Inflationary Cosmology and Particle Physics · Inflationary Cosmology and Particle Physics ......

Inflationary Cosmology and Particle Physics

Qaisar Shafi

Bartol Research InstituteDepartment of Physics and Astronomy

University of Delaware

CambridgeAugust, 2009

1 / 67

Outline

Introduction

Standard Model Inflation

Non-Supersymmetric GUTs and Inflation

Supersymmetry, Supergravity and Inflation

Radiative Corrections and Precision Cosmology

Conclusions

2 / 67

Hot Big Bang Cosmology (SM+GR)

Standard Model (SM) + General Relativity (GR) gives rise tohighly successful ‘Hot Big Bang Cosmology’

Redshift of distant galaxies

Cosmic Microwave Background (CMB)

Primordial nucleosynthesis

3 / 67

Hot Big Bang Cosmology (SM+GR)

But Hot Big Bang Cosmology cannot explain:

WMAP Collaboration 2008

High degree of CMB isotropy

Origin of δTT

‘Flatness’

nB/s ∼ 10−10 (baryon asymmetry)

Cold Dark Matter (CDM)

· · ·

4 / 67

Hot Big Bang Cosmology (SM+GR)

In recent years ΛCDM has become the leading candidate for thedescription of the universe.

WMAP Collaboration 2008

Where does ΛCDM come from?

5 / 67

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?

6 / 67

Physics Beyond the SM

Solar and atmospheric neutrino oscillation experiments requirenew physics beyond the SM

Simplest Extensions1 Just add 2-3 SM singlet (‘right handed’) neutrinos and use

seesaw mechanism to understand why mν ≪ mcharged matter

2 Gauge U(1)B−L global symmetry of SM⇒ requires 3 RH-ν’s to cancel anomalies

Will discuss these two possible extensions of the SM in thecontext of inflation

Gauge hierarchy problem (MW ≪MP )

Electric charge quantization

Strong CP problem (axions)

7 / 67

Beyond the SM

Supersymmetry (Gauge hierarchy problem, LSP dark matter)

Grand Unification (Charge quantization, proton decay, gaugecoupling unification)

Technicolor (Complicated models)

Large Extra Dimensions (Black holes at LHC?)

Warped Extra Dimensions (TeV scale KK excitations)

Universal Extra Dimensions (KK dark matter candidate)

8 / 67

Slow-roll Inflation

The slow-roll parameters may be defined as

ǫ =m2p

2

(

V ′

V

)2, η = m2

p

(

V ′′

V

)

, ξ2 = m4P

(

V ′V ′′′

V 2

)

.

Assuming slow-roll approximation (i.e. (ǫ, |η|, ξ2) ≪ 1), thespectral index ns, the tensor to scalar ratio r and the runningof the spectral index dns/d ln k are given by

ns ≃ 1 − 6ǫ+ 2η, r ≃ 16ǫ, dnsd ln k ≃ 16ǫη − 24ǫ2 − 2ξ2.

The number of e-folds after the comoving scale l0 = 2π/k0

has crossed the horizon is given by

N0 = 1m2p

∫ φ0

φe

(

VV ′

)

dφ.

Inflation ends when ǫ(φe) ∼ 1 or η(φe) ∼ 1.

The amplitude of the curvature perturbation is given by

∆R = 12√

3πm3p

(

V 3/2

V ′

)

φ=φ0

= 4.91× 10−5 (WMAP5 normalization)

9 / 67

Standard Model Inflation?

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

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

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

φ4

(1+ ξφ2

m2P

)2; G(t) = exp[−4

∫ t0 dt

′γ(t′)/(1 + γ(t′))]

where HT = 1√2(0, v + φ), t = log[ φ

Mt] and γ(t) is the

anomalous dimension of the Higgs field.

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

10 / 67

Standard Model Inflation?

