Inflation, Gravity Waves and Dark Matterbartol_udel_edu_01.pdfInﬂation, Gravity Waves and Dark Matter Qaisar Shaﬁ Bartol Research Institute Department of Physics and Astronomy

Inflation, Gravity Waves and Dark Matter

Qaisar Shafi

Bartol Research InstituteDepartment of Physics and Astronomy

University of Delaware

Corfu 2010Greece

Outline

Introduction

Gauge Singlet Higgs Inflation

Standard Model Higgs Inflation (& variants )

Dark Matter Inflation

MSSM Inflation

Supersymmetric Higgs (Hybrid) Inflation

Summary

Evidence for Inflation

TT power spectrum yields:

Scalar spectral index ns

Baryonic and dark matter densities Ωb,Ωc

Deviations from Gaussianity not reflected in power spectrum

Nolta et. al. (WMAP Collaboration), 2008

Temperature Anisotropy and CMB Polarization

Nolta et. al. (WMAP Collaboration), 2008

In addition to temperature anisotropy, one expects the CMB to become polarized viaThompson scattering: Linear polarization results from the velocities of electrons andprotons. With both the velocity field and temperature anisotropies created byprimordial density fluctuations, one expects to see temperature-polarizationcorrelations. (〈TT 〉, 〈TE〉, 〈EE〉, 〈BB〉)

Planck Resolution

Planck Blue Book

Planck and Tests of Inflation

[Efstathiou, Cambridge 2009]

Capable of measuring large scale B-mode polarization in theCMB

Direct measurement of r (tensor-to-scalar ratio)Can measure r & 0.03, which can rule out many inflationmodels predicting r ≪ 1

Refine measurements of ns, δT/T , Ωb,c,Λ, . . .

Measure or further restrict non-Gaussianity in CMB

Detection of non-Gaussian fluctuations can rule out largeclasses of inflation models

Can probe low multipoles (l < 10) ≡ large distances

Inflationary Cosmology

[Guth, Linde, Albrecht & Steinhardt, Starobinsky, Mukhanov, Hawking, . . . ]

Successful Primordial Inflation should:

Explain flatness, isotropy;

Provide origin of δTT ;

Offer testable predictions for ns, r, dns/d ln k;

Recover Hot Big Bang Cosmology;

Explain the observed baryon asymmetry;

Offer plausible CDM candidate;

Physics Beyond the SM?

Slow-roll Inflation

Inflation is driven by some potential V (φ):

Slow-roll parameters:

ǫ =m2p

2

(

V ′

V

)2, η = m2

p

(

V ′′

V

)

.

The spectral index ns and the tensor to scalar ratio r aregiven by

ns − 1 ≡ d ln∆2R

d lnk , r ≡ ∆2h

∆2R

,

where ∆2h and ∆2

R are the spectra of primordial gravity wavesand curvature perturbation respectively.

Assuming slow-roll approximation (i.e. (ǫ, |η|) ≪ 1), thespectral index ns and the tensor to scalar ratio r are given by

ns ≃ 1− 6ǫ+ 2η, r ≃ 16ǫ.

Slow-roll Inflation

The tensor to scalar ratio r can be related to the energy scaleof inflation via

V (φ0)1/4 = 3.3× 1016 r1/4 GeV.

The amplitude of the curvature perturbation is given by

∆2R = 1

24π2

(

V/m4p

ǫ

)

φ=φ0= 2.43 × 10−9 (WMAP7 normalization).

The spectrum of the tensor perturbation is given by

∆2h = 2

3π2

(

Vm4P

)

φ=φ0.

The number of e-folds after the comoving scale l0 = 2π/k0has crossed the horizon is given by

N0 =1m2p

∫ φ0φe

(

VV ′

)

dφ.

Inflation ends when max[ǫ(φe), |η(φe)|] = 1.

Tree Level Gauge Singlet Higgs Inflation

[Kallosh and Linde, 07; Rehman, Shafi and Wickman, 08]

Consider the following Higgs Potential:

V (φ) = V0

[

1−(

φM

)2]2

←− (tree level)

Here φ is a gauge singlet field.

MΦ

V HΦL

Above vevHAV Linflation

Below vevHBVLinflation

WMAP data favors BV inflation.

