
ì r=0.2 ì æ n =0.968
4400 ø ns=0.962
s
ò n =0.956
s
ì n =0.95
s
00..0055 00..11 00..1155 00..22
MM ΑΑ
PPll
Figure 2: Number of e-foldings. Solid lines correspond to the values of n which fit with the current
s
experimental data within 2σ error bars (with no restriction on r). Shaded areas correspond to
the marginalized joint 68% CL regions, given recently in [4] for (n ,r) pairs, mapped by us to the
s
(M α,Ninf) pairs for natural inflation. Gray background corresponds to the Planck TT+lowP,
Pl e
while red and blue colors represent Planck TT+LowP+BKP and Planck TT+LowP+BKP+BAO
respectively (see Ref. [4] for an explanation of these combinations).
and Planck [13] reported the bound r < 0.12, upon generating the curves of Fig. 2 we also allowed
∼
larger values of r. Curves in Fig. 2 and also Table 1 demonstrate that, within natural inflation,
with values n < 0.962 (the previous Planck 2013 value) and r > 0.1 or n < 0.953 and r > 0.05
s ∼ ∼ s ∼ ∼
there is an upper bound on Ninf:
e
Ninf < 55. (18)
e ∼
Turned around this also implies that violating the bound (18), say Ninf 60, indicates larger n
e ∼ s
and/or smaller r. Present Planck 2015 data seems to favor this. Bound (18) (if realized) would
lead to another striking prediction and constraint.
As discussed in Refs. [28], [5], the Neff, guaranteeing causality of fluctuations, should satisfy:
e
k 1016GeV 1/4 4 3γ 1/4
Ninf = 62 ln ln +ln Vi − ln Ve , (19)
e − a0H0 − Vi1/4 Ve1/4 − 3γ ρ1re/h4
where for the scale k we take k = 0.002Mpc−1, while the present horizon scale is a H
0 0
≈
0.00033Mpc−1. The factor γ accounts for the dynamics of the inflaton’s oscillations [29], [30]
after inflation, and can be for our model approximated as γ 1 1 Ve (will turn out to be a pretty
≃ −16V0
good approximation).
To reconcile the first two entries (62 ln k 60.2) of Eq. (19) with the bound of Eq. (18)
− a0H0 ≈
(see also Fig. 2), the remaining entries of Eq. (19) should be significant enough to bring Ninf down
e
7
r M α n 104 dns Ninf V01/4 Vi1/4 Ve1/4 ρ1/4(GeV)
Pl s ×dlnk e 1016GeV 1016GeV 1016GeV reh
0.001 0.962 7.1 53.62 19.8 2.01 0.53 3.14104
− ·
0.04 0.961 7.71 51.47 3.2 2.01 0.54 53.7
0.15 −
0.055 0.959 8.25 49.73 2.77 2.01 0.54 0.31
−
0.065 0.958 8.71 48.39 2.58 2.01 0.55 0.006
−
0.04 0.967 5.43 61.25 2.92 1.92 0.49 2.771014
− ·
0.06 0.965 6.06 57.92 2.45 1.92 0.5 1.391010
− ·
0.125 0.08 0.962 6.95 53.96 2.2 1.92 0.52 1.14105
− ·
0.1 0.959 8.09 49.79 2.04 1.92 0.54 0.5
−
0.11 0.957 8.76 47.72 2 1.92 0.55 0.001
−
0.097 0.966 5.53 60 1.89 1.81 0.49 9.21012
− ·
0.1 0.965 5.68 59.14 1.87 1.81 0.5 7.131011
− ·
0.1 0.12 0.961 6.79 53.58 1.79 1.82 0.52 5104
− ·
0.135 0.957 7.77 49.7 1.74 1.82 0.54 0.52
−
0.143 0.955 8.33 47.76 1.72 1.82 0.55 0.0017
−
0.172 0.958 4.53 59.5 1.35 1.53 0.47 4.81012
− ·
0.19 0.952 5.36 53.48 1.34 1.53 0.49 8.7104
− ·
0.194 0.95 5.55 50.93 1.34 1.53 0.49 44.8
0.05 −
0.2 0.948 5.86 50.51 1.33 1.53 0.5 13.6
−
0.205 0.946 6.12 49.12 1.33 1.53 0.5 0.225
−
0.21 0.944 6.39 47.78 1.33 1.53 0.51 0.0044
−
Table 1: Numerical Results for different values of r and M α. For all cases A1/2 = 4.686 10−5.