ΛHtL*GHtL

33.5 34.0 34.5 35.00.00140

0.00145

0.00150

0.00155

0.00160

0.00165

0.00170

0.00175

0.00180

t = logH�MtL

Mt = 175 GeV, Ξ = 2893, mh = 134.0 GeV

ΛHtL*GHtL

33.0 33.5 34.0 34.5 35.0

0.01846

0.01848

0.01850

0.01852

0.01854

t = logH�MtL

Mt = 172.4 GeV, Ξ = 6379, mh = 134.1 GeV

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

Mt

] and inflation end scale te = log[ φe

Mt

]

The spectral index is

ns ≃ 1 − 2Ne

+ 13

(

∂ψ0(λ G)

(λ G)

)

; ψ0 ≡√

ξ φ0

mP

with

∆2R ≃ λ G

ξ2N2e

72π2 ⇒ ξ ≃(

Ne6√

2 π ∆R

)√λG ∼ 104 (!)

11 / 67

Standard Model Inflation?

Mt =

169G

eV

Mt=

171G

eV

Mt=

172.4G

eV

Mt=

173G

eV

Mt=

175

GeV

122 124 126 128 130 132 134 136

0.960

0.965

0.970

0.975

0.980

0.985

0.990

mh�GeV

n s M t =169 GeV

M t =171 GeV

M t =172.4 GeV

M t =173 GeV

M t =175 GeV

122 124 126 128 130 132 134 1360.00

0.01

0.02

0.03

0.04

0.05

mh�GeVΛ@Φ

0D

Green curve is the tree level result (independent of Higgs mass).Blue and red include quantum corrections.

12 / 67

Challenges for SM inflation

[Burgess, Lee and Trott; Barbon and Espinosa, 09]

Consider gµν = ηµν +hµνmP

, so that the term ξ φ2 R 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 ≃ V1/40 = λ1/4mP

41/4√

ξ≫ Λ = mP

ξ .

Thus it is not clear how reliable are the calculations.

13 / 67

Non-Supersymmetric GUTs and Inflation

[Q. Shafi and A. Vilenkin, 1984]

Consider Coleman-Weinberg (CW) and Higgs Potentials forinflation in Non-Supersymmetric GUTs

V (φ) = A φ4[

ln(

φM

)

− 14

]

+ AM4

4(CW Potential)

V (φ) = V0

[

1 −(

φM

)2]2

(Higgs Potential)

Here φ is a gauge singlet field.

MΦ

V HΦLColeman-Weinberg Potential

MΦ

V HΦLHiggs Potential

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

14 / 67

Non-Supersymmetric GUTs and Inflation

[V. N. Senoguz, Q. Shafi, 2007]

6 8 10 12 14 16 18Log10@V0

1�4HGeVLD

0.92

0.94

0.96

0.98

1

ns

6 8 10 12 14 16 18Log10@V0

1�4HGeVLD

0.92

0.94

0.96

0.98

1

ns

Below vev inflation15 / 67

Non-Supersymmetric GUTs and Inflation

15.0 15.2 15.4 15.6 15.8 16.0 16.20.92

0.94

0.96

0.98

1.00

1.02

Log10@VHΦ0LD

n s

Below vev inflation

16 / 67

Predictions for CW and Higgs Potentials

[M. Rehman, Q. Shafi and J. Wickman, 2008]

0.92 0.94 0.96 0.98 1.00 1.02 1.040.0

0.1

0.2

0.3

0.4

0.5

ns

r

~Φ4

~Φ2

~1-Φ4

~1-Φ2

Higgs Potential HAVL

Higgs Potential HBVL

CW Potential HAVL

CW Potential HBVL

17 / 67

Proton Decay and WMAP5

15.4 15.6 15.8 16.0 16.2 16.40.92

0.93

0.94

0.95

0.96

log10@VHΦ0L1�4�GeVD

n s

~Φ4

~Φ2

~1-Φ4

~1-Φ2

Higgs Potential HAVL

Higgs Potential HBVLCW Potential HAVL

CW Potential HBVL

MX ∼ 2-4V1/40 ∼ 1016 GeV

(CWP)

⇒ τp ∼ 1034-1038 years(proton lifetime)

V1/4

0(GeV) V (φ0)

1/4(GeV) ns r

1.75 × 1016 1.61 × 1016 0.9613 0.05382.× 1016 1.74 × 1016 0.9630 0.0730

2.25 × 1016 1.83 × 1016 0.9637 0.08892.5 × 1016 1.89 × 1016 0.9638 0.1012.75 × 1016 1.93 × 1016 0.9636 0.1103.× 1016 1.96 × 1016 0.9635 0.1174.× 1016 2.03 × 1016 0.9628 0.133

4.× 1016 2.14 × 1016 0.9607 0.1653.× 1016 2.17 × 1016 0.9600 0.174

18 / 67

Magnetic Monopoles and Inflation

[Q. Shafi and A. Vilenkin, 1984; G. Lazarides, Q. Shafi, 1984]

Consider the breaking

G ≡ SO(10) → H → SM

where H ≡ SU(4) × SU(2) × SU(2).