Tree Level Gauge Singlet Higgs Inflation

0.940 0.945 0.950 0.955 0.960 0.965 0.9700.00

0.05

0.10

0.15

ns

r

N0 =50 N0

» 55

N0 =60

QPHΦ 2limit L

Radiative Corrections in Gauge Singlet Higgs Inflation

Consider the following interaction of inflaton φ with someGUT symmetry breaking scalar boson Φ:

Lint = λ2Φ2 φ2 Φ2

Include Radiative Corrections (Quantum Smearing):

V =(

m2M2

4

)

[

1−(

φM

)2]2

+Aφ4[

ln(

φM

)

− 14

]

+ AM4

4 ,

where V (φ = 0) ≡ V0 = m2M2

4 + AM4

4 and A =Nλ4Φ32π2 .

Note that we can use ‘Minkowski space’ CW correctionsprovided the propagating fields have masses ≫ H (Hubbleconstant).

Higgs Inflation, Quantum Smearing and Tensor to Scalar Ratio

[Rehman and Shafi, 2010]

0.940 0.945 0.950 0.955 0.960 0.965 0.9700.00

0.05

0.10

0.15

ns

r

A = 0

A = 10-14.0

A = 10-13.6

A = 10-13.3

CWPHN0 » 55L

CWPHN0=60L

QPHΦ 2limit L•

Note that r & 0.02 if ns & 0.96. Thus, Planck will test Higgsinflation soon!

Quantum Smearing

0.940 0.945 0.950 0.955 0.960 0.965 0.9700.00

0.05

0.10

0.15

ns

r

N0 = 60

QPHΦ 2limit L

Higgs

Quadratic+Radiative Correction

Coleman-Weinberg

Quantum Smearing

16.00 16.05 16.10 16.15 16.20 16.25 16.30 16.350.960

0.962

0.964

0.966

0.968

0.970

log10@HVHΦ0LL14GeVD

n s

N0 = 60

Higgs

Quadratic+Radiative Correction

Coleman-Weinberg

The vacuum energy scale during observable inflation is well below mP . This implies

that the quantum gravity effects are relatively unimportant here.

Standard Model Inflation?

[Bezrukov, Magnin and Shaposhnikov; De Simone, Hertzberg and Wilczek.]

[Barvinsky, Kamenshchik, Kiefer, Starobinsky and Steinwachs.]

[Okada, Rehman and Shafi.]

Consider the following action with non-minimal coupling:

SJ =∫

dx4√−g−|∂H|2 − λ

(

H†H − v2

2

)2+ 1

2m2P R+ ξH†HR

In the Einstein frame the potential turns out to be:

VE(φ) ≃λG4

4φ4

(

1+ ξ G2φ2

m2P

)2 ; G(t) = exp[−∫ t0 dt

′γ(t′)/(1 + γ(t′))]

where HT = 1√2(0, v + φ), t = log[ φMt

] and γ(t) is the

anomalous dimension of the Higgs field.

For large φ values, VE(φ) gives rise to inflation.


Mt = 171.8 GeV

Mt = 173.1 GeV

Mt = 174.4 GeV

33.0 33.5 34.0 34.5 35.0

-0.005

0.000

0.005

0.010

t = logHΦMtL

H¶ΨHΛ

G4 LLΛG

4

mh = 134 GeV

Mt = 171.8 GeV

34.6 34.8 35.0 35.2 35.4 35.6

0.0019

0.0020

0.0021

0.0022

t = logHΦMtL

mh = 128 GeV

λ(t)G(t)4 vs. t, between pivot-scale t0 = log[ φ0

Mt] and inflation end scale

te = log[ φe

Mt]

The spectral index is

ns ≃ 1− 2Ne

+ ψ0

3

(

∂ψ0 (λG4)

(λG4)

)

; ψ0 ≡√ξ φ0mP

with

∆2R ≃ λ

ξ2N2e

72π2 ⇒ ξ ≃(

Ne6√2π∆R

)√λ ∼ 104 (!)


128 130 132 134 1360.94

0.95

0.96

0.97

0.98

0.99

1.00

mh

n sMt = 171.8 GeV

Mt = 173.1 GeV

Mt = 174.4 GeV

The horizontal line is the classical tree level result which isindependent of Higgs mass. All other curves include quantumcorrections with different values of Mt.


0.94 0.95 0.96 0.97 0.980.000

0.001

0.002

0.003

0.004

0.005

0.006

ns

r

Mt = 174.4 GeV

Mt = 173.1 GeV

Mt = 171.8 GeV

r vs. ns. The black circle corresponds to the classical tree levelresult. All other curves include quantum corrections with differentvalues of Mt.


