Grand Unification &the Early Universe

Wilfried BuchmüllerDESY, Hamburg

Bielefeld, May 2015

Forces and matter of the Standard Model naturally fit into the ‘unification’(GUT) group SU(5) [Georgi, Glashow ’74], or larger extensions,

SU(5) =

✓SU(3) ⇤

⇤ SU(2)

◆,

with quarks and leptons represented by 10, 5⇤ and (1 = ⌫c),

0

BBBB@

0 uc uc u d0 uc u d

0 u d0 ec

0

1

CCCCA�

0

BBBB@

dc

dc

dc

⌫e

1

CCCCA� (⌫c) .

Hope: extrapolation of SM possible up to the GUT energy scale ⇤GUT ⇠1015 GeV where weak, electromagnetic and strong forces have equal strength.

Reasons for Grand Unified Theories

Effective interactions strengths (couplings ) depend on energy and momentum transfer; unification at GUT scale in SM and supersymmetric extension; hint for supersymmetry? Search at LHC, DM, ... !!

↵i

Grand Unification & Cosmology

• `First’ model of inflation [Guth, ’80]: inflation driven by false GUT vacuum energy (problem with subsequent phase transition ...)

• GUT baryogenesis: heavy leptoquark decay (gauge couplings too large, not enough departure from thermal equilibrium ... [Kolb, Wolfram ’80])

• Potential problem: formation of topological defects in GUT phase transition, monopoles domain walls, strings, ...

• Theoretical developments: new ways of GUT symmetry breaking, Higgs- type phase transitions only for Abelian subgroups (only cosmic strings; SUSY GUTs and extra dimensions: new inflaton & dark matter candidates; `sphalerons’, allows for leptogenesis, natural for GUT scale heavy neutrinos

• Attractive framework: hybrid inflation with with B-L breaking (difference of baryon and lepton number)[Linde; Dvali, et al; Binetruy, Dvali; Halyo; all ’94]; potential observables: strings, gravity waves, CMB (large field inflation), ...

Lepton number can be violated by masses and couplings of heavy Majorana neutrinos to light neutrinos and charged leptons; after electroweak symmetry breaking, 3 light and 3 heavy neutrinos (seesaw mechanism):

For hierarchical right-handed neutrinos, after electroweak symmetry breaking, light neutrino masses naturally related to the GUT scale:

suggests that neutrino physics probes the mass scale of grand unification(but there are also other examples ... )

N ' ⌫R + ⌫cR , ⌫ ' V T⌫ ⌫L + ⌫cLV

⇤⌫

mN ' M , m⌫ ' �V T⌫ mT

D1

MmDV⌫

Seesaw & Leptogenesis

M3 ⇠ ⇤GUT ⇠ 1015 GeV , m3 ⇠ v2

M3⇠ 0.01 eV

CP violating heavy neutrino decays:

Leptogenesis relates successfully neutrino masses and matter-antimatter asymmetry; CP asymmetry is small: suppressed by small Yukawa couplings, and loop effect (quantum interference); natural explanation of very smallobserved baryon asymmetry

"1 =� (N1 ! H + lL)� �

⇣N1 ! H† + l†L

⌘

� (N1 ! H + lL) + �⇣N1 ! H† + l†L

⌘

' � 3

16⇡

M1

(hh†)11v2FIm

�h⇤m⌫h

†�11

Rough estimate for CP asymmetry (hierarchical heavy neutrinos) and baryon asymmetry:

"1 ⇠ 0.1M1m3

v2F

⇠ 0.1M1

M3⇠ 10�5 . . . 10�6

⌘B ' �csfNB�L ' 10�2"1f

Neutrino masses suggest that leptogenesis is process close to thermal equilibrium, i.e. ; in terms of neutrino masses:

confirmed by solution of Boltzmann equations; good quantitative understanding; significant progress in QFT description

�1 ⇠ H|T=M1

em =(mDm†

D)11M1

⇠ m⇤ =16⇡5/2

3p5

g1/2⇤v2FMP

' 10�3 eV

Sphaleron

cL

τ

L

e

µ

LsLs t

b

b

ν

ν

ν

u

d

d

L

L

LL

L

Leptogenesis & Supersymmetry

Leptogenesis & gravitinos: for (thermal) leptogenesis and typical superparticle masses, thermal production yields observed amount of dark matter:

is natural value; but why is reheating temperature close to minimal LG temperature?

