moduli at finite temperaturevietnam.in2p3.fr/2019/cosmology/transparencies/4_thursday/2_after… ·...

Moduli at finite temperature

Michael Ratz

15th Rencontres du Vietnam Cosmology 2019

Based on:

• W. Buchmüller, K. Hamaguchi, O. Lebedev & M.R. Nucl. Phys.B699, 292-308 (2004)

• B. Lillard, M.R., T. Tait & S. Trojanowski JCAP 1807 no. 07, 056(2018)

Diclaimers and apologiesDiclaimers and apologies• very little citations• many cartoons

Moduli at finite temperature Outline

Overview

+ in most more fundamental theories couplings are field–dependent

+ in particular, in string theory the gauge and Yukawa couplings arefunctions of so–called moduli

g = g(S) and y = y(S)

+ it is extremely hard to test this moduli–dependence at collidersbecause they are usually very weakly coupled

e.g. Bauer, Schell & Plehn (2016)

+ however, the existence of the moduli has dramatic impacts on theearly universe because they are usually very weakly coupled

purpose of this talk:

discuss moduli in the hot early universe

Michael Ratz, UC Irvine 07/August 15 2019 3/ 30

Overview

gauge coupling Yukawa coupling

Overview

moduli (SM singlets)

Overview

Some basics

+ consider field–dependent couplings

+ S is weakly coupled: all couplings are suppressed by Λ (in stringsoften Λ ' MP)

+ S is light (in strings often mS ∼ m3/2?∼ TeV)

+ relatively light, weakly coupled fields decay latey potential threatfor cosmology

+ well–known gravitino problem Khlopov & Linde (1984) ,. . . (many others)

+ moduli problem more model–dependent but also typically moresevere

Some basics

gravitino mass

Some basics

Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .

ρS ∝ a −3

ρrad ∝ a −4

ρS ∝ a −3

ρrad ∝ a −4

ρrad ∝ a−3/2 dilution

ln T 4decay

Moduli problems/effects:• if S decays during or

after BBN your model isdoomed

• if S decays afterbaryogenesis you musthave a very efficientmechanism

• S decays will modifydark matter abundance

the last two issues will bediscussed in Mu-Chun’s talk

ρS ∝ a −3

ρrad ∝ a −4

ρS ∝ a −3

ρrad ∝ a −4

ln T 4decay

ρS ∝ a −3

ρrad ∝ a −4

ρS ∝ a −3

ρrad ∝ a −4

ln T 4decay

ρS ∝ a −3

ρrad ∝ a −4

ρS ∝ a −3

ρrad ∝ a −4

ln T 4decay

ρS ∝ a −3

ρrad ∝ a −4

ρS ∝ a −3

ρrad ∝ a −4

ln T 4decay

Main focus of this talk

+ will show that moduli move in such a way that couplings decrease

gT→large−−−−−−−→ 0 and y

T→large−−−−−−−→ 0

temperature

+ substantial change of couplings when

T ∼ Tcrit =√

“modulusdecay

constant”

mass+ briefly discuss constraints from late–decaying moduli

T→large−−−−−−−→ 0

T ∼ Tcrit =√

“modulusdecay

constant”

+ briefly discuss constraints from late–decaying moduli

T→large−−−−−−−→ 0

T ∼ Tcrit =√

“modulusdecay

constant”

mass+ briefly discuss constraints from late–decaying moduli

Moduli at finite temperature Thermal potential for moduli

Thermal potential for moduli

+ main observation: Vthermal = F

free energy

Vthermal(S) = F(g(S), y(S),T

)temperature+ leading contribution contributions to free energy from gauge and

Yukawa interactions

F = Fnon−interacting + ∆F(1)gauge + ∆F

Yukawa + O(g3, y3, g2y2)

∆F(1)gauge = α2 g2 T 4

∆F(1)

Yukawa =5 |y |2

576T 4 per Weyl fermion

α2 =3

196(N2

C − 1) (NC + 3NF ) for SU(NC) w/ NF fundamentals

crucial: signs are positive

å free energy gets minimized for smaller couplings y and g

free energyVthermal(S) = F(g(S), y(S),T

)temperature

+ leading contribution contributions to free energy from gauge andYukawa interactions

∆F(1)

Yukawa =5 |y |2

α2 =3

196(N2

free energyVthermal(S) = F

(g(S), y(S),T

)temperature

gauge Yukawa

∆F(1)