First breaking produces superheavy monopoles carrying one

unit of Dirac charge; these will be inflated away.

π2(G/H) = Z2;

The second breaking at scale Mc produces monopoles which

carry two units of Dirac magnetic charge. These are

intermediate mass monopoles and they may survive inflation.

19 / 67

Magnetic Monopoles and Inflation

Consider the quartic coupling −cφ2χ†χ, with c ∼ (Mc/M)2.

Here χ vev breaks 4-2-2 to 3-2-1 and φ is the inflaton.

Monopole formation occurs when cφ2 ∼ H2

→ H(t− tχ) ≡ η ∼ 3c/λ.

Initial monopole number density ∼ H3, which gets diluted by

inflation down to H3 exp(−3η); thus,

rM = nM/T3R ∼ (H/TR)3 exp(−3η) . 10−30.

Roughly 25–30 e-folds can yield a flux close to or below the

Parker bound.

20 / 67

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)

21 / 67

Why Supersymmetry?

MSSMΑ1-1

Α2-1

Α3-1

2 4 6 8 10 12 14 16

10

20

30

40

50

60

Log10@L�GeVD

Αi-

1

22 / 67

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 = λ Φn

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)

23 / 67

MSSM Inflation

Hinf ∼ (mφφ0)mP

∼ mφ(mφ

mP)

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 − 4N ≃ 0.92

dnsdlnk ≃ − 4

N2 ≃ −0.002

r ∼ HmP

∼ 10−16

24 / 67

CMSSM and Inflation

CMSSM is a special case of MSSM in which we assume thatSUSY is spontaneously broken in some ‘hidden’ sector and thisinformation 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

25 / 67

Susy Hybrid Inflation

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

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

SUSY hybrid inflation is defined by the superpotential

W = κS (Φ Φ −M2)

inflaton (singlet) = S, waterfall field = (Φ ,Φ) ∈ GR-symmetry

Φ Φ → Φ Φ, S → eiα S, W → eiαW

⇒ W is unique renormalizable superpotential

Some examples of gauge groups:

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

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

G = 4c × 2L × 2R, (Φ ,Φ) = ((4, 1, 2) , (4, 1, 2))

G = SO(10), (Φ ,Φ) = (16 , 16)

26 / 67

Susy Hybrid Inflation

SUSY vacua

|〈Φ〉| = |〈Φ〉| = M, 〈S〉 = 0

Potential

VF = κ2 (M2 − |Φ2|)2 + 2κ2|S|2|Φ|2

0

2

4

ÈSÈ�M

-1

0

1

ÈFÈ�M

0.0

0.5

1.0

1.5

2.0

V�Κ2M 4

27 / 67

Susy Hybrid Inflation

Mass splitting in Φ − Φ

m2± = κ2 S2 ± κ2M2, m2

F = κ2 S2

One-loop radiative corrections

∆V1loop = 164π2 Str[M4(S)(ln M2(S)

Q2 − 32)]

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

)

28 / 67

Susy Hybrid Inflation

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 −(mp

M

)2(

κ2N8π2

)

|F ′′(x0)| x0≫1−−−→ 1 − 1N0

≃ 0.98

r ≃ 4 (1 − ns)F ′(x0)|F ′′(x0)|

29 / 67

Minimal Sugra Hybrid Inflation

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∗j

W ∗ − 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, ...}

30 / 67

Minimal Sugra Hybrid Inflation

Take into account SUGRA corrections, radiative correctionsand soft SUSY breaking terms, the potential is given by

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)] and S ≪ mP .