Simple observation: heavy neutrino decay width (for typical LG parameters)

yields reheating temperature (for gas of decaying heavy neutrinos)

wanted for gravitino DM. Intriguing hint or misleading coincidence?

⌦3/2h2 = C

✓TR

1010 GeV

◆✓100GeV

m3/2

◆⇣ mg

1TeV

⌘2, C ⇠ 0.5

⌦3/2h2 ⇠ 0.1

[WB, Domcke, Kamada, Schmitz ’13, ’14]

�N1 =m1

8⇡

✓M1

vF

◆2

⇠ 103 GeV , em1 ⇠ 0.01 eV , M1 ⇠ 1010 GeV

TR ⇠ 0.2 ·q�0N1

MP ⇠ 1010 GeV

Spontaneous B-L breaking & false vacuum decay

Supersymmetric SM with right-handed neutrinos:

in SU(5) notation: ; B-L breaking:

yields heavy neutrino masses.

Lagrangian is determined by low energy physics: quark, lepton, neutrino masses etc, but it contains all ingredients wanted in cosmology: inflation, leptogenesis, dark matter, ..., all related! [F-term hybrid inflation: Copeland et al. ’94, Dvali et al. ’94].

Technically: Abelian Higgs model in unitary gauge; inflation ends with phase transition (“tachyonic preheating”, “spinodal decomposition”)

WM = huij10i10jHu + hd

ij5⇤i 10jHd + h⌫

ij5⇤in

cjHu + hn

i ncin

ciS1

10 � (q, uc, ec), 5⇤ � (dc, l), nc � (⌫c)

WB�L = ��

✓1

2v2B�L � S1S2

◆

hS1,2i = vB�L/p2

0

1

2

3

1

0

1

0

0.5

Φ MP

H MP

V MP4

S / vB-L

/

/

time-dependent masses of B-L Higgs, inflaton, heavy neutrinos ... (bosons and fermions):

m2� =

1

2�(3v2(t)� v2B�L) , m2

� = �v2(t) , M2i = (hn

i )2v2(t) . . .

0.001

0.01

0.1

1

5 10 15 20 25 30

<φ2 (

t)>1/

2 /v,

n B(t)

time: mt

φ(t) (Tanh)<φ2(t)>1/2/v (Lattice)

nB(x100) (Tanh)nB(x100) (Lattice)

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

0.1 1

Occ

upat

ion

num

ber:

n k

k/m

Bosons: Lattice Tanh

Fermions: Lattice Tanh

Rapid transition from false to true vacuum by fluctuations of ‘waterfall’ B-L Higgs field; production of low momentum Higgs bosons (contain most energy), also other bosons and fermions coupled to B-L Higgs field [Garcia-Bellido, Morales ’02], production of cosmic strings: initial conditions for reheating

Gauge

Higgs +Inflaton

Right-handedNeutrinos

Radiation +B-L asymmetry

Gravitinos

tachyonic preheatingfast process slow process

Schematic view

aRHi aRH aRH

f

SN1nt

N1th

R

B- L

Gé

100 101 102 103 104 105 106 107 1081025

1030

1035

1040

1045

105010-1 100 101 102 103

Scale factor a

ComovingnumberdensityabsNHaL

Inverse temperature M1 êT

Transition from end of inflation to hot early universe (typical parameters), calculated by means of Boltzmann equations:

E

✓@

@t�Hp

@

@p

◆fX(t, p) =

X

i0j0..