[Barvinsky, Kamenshchik, Kiefer, Starobinsky and Steinwachs]

Mt=169

Mt=171

Mt=173

classical

130 140 150 160 170 180 1900.940

0.945

0.950

0.955

0.960

0.965

0.970

Higgs mass m h HGeVL

Spe

ctra

lin

dex

n s

The spectral index ns as a function of the Higgs mass mh for threevalues of the top quark mass.

Type I Seesaw with Three Right Handed Neutrinos

[Okada, Rehman and Shafi, 09]

129 130 131 132 133 134 135 1360.94

0.95

0.96

0.97

0.98

0.99

1.00

mh

n s

MR = 1012.5 GeV

MR = 1013 GeV

MR = 1013.5 GeV

129 130 131 132 133 134 135 1360.94

0.95

0.96

0.97

0.98

0.99

1.00

mhn s

MR = 1012.5 GeV

MR = 1013 GeV

MR = 1013.5 GeV

ns vs. mh for hierarchical (left panel) and inverted-hierarchical(right panel) neutrino mass spectrum for various values of MR andMt = 173.1 GeV.

Type I Seesaw with Three Right Handed Neutrinos

0.94 0.95 0.96 0.97 0.980.000

0.001

0.002

0.003

0.004

0.005

0.006

ns

r

MR = 1013.5 GeV MR = 1013 GeV

MR = 1012.5 GeV

r vs. ns for hierarchical neutrino mass spectrum for various valuesof MR and Mt = 173.1 GeV. The black circle corresponds to theclassical tree level result.

Challenges for SM inflation

[Burgess, Lee and Trott; Barbon and Espinosa; Hertzberg]

Consider gµν = ηµν +hµνmP

, so that the term ξ φ2R yields

ξmP

φ2 ηµν ∂2 hµν + · · ·

This suggests an effective cut-off scale Λ ≈ mPξ ≪ mP .

The energy scale of inflation is estimated to be

Ei ≃ V 1/40 = λ1/4mP

41/4√ξ≫ Λ = mP

ξ .

Thus it is not clear how reliable are the calculations.

Indeed, SM inflation is mired in some controversy (see forinstance, Lerner and McDonald [arXiv:0912.5463]).

Non-Minimal φ4 Inflation

[Okada, Rehman, Shafi, 2010]

Consider the following action in the Jordan frame:

SJ =∫

d4x√−g

[(

m2P+ξφ2

2

)

R− 12(∂φ)

2 − λ4! φ

4]

,

where φ is SM gauge singlet and we require that energy scale

of inflation ≤ Λ ≡ mP /ξ.


-6 -4 -2 0 2

0.01

0.02

0.05

0.10

0.20

0.50

1.00

log10HΞL

V1

4 L

-6 -4 -2 0 2-12

-10

-8

-6

-4

log10HΞL

log 1

0HΛL

V 1/4/Λ and log10(λ) vs. log10(ξ) for tree level non-minimal φ4

inflation with the number of e-foldings N0 = 50 (red dashed curve)and N0 = 60 (green solid curve).


-6 -4 -2 0 2

0.945

0.950

0.955

0.960

0.965

log10HΞL

n s

N0 = 50

N0 = 60

-6 -4 -2 0 2

0.005

0.010

0.020

0.050

0.100

0.200

log10HΞL

r

N0 = 50N0 = 60

ns and r vs. log10(ξ) for tree level non-minimal φ4 inflation withthe number of e-foldings N0 = 50 (red dashed curve) and N0 = 60(green solid curve).


0.94 0.95 0.96 0.97 0.98 0.990.00

0.05

0.10

0.15

0.20

0.25

0.30

ns

r

N0 = 50

N0 = 60

r and vs. ns with N0 = 50 and N0 = 60. The WMAP 1-σ (68%confidence level) bounds are shown in yellow.


-6 -4 -2 0 20

5

10

15

20

log10HΞL

Φ0

mP

N0 = 50

N0 = 60

-6 -4 -2 0 20.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

log10HΞLΦ

em

P

φ0/mP and φe/mP vs. log10(ξ) with Ne = 50 and Ne = 60.