Note: No ‘η problem’ with minimal Kahler potential

[V. N. Senoguz, Q. Shafi ’04; A. D. Linde, A. Riotto ’97]

31 / 67

Minimal Sugra Hybrid Inflation

1.00 1.02 1.04 1.06 1.08 1.10

1

1

1

1

1

1

xl

V�Κ

2 M4

a = +1, x0 = 1.040

1.00 1.02 1.04 1.06 1.08 1.101

1

1

1

1

xl

a = 0, x0 = 1.037

1.00 1.02 1.04 1.06 1.08 1.101

1

1

1

1

1

xl

a = -1, x0 = 1.023

Hybrid vs. Hilltop Solutions (κ = 4 × 10−4)

[L. Boubekeur and D. H. Lyth, 2005]

32 / 67

Minimal Sugra Hybrid Inflation

[M. Rehman, Q. Shafi, J. Wickman, in preparation.]

-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

ns ≃ 1 + 6(

MmP

)2x2

0 +(

mPM

)2[

−κ2N8π2 |F ′′(x0)| + 2

(

m3/2

κ M

)2]

33 / 67

Minimal Sugra Hybrid Inflation

-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

ns ≃ 1 + 6(

MmP

)2x2

0 +(

mPM

)2[

−κ2N8π2 |F ′′(x0)| + 2

(

m3/2

κ M

)2]

34 / 67

Minimal Sugra Hybrid Inflation

a = +1

a = -1

a = -1

-6 -5 -4 -3 -2 -1

15.0

15.5

16.0

16.5

17.0

log10 Κ

log 1

0HM�G

eVL

a = 0a = -1

a = -1

-6 -5 -4 -3 -2 -1

15.0

15.5

16.0

16.5

17.0

log10 Κ

log 1

0HM�G

eVL

35 / 67

Minimal Sugra Hybrid Inflation

a = +1

a = -1

0.92 0.94 0.96 0.98 1.00

15.0

15.5

16.0

16.5

17.0

ns

log@M�G

eVD

a = 0

a = -1

a = -1

0.92 0.94 0.96 0.98 1.00 1.02

15.0

15.5

16.0

16.5

17.0

ns

log@M�G

eVD

36 / 67

Minimal Sugra Hybrid Inflation

a = +1

a = -1

0.92 0.94 0.96 0.98 1.00-14

-12

-10

-8

-6

-4

ns

log 1

0r

r ≃ 4(

mPM

)2[

2(

MmP

)4x3

0 + κ2N8π2 F

′(x0) +am3/2

κM + 2(

m3/2

κM

)2x0

]2

37 / 67

Minimal Sugra Hybrid Inflation

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 ≃ 4(

mPM

)2[

2(

MmP

)4x3

0 + κ2N8π2 F

′(x0) +am3/2

κM + 2(

m3/2

κM

)2x0

]2

38 / 67

Minimal Sugra Hybrid Inflation

a = +1

a = -1

0.92 0.94 0.96 0.98 1.00-8

-7

-6

-5

-4

-3

ns

log 1

0Èd

n s�d

lnkÈ

dnsd ln k ∝ −√

ǫ

39 / 67

Minimal Sugra Hybrid Inflation

a = 0

a = -1

0.92 0.94 0.96 0.98 1.00 1.02

-9

-8

-7

-6

-5

-4

-3

-2

ns

log 1

0Èd

n s�d

lnkÈ

dnsd ln k ∝ −√

ǫ

40 / 67

WMAP and Cosmic Strings

[Battye, Bjorn, Adam]

10-110-210-310-410-5

0.94

0.96

0.98

1.00

1.02

Κ

n s

ns vs. κ for N = 1 susy hybrid inflation with string contribution included. The

contours represent the 68% and 95% confidence levels for a 7 parameter fit to the

WMAP5+ACBAR data [arXiv:astro-ph/0607339].

