Constraining hybrid inflation models with WMAP three-year results

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by  Antonio Cardoso

Constraining hybrid inflation models with WMAP three-year results
Antonio Cardoso∗
Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 2EG, United Kingdom
(Dated: February 5, 2008)
We reconsider the original model of quadratic hybrid inflation in light of the WMAP three-year
results and study the possibility of obtaining a spectral index of primordial density perturbations,
ns,smallerthanonefromthismodel. Theoriginalhybridinflationmodelnaturallypredictsns ≥1
in the false vacuum dominated regime but it is also possible to have ns < 1 when the quadratic
term dominates. We therefore investigate whether there is also an intermediate regime compatible
with thelatest constraints, where thescalar field valueduring the last 50 e-folds of inflation is less
than thePlanck scale.
7 I. INTRODUCTION II. THE HYBRID INFLATION MODEL
0
0
The results from the WMAP three-year data [1] have The potential for the hybrid inflation model is given
2
provided the first indication that the spectral index of by [3]
n
primordialdensityperturbations,n ,issmallerthanone.
a s 1 1 1
J Theoriginalhybridinflationmodel[2]naturallypredicts V(φ,χ)= λ(χ2 χ20)2+ m2φ2+ λ′φ2χ2, (1)
n 1 in the false vacuum dominated regime but it is 4 − 2 2
1 s ≥
1 also possible to have ns < 1 when the quadratic term whereφistheinflaton,χiscalledthe“waterfall”field,λ
dominates [3]. We are therefore interested in whether andλ′ arecouplingconstantsandχ0 andmareconstant

2 there is also an intermediate regime compatible with the masses.
v latestconstraints. Hybridinflationmodelsareattractive It is assumed that χ stays at the origin, which cor-
4 fromatheoreticalpointofviewbecausetheinflatonfield responds to a false vacuum, while φ rolls down from a
7
φisfarbelowthePlankscaleduringinflation,inthefalse initially large (positive) value until it reaches a critical
0
0 vacuum dominated regime, which makes these models value, φc = λ/λ′M, after which χ becomes unsta-
easiertoimplementwithinsupergravity[4]. Thereforewe
1 ble (the effective mass-squared becomes negative) and
6 also investigate whether there is an intermediate regime rapidly rolls dpown towards one of the true minima at
0 where the scalar field value during the last 50 e-folds of χ = χ0. Then φ goes to zero and starts to oscillate
/ inflation is less than the Planck scale. ±
h while χ will reach the true minimum. Inflation will end
p In this paper we study the original hybrid inflation either with the instability or because of the end of slow-
- model with a quadratic potential for the inflaton. We roll,asinthesinglefieldcase,dependingonwhichoccurs
o
want to analyze carefully this model in light of the new
first.
r
t WMAP results, exploringthe space of parametersof the Before φ reaches φ we can write the potential as a
s c
model. We have run a code to calculate numerically the
a function of φ only,
: spectral tilt of the scalar curvature perturbations, ns,
iv the running of the spectral tilt, αs, and the tensor-to- V(φ)=M4+ 1m2φ2, (2)
X scalarratio,r, assumingslow-roll,for aspecific regionof 2
r parameters of the model. We study the cases for which wherewewrotethe falsevacuumenergydensity 1λχ4 as
a we have n < 1 and compare the results obtained with 4 0
s just M4. We define the ratio between the false vacuum
the WMAP three-yearresults. We use the 68%and95%
energy density and the inflaton energy density as
confidence levelcontours fromWMAP only and WMAP
+ SDSS taken from [5]. Recently similar works have M4
usedthe WMAP three-yearresults to put constraintson E(φ)= 1m2φ2. (3)
hybrid inflation models [6, 7, 8]. 2
We also study the inverted hybrid inflation model So if E & 1 we have false vacuum dominated inflation
[9] with a quadratic potential for the inflaton (see Ap- and if E 1 we have almost chaotic inflation [10].