Yukawa =5 |y |2

α2 =3

196(N2

(g(S), y(S),T

)temperature

gauge Yukawa

∆F(1)

Yukawa =5 |y |2

α2 =3

196(N2

(g(S), y(S),T

)temperature

gauge Yukawa

∆F(1)

Yukawa =5 |y |2

α2 =3

196(N2

Yukawa interactions

∆F(1)

Yukawa =5 |y |2

α2 =3

196(N2

Yukawa interactions

∆F(1)

Yukawa =5 |y |2

α2 =3

196(N2

Dilaton destabilizationDilaton destabilizationatat

high temperaturehigh temperature

Moduli at finite temperature Dilaton destabilization at high temperature

Application 1: dilaton destabilization at high temperatureBuchmüller, Hamaguchi, Lebedev & M.R. (2004)

+ typical moduli potential: g2 = 1/(Re S)

vacuumw/ realistic

gauge coupling

run–awayvacuum

+ switch on thermal corrections ∝ 1/(Re S)

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

vacuumw/ realistic

gauge coupling

run–awayvacuum

Critical temperature

vacuumw/ realistic

gauge coupling

run–awayvacuum

O(m2S M2

slope ∼ m2S MP y T 4/MP ∼ m2

bottom–line:

critical temperature T∗ ∼√

vacuumw/ realistic

gauge coupling

run–awayvacuum

O(m2S M2

bottom–line:

vacuumw/ realistic

gauge coupling

run–awayvacuum

O(m2S M2

bottom–line:

vacuumw/ realistic

gauge coupling

run–awayvacuum

O(m2S M2

bottom–line:

Discussion

+ if the dilaton has been destabilized, it will run away and cannot comeback

model–independent contraint:

TR . T∗ ∼√

reheating temperature(maximal temperature

of the radiationdominated universe)

+ model–dependent bounds on the energy density of the universeduring inflation Kallosh & Linde (2004)

+ the bounds can be circumvented by stabilizing the field combinationthat fixes the gauge coupling in a different way (i.e. w/ an infinitebarrier) Kane & Winkler (2019)

Discussion

TR . T∗ ∼√

Discussion

TR . T∗ ∼√

ConstraintsConstraintsonon

flavonsflavons

Moduli at finite temperature Constraints on flavons

Field–dependent fermion masses

+ e.g. Froggatt–Nielsen mechanismFroggatt & Nielsen (1979)

LFN =3∑

i ,j=1

Q i Φ uj +3∑

i ,j=1

Q i Φ dj + h.c.

flavon

+ potential

VS = − µ2S |S|

2 + λS |S|4 + λSΦ|S|2|Φ|2 + U(1)FN breaking terms

å VEV at T = 0

S =1√

2(vS + σ + i ρ)

+ effective potential

Veff(σ,T ) = γ TY T 4 + αT 4σ

m2σ(T )

2σ2 +

3!σ3 +

4σ4 + . . .

α = γ ∂TY∂ε∼ 10−2

LFN =3∑

i ,j=1

Q i Φ uj +3∑

i ,j=1

Q i Φ dj + h.c.

+ potential

VS = − µ2S |S|

å VEV at T = 0

S =1√

2(vS + σ + i ρ)

m2σ(T )

2σ2 +

3!σ3 +

4σ4 + . . .

α = γ ∂TY∂ε∼ 10−2

LFN =3∑

i ,j=1

Q i Φ uj +3∑

i ,j=1

Q i Φ dj + h.c.

+ potential

VS = − µ2S |S|

å VEV at T = 0

S =1√

2(vS + σ + i ρ)

m2σ(T )

2σ2 +

3!σ3 +

4σ4 + . . .

α = γ ∂TY∂ε∼ 10−2

LFN =3∑

i ,j=1

Q i Φ uj +3∑

i ,j=1

Q i Φ dj + h.c.

+ potential

VS = − µ2S |S|

å VEV at T = 0

S =1√

2(vS + σ + i ρ)

m2σ(T )

2σ2 +

3!σ3 +

4σ4 + . . .