δTT =

√

(

δTT

)2

inf+

(

δTT

)2

cs,

(

δTT

)

cs∝ Gµ = 2π

(

MMP

)22.4

ln(2/κ2)

41 / 67

Reheat Temperature and Non-Thermal Leptogenesis

a = 0

a = -1

a = -1

-6 -5 -4 -3 -2 -1

7

8

9

10

log10 Κ

log 1

0HT

r�G

eVL

a = 0

a = -1

a = -1

9 10 11 12 13 14 15

15.0

15.5

16.0

16.5

17.0

log10 Hminf �GeVL

log 1

0HM�G

eVL

Tr & 1.6 × 107 GeV(

1016 GeVM

)1/2(

minf1011 GeV

)3/4(

0.05 eVmν3

)1/2

42 / 67

Reheat Temperature and Non-Thermal Leptogenesis

a = +1

a = -1

a = -1

-6 -5 -4 -3 -2 -1

7

8

9

10

log10 Κ

log 1

0HT

r�G

eVL

a = +1

a = -1

a = -1

11 12 13 14 15

15.0

15.5

16.0

16.5

17.0

log10 Hminf �GeVL

log 1

0HM�G

eVL

Tr & 1.6 × 107 GeV(

1016 GeVM

)1/2(

minf1011 GeV

)3/4(

0.05 eVmν3

)1/2

43 / 67

Non-Minimal Sugra Hybrid Inflation

[M. Bastero-Gil, S. F. King, Q. Shafi; M. Rehman, V. N. Senoguz, Q. Shafi 06]

The Kahler potential including non-minimal terms can beexpanded as

K = |S|2 + |Φ|2 +∣

∣Φ∣

∣

2

+κS|S|44m2

p+ κSφ

|S|2|Φ|2m2p

+ κSφ

|S|2|Φ|2m2p

+ κSS|S|66m4

p+ · · · .

The potential is of the following form

V ≃ κ2M4

(

1 − κS

(

Mmp

)2x2 + γS

(

Mmp

)4x4

2 + κ2N8π2 F (x)

+a(

m3/2x

κM

)

+(

m3/2x

κM

)2)

,

where γS = 1 − 7κS2 + 2κ2

S .

44 / 67

Non-Minimal Sugra Hybrid Inflation

10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1

0.90

0.92

0.94

0.96

0.98

1.00

n s

s = 0 s = 0.002 s = 0.005 s = 0.01 s = 0.02

nS ≃ 1 − 2κS + 6γS

(

Mmp

)2x2

0 −(mp

M

)2(

κ2N8π2

)

|F ′′(x0)|

45 / 67

MSSM µ Problem

In MSSM, the term µHuHd is both supersymmetric and alsogauge invariant, and we require that |µ| ∼ 102–103 GeV inorder to implement radiative EW breaking

µ, in principle, could be of order MGUT /MP ?

In SUSY hybrid inflation models, the R-symmetry forbids the‘bare’ µ term. However, λHuHdS is allowed, λ adimensionless coefficient.

After SUSY breaking, 〈S〉 ∼ −m3/2/κ, so thatµ ∼ −m3/2λ/κ.

46 / 67

Shifted Hybrid Inflation

[R. Jeannerot, S. Khalil, G. Lazarides, Q. Shafi ’2000]

Useful for inflating away troublesome objects.

Shifted hybrid inflation is defined by the superpotential

W = κS (Φ Φ −M2) − S (Φ Φ)2

M2S

Potential

VF = κ2 (M2 − |Φ|2 + ξ |Φ|4)2 + 2κ2|S|2|Φ|2 (M2 − 2 ξ |Φ|2)2

where ξ = M2/κM2S

Inflationary trajectories

|Φ| = 0 and |Φ| =√

1/2 ξ M with |S| > M

47 / 67

Shifted Hybrid Inflation

Monopoles produced in the shifted directions get inflated away

0

1

2

3

4

ÈSÈ�M

-202

ÈFÈ�M

0

1

2

V�Κ2M 4

48 / 67

Shifted Hybrid Inflation

Including 1-loop radiative corrections in the shifted direction

(|Φ| =√

1/2 ξ M)

V ≃ κ2M4ξ

(

1 + κ2N8π2 F (xξ)

)

where xξ = |S|/Mξ and Mξ = M√

1/4 ξ − 1

Same results/predictions are obtained as that of regular

SUSY/SUGRA inflation

Mξ ⇔ M

49 / 67

Chaotic Inflation (Linde)

Λ Φ4

1

2m2Φ2

0.92 0.94 0.96 0.98 1.00 1.020.0

0.1

0.2

0.3

0.4

ns

r

Komatsu et al, WMAP Collaboration 2008

Driven by potentials m2φ2 or λφ4

Where does φ come from?