pendix). ≪
A. The dynamics
Thedynamicsofhybridinflationaregivenbytheequa-
tion of motion of the inflaton (we are assuming that the
“waterfall”fieldstaysattheorigin,soitdoesnotevolve)
and the Friedmann equation,
∗email: [email protected] φ¨+3Hφ˙ = V′(φ), (4)
−
2
H2 = 8π 1φ˙2+V(φ) , (5) the details of reheating at the end of inflation. Given
3m2 2 that φ can be made arbitrarily small by suitable choice
Pl (cid:18) (cid:19) of λ/λc′ we leave φ50, the value of φ when cosmological
where H = a˙/a is the Hubble parameter, a is the scale scales leave the Hubble scale, as a free parameter to be
factor,mPl isthePlanckmass(whichwesetequalto1), determined by observations, subject only to the restric-
a dot represents a derivative with respect to time and a tion that we must have at least 50 e-folds between φ50
primedenotesaderivativewithrespecttothefieldφ. We and φ . If φ does not exist then we can always have at
ǫ ǫ
will use the slow-rollapproximationfor our calculations, least 50 e-folds and any value of φ50 is allowed.
which is given by the conditions
m2 V′(φ) 2
ǫ(φ) Pl 1, (6) B. The perturbations
≡ 16π V(φ) ≪
(cid:18) (cid:19)
2 ′′
m V (φ) The powerspectrum of scalarcurvature perturbations
η(φ) Pl 1. (7)
≡ 8π V(φ) ≪ at horizon crossing (when the comoving scale k equals
the Hubble radius, k = aH, during inflation), which is
Withtheseapproximations,Eqs. (4)and(5)canbewrit- conservedonlargescales in single field inflation, is given
ten as by, to leading order in the slow-rollparameters, [11]
3Hφ˙ V′(φ), (8) H2 2
≃ −
8π R(k)= , (15)
H2 V(φ,χ). (9) P 2πφ˙ (cid:12)
≃ 3m2 (cid:18) (cid:19) (cid:12)∗
Pl (cid:12)
(cid:12)
where the subscript indicates tha(cid:12)t the quantity is to
Thenwecanwrite,forthenumberofe-foldsofexpansion ∗
be evaluated at horizon crossing. By virtue of the slow-
between two field values φ1 and φ2,
roll conditions this formula gives a value of R which is
P
a2 8π φ2 V(φ) nearly independent of k. For the potential in Eq. (2),
N(φ1,φ2)≡lna1 ≃−m2Pl Zφ1 V′(φ)dφ. (10) aEsqssu.m(i6n)g, (s8l)o,w(-9r)o,ll(,2)Eqan.d((1152)),can be written as, using
For the potential in Eq. (2) we have 16π(2M4+m2φ2 )3
50
η(φ) = m2m2Pl , (11) PR = 3 m6Plm4φ250 . (16)
4π(2M4+m2φ2)
So for each values of M and m we can find the values of
ǫ(φ) = m4φ2m2Pl = 1 8π η2φ2,(12) φ50 (there are several possible values for each combina-
4π(2M4+m2φ2)2 2m2 tionof M andm)which satisfy the density perturbation
Pl
8πM4 φ1 2π 2 2 amplitude, for scales of cosmological interest, from the
N(φ1,φ2) ≃ m2m2Pl lnφ2 + m2Pl(φ1−φ2). (13) WofMthAePsptehcrterea-lyteialtrirseszuelrtos)[.1](assumingthattherunning
When the end of slow-roll occurs before the inflaton Thespectraltiltforthescalarcurvatureperturbations,
reaches φ we identify the end of inflation with the con- the running of the spectral tilt and the tensor-to-scalar
c
dition ǫ = 1, which occurs for the value of the inflaton ratio can be written as, to leading order in the slow-roll
field [3] parameters, [11]
φǫ = √m1P6lπ 1+s1− m322Pπl Mm24!. (14) αnss == 116−ǫ∗6ηǫ∗∗−+224ηǫ∗2∗,+2ξ∗2, ((1187))
r = 16ǫ∗, (19)
There is a second root at a smaller φ, below which ǫ
whereξ isahigherorderslow-rollparameterandisequal
becomessmallerthanunityagain,butnumericallyithas
to zero for our specific potential, Eq. (2). Then for this
beenfoundthatslow-rollisnotre-establishedbeforeφ=
potential we can write
0 [3].