X

ij..

CX(Xi0j0.. $ ij..)

yields correct baryon asymmetry and dark matter abundance

Abelian Higgs string: Gµ/c2 < 3.2⇥ 10�7, consistent with GUT scale strings

Relic gravitational waves are window to very early universe; contributions from inflation, preheating & cosmic strings (see Maggiore ’07, Dufaux et al ’10, Hindmarsh ’11); cosmological B-L breaking: prediction of GW spectrum with all contributions! Perturbations in flat FRW background:

determined by linearized Einstein equations,

yields spectrum of GW background:

ds

2 = a

2(⌧)(⌘µ⌫ + hµ⌫)dxµdx

⌫, hµ⌫ = hµ⌫ � 1

2⌘µ⌫h

⇢⇢

h00µ⌫(x, ⌧) + 2

a0

ah0µ⌫(x, ⌧)�r2

x

hµ⌫(x, ⌧) = 16⇡GTµ⌫(x, ⌧)

⌦GW (k, ⌧) =1

⇢c

@⇢GW (k, ⌧)

@ ln k,

Z 1

�1d ln k

@⇢GW (k, ⌧)

@ ln k=

1

32⇡G

Dhij (x, ⌧) h

ij (x, ⌧)E

Gravitational Waves

Result: GWs from inflation, preheating and cosmic strings (Abelian Higgs), for typical parameters consistent with leptogenesis and dark matter; similar spectrum from inflation and cosmic strings, but different normalization!

inflation

AH cosmic strings

f0 feq fRH fPH

fPHHsL

fPHHvL

preheating

10- 20 10-15 10-10 10-5 100 105 1010

10- 25

10- 20

10-15

10-10

10-5 100 105 1010 1015 10 20 10 25

f @HzD

WGWh2k @Mpc-1 D

Observational prospects (for typical parameters); ‘soon’: Advanced Ligo (6) [100 Hz], eLISA [0.01 Hz]; ‘later’: Einstein Telescope [100 Hz], BBO/DECIGO [0.01 Hz]; eventually determination of reheating temperature (leptogenesis)?!

H7LH8L

H3L

H5LH6LH1LH2L

H4L

inflation

AH cosmic strings

NG cosmic strings

preheating

10- 20 10-15 10-10 10-5 100 105 1010

10- 25

10- 20

10-15

10-10

10-5

10010-5 100 105 1010 1015 10 20 10 25

f @HzD

WGWh2

k @Mpc-1 D

Inflationary potential can be strongly affected by supersymmetry breaking (supergravity correction [WB, Covi, Delepine ’00]):

linear term turns hybrid inflation into two-field model in complex plane; strong effect on inflatonary observables, now dependent on trajectory, i.e. initial conditions! Inflation consistent with Planck data now possible (otherswise very difficult)

V (�) = V0 + VCW(�) + VSUGRA(�) + V3/2(�) ,

V0 =�2v4

4,

VCW(�) =�4v4

32⇡2ln

✓|�|

v/p2

◆+ . . . ,

VSUGRA(�) =�2v4

8M4P

|�|4 + . . . ,

V3/2(�) = ��v2m3/2(�+ �⇤) + . . .

Hybrid Inflation in the Complex Plane

F-term hybrid inflation ( ) with SUSY breaking: resulting linear term in inflaton potential leads to two-field dynamics of complex inflaton in field space, observables depend on inflationary trajectory, range for spectral index consistent with

N = 0N = 50

VHs, tL

m3ê2 = 50 TeV

Hv, 0LH-v, 0L

Hs*, t*L

Hs f , t f L

Æ

ÆÆ

ÆV

,j<

0

V,j>

0

-3 -2 -1 0 10.0

0.5

1.0

1.5

2.0

s @vD

t@vD

ns ' 0.96

� ⌧ 1

Parameter scan: relations between scale of B-L breaking, gravitino mass and scalar spectral index (correct in green band) cosmic string bound automatically fulfilled! Prediction for tensor-scalar ratio: ; further predictions: characteristic spectrum of gravitational waves, connection between inflation andSUSY searches at the LHC! BUT WHAT ABOUT BICEP2?