[Okada, Rehman, Shafi, 2010]

Consider the following interaction of inflaton φ with the righthanded Majorana neutrino N :

Lint = 12 yN φN N

Include Radiative Corrections (Quantum Smearing):

VE(φ) =14!λφ4−κφ4 ln(φ/µ)(

1+ ξ φ2

m2P

)2 ,

where κ = y4N/(4π)2.


0.940 0.945 0.950 0.955 0.960 0.965 0.9700.00

0.05

0.10

0.15

0.20

0.25

ns

r

Κ = 0

Ξ = 0Κ = 10-13.50

Κ = 10-13.30

Κ = 10-13.26

Κ = 10-13.25

r vs. ns for the radiatively corrected non-minimal φ4 potential with the

number of e-foldings N0 = 60. The black dots represent the meeting

points of the hilltop and the φ4 solutions and correspond, for a given ξ,

to the maximum value of κ.


0.940 0.945 0.950 0.955 0.960 0.965 0.9700.00

0.05

0.10

0.15

0.20

0.25

ns

r

Κ = 0

Ξ = 0Ξ = 10-3

Ξ = 10-2.2

Ξ = 1

Ξ = 200

r vs. ns for the radiatively corrected non-minimal φ4 potential with the

number of e-foldings N0 = 60. The black dots represent the meeting

points of the hilltop and the φ4 solutions and correspond, for a given ξ,

to the maximum value of κ.


-6 -4 -2 0 20.940

0.945

0.950

0.955

0.960

0.965

log10HΞL

n s

Κ = 0

Κ = 10-13.5

Κ = 10-13.3

Κ = 10-13.26

Κ = 10-13.25

-16 -14 -12 -10 -80.940

0.945

0.950

0.955

0.960

0.965

log10HΚL

n s

Ξ = 0

Ξ = 10-3

Ξ = 10-2.2

Ξ = 1

Ξ = 200

ns vs. log10(ξ) and log10(κ) with N0 = 60.


-6 -4 -2 0 20.002

0.005

0.010

0.020

0.050

0.100

0.200

log10HΞL

r

Κ = 0Κ = 10-13.5

Κ = 10-13.3

Κ = 10-13.26

Κ = 10-13.25

-16 -14 -12 -10 -8

0.001

0.002

0.005

0.010

0.020

0.050

0.100

0.200

log10HΚL

r

Ξ = 0 Ξ = 10-3

Ξ = 10-2.2

Ξ = 1

Ξ = 200

r vs. log10(ξ) and log10(κ) with N0 = 60.

r & 0.002, for ns ≥ 0.96


-6 -4 -2 0 2-12

-10

-8

-6

-4

log10HΞL

log 1

0HΛL

Κ = 0

Κ = 10-13.5

Κ = 10-13.3

Κ = 10-13.26

Κ = 10-13.25

-16 -14 -12 -10 -8-12

-10

-8

-6

-4

log10HΚL

log 1

0HΛL

Ξ = 0

Ξ = 10-2.2

Ξ = 10-3

Ξ = 1

Ξ = 200

log10(λ) vs. log10(ξ) and log10(κ) with N0 = 60.


-6 -4 -2 0 20

5

10

15

20

log10HΞL

Φ0

mP

Κ = 0

Κ = 10-13.5

Κ = 10-13.3

Κ = 10-13.26

Κ = 10-13.25

-16 -14 -12 -10 -80

5

10

15

20

log10HΚLΦ

0m

P

Ξ = 0

Ξ = 10-3

Ξ = 10-2.2

Ξ = 1

Ξ = 200

φ0/mP vs. log10(ξ) and log10(κ) with N0 = 60.

(B-L) Inflation + νR Dark Matter

Consider the gauge group SM × U(1)B−L with a SM gaugesinglet inflaton φ charged under B − L.

This simple extension of SM naturally requires the presenceof three right handed (RH) neutrinos due to gauge anomalycancellations. With Z2 parity one of the three RH neutrinoscan be cold dark matter. [Okada, Seto, 2010]

The origin of the baryon asymmetry can be explained throughresonant leptogenisis with TeV scale RH neutrinos.


[Okada, Rehman and Shafi 2010]

Using non-minimal φ4 inflation discussed in previous slides,the inflationary effective potential, in the leading-logapproximation, can be written as

V (φ) ≃(

λ(TeV )4 +

96 g4B−L

16π2 ln[

φTeV

])

φ4

where gB−L is the value of the B − L gauge coupling.