We see that if 32πM4/m2 m2 > 1 then φ does not
Pl ǫ m2 m2(M4 m2φ2 )
exist at all, so in this case inflation has to end by insta- n = 1+ Pl − 50 , (20)
bility. If 32πM4/m2 m2 6 1 it will end by instability s π(2M4+m2φ250)2
Pl
when φ = φc if φc > φǫ or by the end of slow-roll when α = m4Plm6φ250(4M4−m2φ250), (21)
φ=φǫ if φǫ >φc. s 2π2(2M4+m2φ2 )4
50
We assume that cosmological scales exit the Hubble
4m2 m4φ2
scale at least 50 e-folds before the end of inflation. In r = Pl 50 . (22)
practice the actual number of e-folds is dependent upon π(2M4+m2φ250)2
3
C. Results and discussion Fortherangeofparametersforwhich(n ,r)isinsidethe
s
95% confidence level contour we have
Wehaverunacodetocalculatenumericallyφ50,Eand 0.95.n .1 and 0.r .0.3. (26)
the parameters n , r and α at Hubble crossing (when s
s s
φ = φ50), assuming slow-roll and using the definitions
and results from the previous sections, for m between
10−4 and10−8 andM between 10−2 and10−5. For each
valuesofmandM wehaveselectedthevalueofφ50 such
that the amplitude of the scalar curvature perturbations
Eq. (16)obeystheWMAPthree-yearresults[1]. Wedis-
card the cases for which the number of e-folds between
φ50 andφǫ issmallerthan50andthosewithabluespec-
trum, i.e., we require n < 1. The energy density ratio,
s
Eq. (3), is evaluated at φ50.
We first note that as one goes from larger to smaller
values of M the density of points selected by the code
gets larger because the selection of the values of M is
logarithmic, as one can see looking to the plot in Figure
1.
10-5
10-4
M
10-3
10-2
10-4 10-5 10-6 10-7 10-8
m
FIG. 2: Plots of r as a function of ns for 0.9 < ns < 1 (the
blackdotsrepresentthedifferentvaluesofM andm),forthe
hybridinflationmodel,withthe68%(yellow)and95%(blue)
FIG.1: ContourplotofnsasafunctionofmandM for0.9< confidence level contours from WMAP (top) and WMAP +
ns <1, for the hybrid inflation model. The red, green, blue, SDSS(bottom), taken from [5].
yellow and light blue contours (from left to right) represent,
respectively, ns equal to0.91, 0.93, 0.95, 0.97 and 0.99.
Observing the plot in Figure 3 we see that the closer
one gets to large values of M the larger are the values
Analyzing the top plot in Fig. 2 we see that there is a
of r (i.e., the energy scale of inflation increases when M
small range of parameters for which (n ,r) is inside the
s
increases) and α . Analyzing the plot in Figure 4 we
68% confidence level contour from WMAP only [5]. For s
this range we find see that the closer one gets to small values of φ50 the
larger are the values of r (and so also the values of α
s
0.9525.ns .0.9975 and 0.05.r .0.35. (23) get larger). We also note that φ50 is never much smaller
than 1, i.e., the Planck mass. Looking to the plot in
Fortherangeofparametersforwhich(n ,r)isinsidethe
s
Figure 5 we note that the energy density ratio is never
95% confidence level contour we find
muchlargerthan1,whichmeansthatthereisneverareal
0.94.ns .1 and 0.r .0.55. (24) falsevacuumdominationoftheenergydensity,otherwise
we would have n 1. We also see that the closer one
Considering the results from WMAP + SDSS [5] we see s ≥
gets to the false vacuum domination cases (which occur
that, looking to the bottom plot in Fig. 2, the range of
for large values of M) the larger are the values of r and
parametersforwhich(n ,r) isinside the 68%confidence
s thelarger(andpositive)arethevaluesofthetherunning
level contour is smaller than in the previous case. For
α50.
this range we find
Fromthisanalysisweseethatwhenwehavefalsevac-
0.96.ns .0.995 and 0.05.r .0.175. (25) uum domination and small values of φ50, i.e., when we
4
r r
1 1
0.5 0.5
0.2 0.2
Α E
1.´10-6 0.00001 0.0001 0.001 0.01 0.1 s 0 0.25 0.5 0.75 1 1.25 1.5 1.75
0.1 0.1
0.05 0.05
0.02 0.02
FIG. 3: Plot of r as a function of αs for 0.945 <ns <0.955 FIG. 5: Plot of r as a function of E for 0.945 < ns < 0.955
(red), 0.965 < ns < 0.975 (green) and 0.985 < ns < 0.995 (red), 0.965 < ns < 0.975 (green) and 0.985 < ns < 0.995
(blue),for thehybrid inflation model. (blue),for thehybridinflation model.