Cosm

icstring

bound

m3ê2 < 10 MeV As > Asobs

q f= 0

q f≥ pê 32

Fine-tuned phaseq f close to 0

m3ê2

10 MeV

100 MeV

1 GeV

10 GeV

100 GeV

1 TeV

10 TeV

100 TeV 1 PeV 10 PeV

1014 1015 101610-5

10-4

10-3

10-2

v @GeVD

l

r . 2⇥ 10�6

0 50 100 150 200 250 300−0.01

0

0.01

0.02

0.03

0.04

0.05

Multipole

BB l(

l+1)

Cl/2π

[µK2 ]

BKxBK(BKxBK−αBKxP)/(1−α)

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

r

L/L pe

ak

Fiducial analysisCleaning analysis

Joint BICEP2/Keck/Planck analysis (2015)

upper: BICEP2/Keck spectrum before and after subtraction of dust contribution

lower: likelihood fit; primordial BB signal almost 2σ effect; favoured value of r about 0.5

More data this year, BB signal not yet excluded

Version of supersymmetric D-term inflation; superpotential and Kahler potential for waterfall fields and inflaton (shift symmetry, [Yanagida et al ’00]):

Waterfall transition below critical point

can be super-Planckian for small Yukawa couplings (regimeused in following discussion, needed for Lyth bound); theoretical question: FI-term [recent work: Wieck, Winkler ’14; Domcke, Schmitz, Yanagida ’14]

'c =g

�

p2⇠ ,

�/g ⌧ 1

K =1

2(�+ �†)2 + |S+|2 + |S�|2 ,

W = � �S+S� , VD =g2

2

�|S+|2 � |S�|2 � ⇠

�2

⇠

[WB, Domcke, Schmitz ’14; WB, Ishiwata ’14]

Subcritical Hybrid Inflation

Inflaton potential above critical point (waterfall fields integrated out):

with . Below critical point scalar potential for inflaton and waterfall fields ( ):

Note, potential quadratic in inflaton field (consequence of shift symmetry!) Below critical point waterfall transition (spinodal decom-position; tachyonic preheating [Felder et al ’00; ...]), rapid transition to global minimum. Following discussion: tachyonic growth of fluctuations for continuing exponential expansion!

Inflation can continue after critical point [Clesse ’10; Clesse, Garbrecht ’12], mostly models with small field inflation]; here: large field inflation after critical point, with final phase of “chaotic inflation” after tachyonic growth of fluctuations.

V (') = V0

✓1 +

g

2

8⇡2(lnx+ . . .)

◆,

x = �

2'

2/(2g2⇠)

V (', s) =g2

8(s2 � 2⇠)2 +

�2

4s2'2 +O(s4'2) .

MP = 1

jcj*jf

0 5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

0 10 50 400

Inflaton field j @MplD

V infHjLêM G

UT

4

Number of e-folds

Below critical point inflaton potential changes from plateau to quadratic potential of “chaotic inflation”; relevant fluctuations are generated at

'c

'⇤

Computation of quantum fluctuations beyond critical point requires mode expansion of field operators in de Sitter background:

only inflaton field has classical part; close to the critical point ( ):

with given by velocity of inflaton field at critical point

Tachyonic Growth of Fluctuations

'(t, ~x) = '(t) + e

� 32Ht

Zd

3k

(2⇡)3/2

⇣a'(~k)'k(t)e

i~k~x + a

†'(~k)'

⇤k(t)e

�i~k~x⌘,

s(t, ~x) = e

� 32Ht

Zd

3k

(2⇡)3/2

⇣as(~k)sk(t)e

i~k~x + a

†s(~k)s

⇤k(t)e

�i~k~x⌘

t('c) = 0

'(t) ' 'c + 'ct ,

V (s; t) ' 1

2g2⇠2 � 1

2D3t s2 +O(s4; t2) ,

D3 =p2⇠g�|'c|

Analytic estimate of variance due to growth of quantum fluctuations[Asaka, WB, Covi ’01]:

for large times, , one has , i.e. fluctuations are generatedwithin horizon.