0.95 0.96 0.97 0.98 0.99

0.010

0.100

0.050

0.020

0.200

0.030

0.300

0.015

0.150

0.070

ns

rΛHTeVL = 0

N0 = 50

N0 = 60


0.001 0.01 0.1 1

0.95

0.96

0.97

0.98

0.99

gB-LHΦ0L

n s

ΛHTeVL = 0

N0 = 50

N0 = 60

10-6 10-4 0.01 1 100 104

0.95

0.96

0.97

0.98

0.99

Ξn s

ΛHTeVL = 0

N0 = 50

N0 = 60


0.001 0.01 0.1 1

0.010

0.100

0.050

0.020

0.200

0.030

0.300

0.015

0.150

0.070

gB-LHΦ0L

r

ΛHTeVL = 0

N0 = 60

N0 = 50

10-6 10-4 0.01 1 100 104

0.010

0.100

0.050

0.020

0.200

0.030

0.300

0.015

0.150

0.070

Ξr

ΛHTeVL = 0

N0 = 60

N0 = 50


0.001 0.01 0.1 1

0.01

0.1

1

10

100

gB-LHΦ0L

V1

4 L

ΛHTeVL = 0

N0 = 50

N0 = 60

10-6 10-4 0.01 1 100 104

0.01

0.1

1

10

100

ΞV

14 L

ΛHTeVL = 0

N0 = 50

N0 = 60


0.001 0.01 0.1 1

0.001

0.01

0.1

gB-LHΦ0L

g B-

LHT

eVL

ΛHTeVL = 0

N0 = 50

N0 = 60


[Kofman, Linde, Starobinsky 94; Liddle, Lopez 09; Okada, Shafi 2010]

Can the inflaton also be a dark matter particle?

Looks possible if we augment the SM with a gauge singletscalar field φ with mass . 1 TeV, and use non-minimal φ4

inflation discussed in previous slides (see also Lerner and McDonald

arXiv:0909.0520v3 [hep-ph]).

An important role in this story is played by the interaction

g2|H|2φ2,

where g2 ∼ 0.1 from dark matter considerations (Kanemura,

Matsumoto, Nabeshima and Okada, arXiv:1005.5651 [hep-ph]).


Thus, in this example of ’inflaton-dark matter unification’, theφ quartic coupling cannot be much smaller than ∼ 10−3 or so.This allows one to make rather precise predictions for ns andr.

Dark matter φ particles compatible with WMAP bounds arisefrom preheating and subsequent Treheat ∼ 107 GeV. Theenergy density in the remnant φ oscillations turn out to bequite negligible.

Stability of φ particle is ensured by an unbroken Z2 symmetry.


0.960 0.965 0.970 0.975 0.9800.002

0.003

0.004

0.005

0.006

0.007

0.008

ns

r

r vs. ns with N0 = 60 (solid curve) and N0 = 50 (dashed curve)e-foldings. Both curves lie within the WMAP 1-σ (68% confidencelevel) bounds.


0.960 0.965 0.970 0.975 0.9800

10

20

30

40

ns

V14ß

V 1/4/Λ vs. ns with N0 = 60 (solid curve) and N0 = 50 (dashedcurve) e-foldings.

SM Singlet Dark Matter

[Kanemura, Matsumoto, Nabeshima and Okada 2010]

10310210

1

10−1

10−2

10−3

10−4

Dark Matter Mass (GeV)

Cou

plin

g co

nsta

nt (

g)

SuperCDMS

XENON100

XMASS

WMAP

LHC

m = 120 GeVh

LHCCDMS &XENON

Constraints on the nightmare scenario from WMAP, Xenon100 first data, and CDMS II experiments. Expected

sensitivities to detect the signal of the dark matter at XMASS, SuperCDMS, Xenon100, and LHC experiments are

also shown.

Why Supersymmetry?

Resolution of the gauge hierarchy problem

Unification of the SM gauge couplings at

MGUT ∼ 2× 1016 GeV

Cold dark matter candidate (LSP)

Other good reasons:

Radiative electroweak breaking

String theory requires susy

Leading candidate is the MSSM (Minimal Supersymmetric Standard Model)

Why Supersymmetry?

MSSMΑ1-1

Α2-1

Α3-1

2 4 6 8 10 12 14 16

10

20

30

40

50

60

Log10@LGeVD

Αi-

1

MSSM Inflation

[Allahverdi, Enqvist, Garcia-Bellido, Mazumdar]

Numerous flat dimensions exist in MSSM.