r
r, and the running of the spectral tilt, α , become large
s
(with a positive running), which is in contradiction with
1
the WMAP three-year results [5]. This agrees with the
0.5 results in [6, 7, 8]. We found a lower bound on the al-
lowedvaluesofr,whichmightbeanobservationalsignal
0.2 forhybridinflationbecauseifthereisanupperboundon
Φ
50
0 2 4 6 8 thespectralindex,n <1,thenwefindalowerboundon
s
0.1
the tensor-to-scalarratio. This lowerbound corresponds
0.05 to chaotic inflation, whichis in goodagreementwith the
newWMAPresults[5]. Wealsosawthatitisdifficultto
0.02 getφ50 smallerthanthe Planckscalebecause decreasing
φ50 requires us to increase the vacuum energy scale, M,
and this increases the tensor-to-scalarratio, r.
The inverted hybrid inflation model (see Appendix) is
FIG. 4: Plot of r as a function of φ50 for 0.945<ns <0.955
inmuchbetteragreementwiththeWMAPthree-yearre-
(red), 0.965 < ns < 0.975 (green) and 0.985 < ns < 0.995
sults[5]thantheoriginalhybridinflationmodelwhenwe
(blue),for thehybrid inflation model.
consider the false vacuum dominated regimes, especially
because there is no lower bound on the allowed values
of r. Moreover, in the inverted hybrid model there is no
are more distant from chaotic inflation, then the values
of the the running α50 are large and positive, which is problemobtainingsmallvaluesforφ50 ≪1,whichmakes
these model easier to implement within supergravity[4].
in contradiction with the WMAP three-year results [5].
Also the values of the tensor-to-scalarratio r tend to be
very large for these cases, which is also in disagreement
Acknowledgements
with the observations.
When M goes to very small values we recover the re-
sults for chaotic inflation, which correspondto the lower I wouldliketo thank DavidWands for veryuseful dis-
limit for r along the values of n in Fig. 2. For these cussionsandcommentsonthiswork. Iwouldalsoliketo
s
cases, for which φ50 is very large and the energy den- thankthe authorsofRef. [5]forpermissiontoreproduce
sity ratio very small, we have small values for α50 and r, likelihood contours from their work. I am supported by
which is in very goodagreementwith the WMAP three- FCT (Portugal)PhD fellowship SFRH/BD/19853/2004.
year results [5].
APPENDIX: THE INVERTED HYBRID
INFLATION MODEL
III. CONCLUSIONS
If we invert the sign of the inflaton energy density in
We have found that there is an intermediate regime
Eq. (2) we get
for the original hybrid inflation model compatible with
ns < 1, but as one approaches the false vacuum domi- 4 1 2 2
nated limit within this regime the tensor-to-scalarratio, V(φ)=M m φ , (27)
− 2
5
which corresponds to the potential for the inverted hy- is no lower bound on the allowedvalues of r, in contrast
brid inflation model with a quadratic potential for the with the original hybrid inflation model. For this range
inflaton. A particular case of this model are the small- we find
field inflation models [11], for which the potential in Eq.
(27) corresponds to a Taylor expansion about the origin 0.935.ns .0.9875 and 0.r.0.0775. (28)
and higher order terms are required to provide a poten-
tial minimum at some φ=0, as requiredto connect to a Fortherangeofparametersforwhich(ns,r)isinsidethe
reheatingstage(whichis6necessarytomakethemodelre- 95% confidence level contour we get
alistic). So the inflaton starts with an initial small (pos-
itive) value and rolls down towards the minimum at a 0.9175.ns .1 and 0.r .0.0775. (29)
larger value, then it starts to oscillate and inflation ends
Observing the bottom plot in Fig. 7 we can see that
with the end of slow-roll.