One promising candidate isright handed sneutrino:

WN ⊃ mN2

⇒ V ⊃ m2|N |2 ; m ∼ 1013 GeV

Challenge: Does this survive SUGRA corrections? (since N > MP

during inflation)Answer: With N > MP during inflation, SUGRA correctionsoverwhelm this scenario:W = mΦ2 ⇒ large vev of W during inflation ⇒ potentialnegative/no inflation

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Chaotic Inflation

Shift symmetry imposed on the inflaton (removes the nastyexpotential factor in the inflaton potential)K(Φ,Φ∗) invariant for Φ → Φ + i cM , so K is a function ofΦ + Φ∗, i.e., K(Φ + Φ∗)However, with W = mΦ2, chaotic inflation will not take placedue to the large vev of W .

To cure this introduce a field X with W = mX Φ. Duringinflation, X is assumed to be stablized at the origin, so thatW vanishes. (Kawasaki, Yamaguchi, Yanagida)

Why is W = mX Φ the only relevant term?

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Chaotic Inflation

Gravitino Problem

If Φ transforms under a shift symmetry, the following term isallowed:

K = c (Φ + Φ∗), with c ∼ O(1) in Planck units

The inflaton Φ decays into susy breaking sector as well as theMSSM sector, leading to thermal and non-thermal gravitinooverproduction.

One possible solution: Introduce a discrete symmetry whichsuppresses/eliminates the linear term.

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Chaotic Inflation

If Φ is RH sneutrino, it is charged under the shift symmetry. Itdoes not have a Majorana mass, so it is not clear howleptogenesis/ν oscillations work out.

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Radiative Corrections and Precision Cosmology

Chaotic: potential is quadratic (V ∼ m2φ2) or quartic(V ∼ λφ4)

Simplest (renormalizable) potential

Predicts significant primordial gravity waves

Hybrid: inflation driven by φ field, vacuum energy providedby second scalar field χ

V (φ, χ) = κ2(

M2 − χ2

4

)2+ m2φ2

2 + λ2χ2φ2

4

Compelling and robust

Translates well to SUSY

Non-SUSY model results in ns > 1 (unless in chaotic-likeregime) — SUSY case has ns < 1

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Tree Level Inflation Models vs. WMAP5

Λ Φ4

1

2m2Φ2

0.92 0.94 0.96 0.98 1.00 1.020.0

0.1

0.2

0.3

0.4

ns

r

Φ�

0 < 0.67

Φ�

0 = 1

Φ�

0 >> 1

Not Allowed Flat Potential

0.92 0.94 0.96 0.98 1.00 1.02

0.1

0.2

0.3

0.4

nsr

Komatsu et. al. (WMAP Collaboration), 2008

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Radiative Corrections in Chaotic Inflation

Quadratic: V (φ) = 12m

2φ2 ±Aφ4 ln φ2

µ2

Quartic: V (φ) = λφ4 ±Aφ4 ln φ2

µ2

Fermion-dominated corrections (−) (say from right handedneutrinos)

Φ

V HΦL

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Radiative Corrections in Chaotic Inflation — Results

[V. N. Senoguz and Q. Shafi, 2008]

Λ Φ4

1

2m2Φ2

0.92 0.94 0.96 0.98 1.00 1.020.0

0.1

0.2

0.3

0.4

ns

r

57 / 67

Radiative Corrections in Hybrid Inflation

[M. Rehman, Q. Shafi and J. Wickman, 2009]

V (φ, χ) = κ2

(

M2 − χ2

4

)2

+m2φ2

2+λ2χ2φ2

4±Aφ4 ln

(

φ

φc

)

,

where φ ≡ ‘inflaton’ field, χ ≡ ‘waterfall’ field (χ = 0 duringinflation)

0.92 0.94 0.96 0.98 1.00-5

-4

-3

-2

-1

ns

log 1

0r

ΦP = 0.25

ΦP = 1

ΦP = 2.5

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Radiative Corrections & Precision Cosmology

Radiatively-corrected inflation models;

Chaotic Inflation

Permits low values of r (e.g. 0 . r . 0.2 to agree withWMAP5 2σ bound)

Hybrid Inflation

Red spectrum (ns < 1) can be produced for Planckian orsub-Planckian values of the inflaton

Better agreement with WMAP5

Generically 2 solutions of ns for each A

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Brane Inflation

In the simplest brane inflation models the inflaton φ is a fieldwhich parametrizes the distance r between two brane worldsembedded in the extra dimensions.