Utilize UDD and LLE.

By suitable tuning of soft susy breaking parameters A andmφ, an inflationary scenario may be realized.

Flat directions lifted by higher dimensional operators

W = λΦn

mn−3P

, Φ is flat direction superfield.

V = 12m

2φφ

2 +A cos(nθ + θA)λn φn

nmn−3P

+ λ2n φ2(n−1)

m2(n−3)P

For A2 ≥ 8(n − 1)m2φ, there is a secondary minimum at

φ = φ0 ∼ (mφmn−3P )

1(n−2) ≪ mp, with

V ∼ m2φφ

20 ∼ m2

φ(mφmn−3P )

2n−2 (this can drive inflation)

MSSM Inflation

Hinf ∼ (mφφ0)mP

∼ mφ(mφmP

)1

(n−2) ≪ mφ

To implement realistic inflation, one wants both the first andsecond derivatives of V to vanish at φ0:

Thus

V (φ) ∼ V (φ0) +13!V′′′(φ0)(φ− φ0)3 + · · ·

Using slow roll approximations (take n = 6, φ0 ∼ 1014 GeV,mφ ∼ 1− 10 TeV)

ns ∼ 1− 4N0≃ 0.93

dnsdlnk ≃ − 4

N20≃ −0.001

r ∼ HmP∼ 10−16

MSSM Inflation

[Allahverdi, Mazumdar 2006]

Now consider a small deviation from the saddle pointparametrize by α:

A2

8(n−1)m2φ≡ 1 +

(

n−22

)2α2 .

For α2 6= 0, the saddle point becomes a point of inflectionwhere V ′′(φ0) = 0, and the expressions for ns is modified as

ns = 1− 4√∆2 cot[N0

√∆2],

where

∆2 ≡ n2(n− 1)2α2N20

(

MPφ0

)4.

For α2 > 0, we can obtain values of spectral index in the range

0.93 ≤ ns ≤ 1,

for

0 ≤ ∆2 ≤ π2

4N20.

MSSM Inflation

[Allahverdi, Dutta, Mazumdar 2007]

200

400

600

800

250 500 750 1000m1/2[GeV]

m0[

GeV

]

A0=0, µ>0tanβ=10

mh≤

114

GeV

m χ0>m τ

aµ<11×10-10

ns =1, δ

H =2.03x10 -5ns =1, δ

H =1.91x10 -5

ns =0.98, δ

H =1.91x10 -5ns =0.96, δ

H =1.91x10 -5

We show the dark matter allowed region narrow blue corridor, (g-2)µ region (light blue) for aµ ≤ 11 × 10−8 ,

Higgs mass ≤ 114 GeV (pink region) and LEPII bounds on SUSY masses (red). We also show the the dark matter

detection rate by vertical blue lines.

CMSSM and Inflation

CMSSM (Constrained MSSM) is a special case of MSSM inwhich we assume that SUSY is spontaneously broken in some‘hidden’ sector and this information is transmitted to our(visible) sector via gravity.

One employs universal soft SUSY breaking terms (say atMGUT ); these include scalar masses, gaugino masses, trilinearcouplings, etc.

In our discussion of SUSY hybrid inflation we will follow thisminimal SUGRA scenario for the soft terms; note, however,that their magnitudes can be ≫ TeV.

Supersymmetric Higgs (Hybrid) Inflation

[Dvali, Shafi, Schaefer; Copeland, Liddle, Lyth, Stewart, Wands ’94]

[Lazarides, Schaefer, Shafi ’97][Senoguz, Shafi ’04; Linde, Riotto ’97]

Attractive scenario in which inflation can be associated withsymmetry breaking G −→ H

Simplest inflation model is based on

W = κS (ΦΦ −M2)

S = gauge singlet superfield, (Φ ,Φ) belong to suitablerepresentation of G

Need Φ ,Φ pair in order to preserve susy while breakingG −→ H at scale M ≫ TeV, susy breaking scale.