there is alsoawide rangeof parametersfor which(n ,r)
All the equations obtained before for the original hy- s
is inside the 68% confidence level contour from WMAP
brid inflation model, except Eq. (14), are still valid but
with m2 replaced by m2. To find the value of φ we + SDSS [5]. For this range we get
ǫ
−
search for the solutions of Eq. (12) with ǫ equal to 1
0.935.n .0.9925 and 0.r.0.0775, (30)
(there are two different solutions for this equation but s
the results do not depend on which solution we choose).
andfortherangeofparametersforwhich(n ,r)isinside
s
the 95% confidence level contour we find
1. Results and discussion
0.92.n .1 and 0.r .0.0775. (31)
s
In this case we have run the same code (with the ap-
propriate changes in the equations) but for m between
10−4 and10−10 andM between10−2 and10−6. The en-
ergy density ratio, Eq. (3), is evaluated 50 e-folds after
the inflaton has the value φ50.
Note that also in this case the selection of the values
ofM is logarithmic,as onecansee lookingto the plotin
Figure 6.
10-6
10-5
M 10-4
10-3
10-2
10-4 10-5 10-6 10-7 10-8 10-9 10-10
m
FIG. 6: Contour plot of ns as a function of m and M for
0.9 < ns < 1, for the inverted hybrid inflation model. The
red, green, blue, yellow and light blue contours (from left to
right)represent,respectively,nsequalto0.91,0.93,0.95,0.97
and 0.99.
FIG. 7: Plots of r as a function of ns for 0.9 < ns < 1 (the
Analyzing the top plot in Fig. 7 we see that there is a blackdotsrepresentthedifferentvaluesofM andm),forthe
muchwiderrangeofparametersforwhich(n ,r)isinside inverted hybrid inflation model, with the 68% (yellow) and
s
the 68% confidence level contour from WMAP only [5] 95% (blue) confidence level contours from WMAP (top) and
WMAP + SDSS(bottom), taken from [5].
thanin the originalhybridcase,especially because there
6
Looking to the plot in Figure 8 we see that the closer 1, which means that there is a large false vacuum domi-
onegetstolargevaluesofM thesmallerarethevaluesof nation of the energy density. Therefore we can conclude
r and α (although we note that they are always small, that this model is in much better agreement with the
s
even in the cases for which M is small). Analyzing the WMAP results than the original hybrid inflation model
plot in Figure 9 we see that the closer one gets to small when we consider the false vacuum domination regimes,
values of φ50 the smaller are the values of r (and so also especiallybecausethereisnolowerboundontheallowed
the values of αs get smaller). We note that here φ50 values of r. Moreover,in this model there is no problem
can be much smaller than the Planck mass. Observing obtaining small values for φ50 1, which makes these
≪
the plot in Figure 10 we see that the closer one gets to model easier to implement within supergravity [4].
themorefalsevacuumdominationcases(whichoccurfor
large values of M) the smaller are the values of r (and r
so also the values of α ). In this case we note that the
s
energy density ratio can be very large, so we can have a 0.01
strong vacuum domination.
0.0001
r
1.´10-6
0.01
1.´10-8
0.0001
E
1.´10-6 1 100 10000 1.´106 1.´108
1.´10-8
FIG. 10: Plot of r as a function of E for 0.945<ns <0.955
Α (red), 0.965 < ns < 0.975 (green) and 0.985 < ns < 0.995
1.´10-11 1.´10-9 1.´10-7 0.00001 s (blue),for theinverted hybridinflation model.
FIG. 8: Plot of r as a function of αs for 0.945 <ns <0.955
(red), 0.965 < ns < 0.975 (green) and 0.985 < ns < 0.995
(blue),for theinverted hybridinflation model.
r
0.01
0.0001
1.´10-6
1.´10-8
Φ
50
0.001 0.1 10 1000 100000.
FIG. 9: Plot of r as a function of φ50 for 0.945<ns <0.955
(red), 0.965 < ns < 0.975 (green) and 0.985 < ns < 0.995
(blue),for theinverted hybridinflation model.
Fromthepreviousanalysisweseethatfortheinverted
hybrid inflation model we can have, specially for large
M, very small values for r and α , which is in very good
s
agreement with the WMAP three-year results [5]. For
thesecasesφ50 isalsoverysmall(itcanbemuchsmaller
than 1, i.e., the Planck mass)and E is much largerthan
7
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