If the distance between these two branes is much bigger thanthe string scale, the potential between them is essentiallygoverned by the infrared bulk supergravity. Assuming theeffects of higher string excitations are decoupled, the potentialis expected to be

V ∼M4s

(

a− b(Msr)N−2

)

,

where Ms is the string scale, and a, b are constants.

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Brane Inflation

Thus, in this picture inflation in four dimensions is driven bybrane motion in the extra-dimensional space.

Positive powers of φ are absent because the inter-braneinteraction falls off with the distance.

For a D3 brane - anti D3 brane system, assuming the extradimensions have been suitably compactified (say T 6) andremain frozen during inflation,

V (φ) ∼M4s

(

1 − αφ4

)

, φ =√T3 r, α = T3

16 π2 .

(contributions from dilaton, graviton and RR field)

In this caseδTT ∝Mc/MP , Mc ∼ 1012 GeV (intermediate compactification scale),

ns ≃ 1 − 2N0

≈ 0.96.

NOTE: No very compelling/realistic model so far!!

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D-brane Inflation

Brane-Antibrane Dvali & Tye; Dvali,Shafi,Solganik;Burgess,Majumdar,Nolte,Quevedo,Rajesh,Zhang; Alexander;.

Branes at Angles Garcia-Bellido, Rabadan, Zamora;Blumenhagen, Kors, Lust, Ott; Dutta, Kumar, Leblond.

D3-D7 Dasgupta,Herdeiro,Hirano, Kallosh; Hsu,Kallosh,Prokushkin; Hsu & Kallosh; Aspinwall & Kallosh; Haack,Kallosh, Krause, Linde, Lust, Zagermann.

warped brane-antibraneKachru,Kallosh,Linde,Maldacena,L.M.,Trivedi; Firouzjahi &Tye; Burgess,Cline,Stoica,Quevedo; Iizuka & Trivedi,...

DBI Silverstein & Tong; Alishahiha,Silverstein,Tong; Chen;Chen; Shiu & Underwood; Leblond & Shandera,...

Monodromy Silverstein & Westphal.

Adapted from McAllister, 2008

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D-brane Inflation

Closed string inflation models, e.g.:

Racetrack Blanco-Pillado, Burgess, Cline, Escoda,Gomez-Reino, Kallosh, Linde, Quevedo.

Kahler moduli Conlon & Quevedo.

N-flation Dimopoulos, Kachru, McGreevy, Wacker; Easther &L.M.

Adapted from McAllister, 2008

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Chaotic inflation in string theory

[Silverstein, Westphal, 2008]

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Brane Inflation and Cosmic Strings

Prior to brane annihilation the associated gauge symmetry isU(1) × U(1). One linear combination gives rise to D-strings,while the orthogonal combination is associated with F(fundamental)-strings.

It has been argued that a substantial fraction of energy of theannihilating branes is used up in the production of a networkof cosmic D and F strings.

If one assumes that brane inflation gives rise to the observedinhomogeneities, estimates suggest that

10−11 . Gµ . 10−6.

With some(?) luck it may be possible to observe theseprimordial strings with LIGO/LISA.

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Conclusions

One of the most important challenges in our field is to find a“Standard Model of Inflationary Cosmology”.

A large class of realistic inflation models, based on grandunification and/or low scale supersymmetry are consistentwith the data currently available from WMAP and otherobservations.

Non-supersymmetric GUT inflation models typically predict an‘observable’ value for the tensor to scalar ratio r.

Supersymmetric hybrid inflation, on the other hand, predicts a‘tiny’ r value (. 10−4), even though the symmetry breakingscale associated with inflation is comparable to MGUT .

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Conclusions

Hopefully, the PLANCK Satellite will soon shed light on thisfundamental parameter r.

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?

Is the LSP stable? Mass?

To achieve truly significant progress in particle physics andcosmology we need both the LHC and the PLANCK Satellite.They hold the key to future developments in these fields.

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