Pl s
×
(at least) to 55. The 3rd and 4th entries on the r.h.s. of Eq. (19) can be calculated with help
≈
of another observable - the amplitude of curvature perturbation A , which according to the Planck
s
measurements [4], [5], should satisfy A1/2 = 4.686 10−5 (this value corresponds to the ΛCDM
s
×
model). Generated by inflation, this parameter is given by:
1 3/2 4√6 1/2 M α
A1/2 = V V0 Pl . (20)
s √12π M3 ′ ≃ 3π M2 r(1+8(M α)2/r)1/2
(cid:12)(cid:12) PlV (cid:12)(cid:12)φiΘ Pl Pl
(cid:12) (cid:12)
In order to obtain the observed v(cid:12)alue of A(cid:12) 1/2, for typical r = 0.12 and M α 0.1 we need to have
s Pl
1/4 10−2M . This, on the other hand, gives 1/4 0.01M and 1/4 ∼ 2 10−3M . Using
V0 ∼ Pl Vi ∼ Pl Ve ∼ · Pl
these values in (19) we see that the sum of the 3rd and 4th terms is 3.4. Thus, the last term should
≈
be responsible for a proper reduction of Ninf. Namely, during the reheating process, the universe
e
should expand by nearly 10 (or even more) e-foldings. This means that, for this case, the model
should have a significant reheat history with ρ1/4 100 GeV.12 Within the scenario of natural
reh ∼
inflation, this has not been appreciated before.13 For lower r and appropriate values of M α (and
Pl
n ) the reheating temperature can be big. The concise numerical results (compared to the rough
s
1/4
evaluation below Eq. (20)) are given in Table 1, where we considered cases with ρ not smaller
reh
12The reheating process can continue even till the epoch of nucleosynthesis. In this case one should have ρ1/4
few 10−3 GeV. reh ∼
13×See however some recent analysis in Ref. [21].
8
than 10−3 GeV, and Ninf 62. The values of the spectral index running dns = 16ǫ η 24ǫ2 2ξ
e ≤ dlnk i i− i − i
are also presented. The first three row-blocks correspond to the values of n within 2σ ranges of the
s
current experimental data. The first three cases of the bottom block correspond to the n within
s
3σ range, while the last three lines of this block have lower values of n (beyond the 3σ deviation).
s
Since the issue for the value of r is not fully settled yet, we have included moderately large values of
r( 0.15). At the bottom block of the table we gave results for r = 0.05, which corresponds to the
≤
peak of the r’s likelihood curve presented by the joint analysis of BICEP2/Keck and Planck [13].
Note that the results presented here are consistent with the analysis for natural inflation carried
out before [8] (see 2nd citation of this Ref.).
Below we will show that within our scenario of natural inflation, a low reheat temperature is
realized naturally.
3 Natural inflation from 5D SUGRA
In order to address the details of inflaton decay, related to the reheat temperature, we need to
specify the underlying theory natural inflation emerged from. A very good candidate is a higher
dimensional construction [6]. Here we present a 5D conformal SUGRA realization [1] of this idea,
using the off-shell superfield formulation developed in Refs. [19], [20].14
Lagrangian couplings, for the bulk H = (H, Hc) hypermultiplets’, components are:
e−1 (H) = d4θ(T+T†) H†H +Hc†Hc + d2θ 2Hc∂ H +g Σ (eiθˆ1H2 e−iθˆ1Hc2) +h.c. (21)
(4)L y 1 1 −
Z Z (cid:16) (cid:17)
(cid:0) (cid:1)
where the odd fields V are set to zero.15 Σ is the Z even 5th component of the 5D U(1) vector
i 1 2
supermultiplet. With the parity assignments
Z : H H , Hc Hc , (22)
2
→ → −
the KK decomposition for H and Hc superfields is given by
+∞ +∞
1 1 ny 1 ny
H = H(0) + H(n)cos , Hc = H(n)sin . (23)
2√πR √2πR R √2πR R
n=1 n=1
X X
With these decompositions, and steps given in Appendix A, we can calculate the mass spectrum
of KK states, their couplings to the inflaton and with these, the one loop order inflation potential
(dropping higher winding modes) having the form of (1) with
3
α = πg R, = and = 1 cos(πR F ) . (24)
4 V0 16π6R4B B − | T|
The 4D inflaton field φ is related to the 5D U(1) gauge field A1 as:
Θ 5
φ = √2πRA1 . (25)
Θ 5
14Forthecomponentformalismof5DconformalSUGRAseethepioneeringworkbyFujita,KugoandOhashi[18].