R-symmetry

ΦΦ→ ΦΦ, S → eiα S, W → eiαW

⇒ W is a unique renormalizable superpotential

Susy Higgs (Hybrid) Inflation

Some examples of gauge groups:

G = U(1)B−L, (Supersymmetric superconductor)

G = SU(5)× U(1), (Φ = 10), (Flipped SU(5))

G = 3c × 2L × 2R × 1B−L, (Φ = (1, 1, 2,+1))

G = 4c × 2L × 2R, (Φ = (4, 1, 2)),

G = SO(10), (Φ = 16)


Tree Level Potential

VF = κ2 (M2 − |Φ2|)2 + 2κ2|S|2|Φ|2

Susy vacua

|〈Φ〉| = |〈Φ〉| =M, 〈S〉 = 0

0

2

4

ÈSÈM

-1

0

1

ÈFÈM

0.0

0.5

1.0

1.5

2.0

VΚ2M 4


Take into account radiative corrections (because during inflationV 6= 0 and susy is broken by FS = −κM2)

Mass splitting in Φ− Φ

m2± = κ2 S2 ± κ2M2, m2

F = κ2 S2

One-loop radiative corrections

∆V1loop = 164π2Str[M4(S)(ln M

2(S)Q2 − 3

2)]

In the inflationary valley (Φ = 0)

V ≃ κ2M4(

1 + κ2N8π2 F (x)

)

where x = |S|/M and

F (x) = 1

4

(

(

x4 + 1)

ln(x4

−1)x4 + 2x2 ln x2

+1

x2−1

+ 2 ln κ2M2x2

Q2 − 3

)


[Dvali, Shafi, Schaefer ’94]

Tree Level plus radiative corrections:

-5 -4 -3 -2 -10.92

0.94

0.96

0.98

1.00

1.02

log10 Κ

n s

0.980 0.985 0.990 0.995 1.000 1.005

-16

-14

-12

-10

-8

-6

-4

ns

log 1

0r

ns ≈ 1− 1N0≈ 0.98

δT/T ∝ (M/MP )2 ∼ 10−5 −→ attractive scenario (M ∼MG)

Full Story

Also include supergravity corrections + soft susy breaking terms

The minimal Kahler potential can be expanded as

K = |S|2 + |Φ|2 +∣

∣Φ∣

∣

2

The Sugra scalar potential is given by

VF = eK/m2p

(

K−1ij DziWDz∗jW ∗ − 3m−2p |W |2

)

where we have defined

DziW ≡ ∂W∂zi

+m−2p∂K∂ziW ; Kij ≡ ∂2K

∂zi∂z∗j

and zi ∈ Φ,Φ, S, ...

Full Story

[Senoguz, Shafi ’04; Jeannerot, Postma ’05]

Take into account sugra corrections, radiative corrections andsoft susy breaking terms:

V ≃κ2M4

(

1 +(

Mmp

)4x4

2 + κ2N8π2 F (x) + a

(

m3/2x

κM

)

+(

m3/2x

κM

)2)

where a = 2 |2−A| cos[argS + arg(2−A)], x = |S|/M andS ≪ mP .

Note: No ‘η problem’ with minimal (canonical) Kahler potential !

Results

[Rehman, Shafi, Wickman, 2009]

-6 -5 -4 -3 -2 -10.92

0.94

0.96

0.98

1.00

log10 Κ

n s

a = +1

a = -1

a = -1

Results

-6 -5 -4 -3 -2 -10.92

0.94

0.96

0.98

1.00

1.02

log10 Κ

n s

a = 0

a = -1

a = -1

Results

a = 0a = -1

0.92 0.94 0.96 0.98 1.00 1.02

-16

-14

-12

-10

-8

-6

-4

ns

log 1

0r

r . 10−4 within 2-σ bounds of WMAP data

Supersymmetric Flipped SU(5)

[Antoniadis, Ellis, Hagelin, Nanopoulos, 1987]

[For recent discussion and additional references see Nanopoulos, hep-ph/0211128; Li,

Nanopoulos and Walker, arXiv:0910.0860 [hep-ph]]

Flipped SU(5) ≡ SU(5)×U(1)XChiral superfields are arranged as

101 =

(

dc Qνc

)

, 5−3 =(

uc

L

)

, 15 = ec

Compared to standard SU(5), these multiplets correspond tothe interchange

uc ←→ dc, ec ←→ νc

Properties of Flipped vs. Standard SU(5)

Flipped SU(5) (Minimal) SU(5) (Minimal)

Low scale susy Yes YesDoublet-triplet splitting Yes Fine tuningDimension 5 proton decay Eliminated! Challengingµ problem No Fine tuningInflation Easy DifficultDimension 6 proton decay τp ∼ 10