For typical parameters ( ), velocity of inflaton field and Hubble parameter

'c = �@'V

3H

����'c

= �g2� ln 2

4p3⇡2

p⇠ .

g2 = 1/2, � = 5⇥ 10�4,p⇠ = 2.8⇥ 1016 GeV

Hc ⌘ H('c) = 9.1⇥ 1013 GeV , 'c ' �21H2c , D ' 1.5Hc .

hs2(t)i = hs2(t)ius � hs2(0)ius

'Z kb(t)

0dk

k2

2⇡2e�3Hct |sk(t)|2

'3

2/3�

2�13

�

192⇡3D2

(Dt)3/2 exp

✓4

3

(Dt)3/2 +Ht

◆;

Dt > 1 pb(t) > Hc

Beyond decoherence time ( ),

soft modes ( ) generate classical waterfall field. Backreaction due to self-interaction of waterfall field can then be taken into account by means of classical field equations:

Classical waterfall field, with matching

quickly reaches local minimum; approach of global minimum much later,in final stage generation of effective mass term for inflaton.

|sk(tdec)⇡sk(tdec)| ⌘ Rdec � ~ ⌘ 1

tdec ⇠1

D

�34 ln(2Rdec)

�2/3,

'+ 3H'+1

2�2s2' = 0 ,

s+ 3Hs�✓g2⇠ � �2

2'2

◆s+

g2

2s3 = 0 .

s(tdec) = hs2(tdec)i1/2 ,

k < kb(t)

tdec sHtL

smin

without backreaction

with backreaction

matching

1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

2.5

Time t @1 ê HcD

100<s2HtL>

1ê2ês 0

waterfall fields approaches time-dependent minimum due to backreaktion

at the end of inflation standard picture: inflaton oscillations, reheating , etc ...

0 100 200 300 400 500 600 700 800 900

0

5

10

15

0100200300400500600 -1 -2

Time t @1 ê HcD

Waterfallfield

sAM P

lë103E

Number of e-folds Ne

0 100 200 300 400 500 600 700 800 900

0

5

10

15

20

0100200300400500600 -1 -2

Time t @1 ê HcDInflatonfieldj@M P

lD

Number of e-folds Ne

Effective inflaton potential below critical point depends on only two variables,

where FI-term and Yukawa coupling are replaced by inflationary energy scale, and relative couplings strength (critical inflaton field),

(remark: very similar to chaotic inflation in string models, corrected bymoduli effects [WB, Dudas, Hurtier, Westphal, Wieck, Winkler ’15])

Minf =

✓g⇠p2

◆1/2

, � =�pg; '2

c =2p2M2

inf

�2

Vinf(') = V (', smin('))

= 2M4inf

'2

'2c

✓1� 1

2

'2

'2c

◆, ' 'c

“Flattened” Chaotic Inflation

Remarkably, the Planck data

determine the energy scale of inflation rather precisely,

Ne = 60

Ne = 50

MGUT

3 4 5 6 7 81

2

3

4

5

l @10-4D

Minf@10

16GeVD

Ne = 60

Ne = 50

3 4 5 6 7 810

15

20

25

30

35

40

l @10-4D

j c@M p

lDns = 0.9603± 0.0073 , r < 0.11 (95%CL)

1.6 (1.9) �

Minf1016 GeV

� 2.4 (2.2)

1σ & 2σ regions for different data sets (Planck 2013) compared with “natural inflation” and “subcritical hybrid” inflation (green); prediction of subcritical hybrid inflation for bound on r for 60 (50) e-folds:

Models in their simplest form disfavoured by Planck 2015 data; modifications possible, more data this year ...