Note also, that the component off shell 5D SUGRA formulation, discussed by Zucker [22], was used in many
phenomenologically oriented papers.
15The bulk hypermultiplet action of Eq. (21), derived from 5D off shell SUGRA construction [19], including
coupling with a radion superfield T, in a rigid SUSY limit coincides with the one given in Ref. [32].
9
Sincethemodeliswelldefined, wealsocanwritedowntheinflatoncouplingwiththecomponents
of H. The latter, having a coupling with the SM fields, would insure the inflaton decay and the
reheating of the Universe. In our setup, we assume that all MSSM matter and scalar superfields
are introduced at the y = 0 brane. Since H is even under orbifold parity and a singlet under all SM
gauge symmetries, it can couple to the MSSM states through the following brane superpotential
couplings
= √2πR d2θdyδ(y)λlh H +h.c. (26)
H−br u
L
Z
where l and h are 4D N = 1 SUSY superfields corresponding to lepton doublets and up type higgs
u
doublet superfields respectively. In Eq. (26), without loss of generality, only one lepton doublet
(out of three lepton families) is taken to couple with the H,
+∞ +∞
1 1
λ ( ψ(0) + ψ(n))(lh +˜lh˜ )+( H(0) + H(n))lh˜
LH−br ⊃ − √2 H H u u √2 u
n=1 n=1
X X
+∞
1
(0) (n) ˜
( F + F )lh +h.c. (27)
− √2 H H u!
n=1
X
˜ ˜
where l now denotes the fermionic lepton doublet and h an up-type higgs doublet. States l and h
u u
stand for their superpartners respectively. H(n) and ψ(n) in Eq. (27) indicate scalar and fermionic
H
components of the superfield H.16
Upon eliminating all F-terms and heavy fermionic and scalar states (in the H and Hc super-
fields), we can derive effective operators containing the inflaton linearly. As it will turn out within
˜
themodelconsidered (see discussion inAppendix A.1), thel statesareheavier thantheinflatonand
˜
operators containing l are irrelevant for the inflaton decay. Thus, the effective operators, needed
to be considered, are
φ C (lh )2 +C (lh˜ )2 +h.c. +C φ (lh˜ )(¯lh¯˜ ). (28)
Θ 0 u 1 u 2 Θ u u
(cid:16) (cid:17)
These terms should be responsible for the inflaton decay. Derivation and form of the C-coefficients
are given in Appendix A.
3.1 Inflaton Decay and Reheating
As was mentioned above and shown in Appendix A.1, the slepton states ˜l have masses 1 F
2| T| ∼
1/(2R) and thus are heavier than the inflaton. Indeed, the latter’s mass, obtained from the poten-
tial, is:
g √3(1 cos(πR F ))1/2 1
4 T
M = − | | . (29)
φΘ 4π2R ≪ R
˜
(g 1 for successful inflation). Thus the inflaton decay in channels containing l is kinematically
4
≪
forbidden. Anticipating, we note that the preheating process by inflaton decay in heavy states
16In Eq. (27) we have omitted HF h and H˜lF type terms, which because of the smallness of the µ
l u hu
term( few TeV) and suppressed lepton Yukawa couplings(∼< 10−2) can be safely ignored in the inflaton decay
∼ ×
proccess.
10