34–1036 yrs τp ∼ 1035–1036 yrs

Monopole problem No YesSeesaw mechanism Automatic NoCharge quantization No YesUnification of gauge couplings Can be arranged YesCDM Yes Yes

Minimal Hybrid Inflation and Flipped SU(5)

[Rehman, Shafi, Wickman, 2009]

Consider susy hybrid inflation in flipped SU(5), where Φ is a10-plet

Allowing the soft mass squared to vary, the potential appearsasV ≃κ2M4

(

1 +(

Mmp

)4x4

2 + κ2N8π2 F (x) + a

(

m3/2x

κM

)

+(

MS xκM

)2)

If M2S < 0 the soft susy breaking mass squared term drives

the spectral index toward red-tilted values

The minimal model consistent with ns = 0.96 – 0.97 leads topredictions of the proton lifetime of order 1034–1036 years

Minimal Hybrid Inflation and Flipped SU(5)

a = 0

a = +1

a = -1

4 6 8 1014.0

14.5

15.0

15.5

16.0

16.5

log10HÈMSÈGeVL

log 1

0HMG

eVL

ns = 0.967,m32 = 1 TeV,N = 10,MS2 < 0

a = 0

a = +1

a = -1

4 6 8 10

-8

-7

-6

-5

-4

-3

-2

-1

log10HÈMSÈGeVL

log 1

0HΚL

ns = 0.967,m32 = 1 TeV,N = 10,MS2 < 0

log10(M/GeV) and log10(κ) vs. log10(|MS|/GeV) in the flippedSU(5) model (N = 10), with ns fixed at the central value 0.967.

Non-Minimal SUSY Hybrid Inflation and Tensor Modes

As we have seen, the minimal SUSY hybrid inflation modelyields r values . 10−4

A more general analysis with a non-minimal Kahler potentialcan lead to larger r-values

r

10-6

10-5

10-4

10-3

10-2

10-1

ns

0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1

(Planck)

Dark Matter and Flipped SU(5)

[Gogoladze, Khalid, Raza, Shafi, 2009]

Plots of the χ01-nucleon spin-independent cross-section versus mχ0

1

for the CMSSM,

FSU(5)-UH, NUHM2 and FSU(5)models for tan β = 10. Also shown are current limits

from CDMS II (solid black line), XENON10 (dashed black) and projected reach of

SuperCDMS (solid red) and XENON100 (dashed red).

Dark Matter and Flipped SU(5)


Plots of the χ01-nucleon spin-independent cross-section versus mχ0

1

for the CMSSM,

FSU(5)-UH, NUHM2 and FSU(5)models for tan β = 50. Also shown are current limits

from CDMS II (solid black line), XENON10 (dashed black) and projected reach of

SuperCDMS (solid red) and XENON100 (dashed red).

SU(4)c × SU(2)L × SU(2)R DM with Yukawa Unification


Plots in the σSI - mχ01

and σSD- mχ01

planes. In the σSI - mχ01

plane we show the current bounds (black

lines) and future reaches (red lines) of the CDMS (solid lines) and Xenon (dotted lines) experiments. In the σSD -

mχ01

plane we show the current bounds from Super K (black line) and IceCube (dotted red line) and future reach

of IceCuce DeepCore (red solid line).

Summary

The predictions of r (primordial gravity waves) for variousmodels of inflation are as follows:

Gauge Singlet Higgs Inflation:

r & 0.02 for ns ≥ 0.96

SM Higgs Inflation:

r ∼ 0.003 (ns ∼ 0.968)

Non-Minimal φ4 Inflation:

r & 0.002 for ns ≥ 0.96

Dark Matter Inflation:

0.003 . r . 0.007

MSSM Inflation:

r ∼ 10−16 with 0.93 . ns . 1

Susy Higgs (Hybrid) Inflation:

r . 10−4 (minimal), r . 0.05 (non-minimal)

Results from PLANCK are eagerly awaited!

Summary

In addition, we expect the LHC to provide answers to thefollowing long-standing and very basic questions:

Is there a ‘SM Higgs boson’ and what is its mass?

Is there low scale supersymmetry? LSP dark matter?

Documents

Inflation, Gravity Waves and Dark Matterbartol_udel_edu_01.pdfInﬂation, Gravity Waves and Dark Matter Qaisar Shaﬁ Bartol Research Institute Department of Physics and Astronomy