0.00

0.05

0.10

0.15

0.20

0.25

Tens

or-to

-Sca

larRa

tio(r)

0.94 0.96 0.98 1.00Primordial Tilt (ns)

Subcriticalhybrid inflation

Planck+WP+BAOPlanck+WP+highLPlanck+WPNatural InflationNe = 50

Ne = 60

r > 0.049 (0.085)

Conclusions

Strong theoretical motivation for unification of forces beyondcurrent microscopic theory (SM)

Possible cosmic signatures of grand unification: cosmic strings,gravitational waves, B-modes in CMB; several indirect hints: leptogenesis, nonthermal dark matter, ...

Hybrid inflation attractive framework for describing very early universe (production of entropy, dark matter, leptogenesis,...)

Hints for GUT scale already in subcritical hybrid inflation together with Planck data; prediction: lower bound on tensor-to-scalar ratio, in reach of upcoming experiments!

BACKUP SLIDES

Connection to particle physics in very early universe, before formation of CMB: nucleosynthesis, formation of hadrons, generation of mass (electroweak phase transition), dark matter, leptogenesis ... preceeded by inflation (?) which generates initial conditions of hot early universe;important: global picture of very early universe!

[adapted from Schmitz ’12]

Time evolution of temperature: intermediate plateau (“maximal temperature”),determined by neutrino properties! Yields gravitino abundance when combinedwith ‘standard formula’

aRHi aRH aRH

f

100 101 102 103 104 105 106 107 108107

108

109

1010

1011

101210-1 100 101 102 103

Scale factor a

THaL@Ge

VD

Inverse temperature M1 êT

2¥1010

3¥1010

5¥1010

7¥1010

7¥10101¥10112¥1011

10-5 10-4 10-3 10-2 10-1 100

5

10

20

50

100

200

500

mé 1 @eVD

mGé@Ge

VD

M1 @GeVD such that WGé h2 = 0.11

v B-L=5.0¥1015GeV

mgé=1TeV

M1 @GeVDhBnt > hB

obs

hB < hBobs

Whé > WDMobs

Wwé > WDMobs

wé

hé

Gé

10-5 10-4 10-3 10-2 10-1 100500

1000

1500

2000

2500

3000

mé 1 @eVD

mLSP@Ge

VD

100¥mGé@TeV

D

parameter scans: successful leptogenesis and gravitino DM (left) or neutralino DM (right) [non-thermally produced in decays of thermally produced gravitinos] constrains neutrino and superparticle masses; can be tested at LHC!

Slow-Roll Inflation

ns � 1 = 2⌘⇤ � 6✏⇤ , nt = �2✏⇤

✏ =M2

pl

2

✓@�V

V

◆2

, ⌘ = M2pl

@2�V

V

r =�2

s

�2t

=2

3⇡2�s

V (�⇤)

M4pl

, V 1/4 ⇡⇣ r

0.01

⌘1/41016 GeV

Inflation is driven by slowly varying potential energy of inflaton field;

small slow-roll parameters determine spectral indices:

V (�)

Tensor-to-scalar ratio yields energy density during inflation:

Tensor-to-scalar ratios require super-Planckian field values [Lyth ’97; Antusch, Nolde ’14]; theoretical challenge, ongoing theory activities...

r & 0.005

Beyond local spinodal time waterfall field tracks local minimum; asymmetric trajectory in field space:

V Hj, sL

0 5 10 15 20

0

5

10

15

0500600700800

0

100

200

500

Inflaton field j @MPlD

Waterfallfield

sAM P

lë103E

Time t @1 ê HcD

Numberofe-foldsNe

tdec

22.714 22.715 22.7160.00.51.01.52.0

012

j @MPlD

sAM P

lë104E

Time t @1 ê Hc D