α = γ ∂TY∂ε∼ 10−2

Flavon dynamicsLillard, M.R., Tait & Trojanowski (2018)

+ the flavon gets driven away from its T = 0 minimum until it getsstopped by the mass term or Hubble friction

∆σ ' − αT 4

Λ m2eff

where m2eff = 6H2 + m2

+ as the temperature decreases, the flavon undergoes oscillationsaround the T = 0 minimum, which behave like nonrelativistic matter

Flavon dynamicsLillard, M.R., Tait & Trojanowski (2018)

+ the flavon gets driven away from its T = 0 minimum until it getsstopped by the mass term or Hubble friction

∆σ ' − αT 4

Λ m2eff

where m2eff = 6H2 + m2

+ as the temperature decreases, the flavon undergoes oscillationsaround the T = 0 minimum, which behave like nonrelativistic matter

Thermal moduli potential (cartoon)

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

S〈S〉T =0

〈S〉T

Flavon oscillations

-u4 ±u

32 DΩ

0.1 0.2 0.5 1.0 2.0 5.0-1.0

Temperature HuL

ω =Λσ

MH =MP

1.66√

g∗' 1.4 · 1017 GeV

u = T/T∗ T∗ =√

mσ MH

Flavon oscillations

-u4 ±u

32 DΩ

0.1 0.2 0.5 1.0 2.0 5.0-1.0

Temperature HuL

ω =Λσ

MH =MP

1.66√

g∗' 1.4 · 1017 GeV

u = T/T∗ T∗ =√

mσ MH

Flavon oscillations

ΩHuL-u

4 ±u32 DΩ

0.1 0.2 0.5 1.0 2.0 5.0-1.0

Temperature HuL

Flavon oscillations

ΩHuL-u

-u4 ±u

32 DΩ

0.1000.050 0.2000.030 0.1500.070-0.003

-0.002

-0.001

Temperature HuL

ρS ∝ a −3

ρrad ∝ a −4

ρS ∝ a −3

ρrad ∝ a −4

ln T 4decay

ρS ∝ a −3

ρrad ∝ a −4

ρS ∝ a −3

ρrad ∝ a −4

ln T 4decay

ρS ∝ a −3

ρrad ∝ a −4

ρS ∝ a −3

ρrad ∝ a −4

ln T 4decay

ρS ∝ a −3

ρrad ∝ a −4

ρS ∝ a −3

ρrad ∝ a −4

ln T 4decay

ρS ∝ a −3

ρrad ∝ a −4

ρS ∝ a −3

ρrad ∝ a −4

ln T 4decay

BBN constraints

+ BBN is a highly predictive and constrained theory explaining theprimordial abundance of light elements cf. talk by Evan Grohs on Tuesday

+ however, late–decaying particles screw up these abundances

highly

BBN constraints

highly

BBN constraints

highly

BBN constraints

highly

BBN constraints

10 102

TR = 109 GeV

TR = 107 GeV

TR = 105 GeV

10 −7

10 −5

10 −3

mσ [GeV]

dilution factor

BBN constraints

10 102

TR = 109 GeV

TR = 107 GeV

TR = 105 GeV

10 −7

10 −5

10 −3

mσ [GeV]

dilution factor

Moduli trappingModuli trappingatat

symmetry–enhanced points?symmetry–enhanced points?

Moduli at finite temperature Moduli trapping at symmetry–enhanced points?

Moduli trapping at symmetry–enhanced points?

+ original belief: moduli deviate from the T = 0 minimum of thepotential in the early universe

+ it has been argued that moduli may get trapped atsymmetry–enhanced points e.g. Kofman, Linde, Liu, Maloney, McAllister & Silverstein (2004)

+ this mechanism may conceivably prevent the moduli from deviatingfrom their T = 0 minimum during inflation

+ however, thermal effects will always drive the moduli that determinegauge and Yukawa couplings away from their T = 0 minima at highT

bottom–line:

moduli problems more severe than generallyappreciated

bottom–line:

SummarySummary

Moduli at finite temperature Summary

Summary

+ moduli potential receives temperature–dependent corrections

∆V = F(g(s), y(S)

+ bound on reheating temperature in standard mechanisms of dilatonfixing: TR .

√mS MP

+ cosmological constraints on models with “light” flavons

moduli problem more severe than generally appreciated

Summary

∆V = F(g(s), y(S)

)+ bound on reheating temperature in standard mechanisms of dilaton

fixing: TR .√

Summary

∆V = F(g(s), y(S)

fixing: TR .√

Summary

∆V = F(g(s), y(S)

fixing: TR .√

Summary

inflation ends

baryogenesis

nucleosynthesist = 1sec

today (August 15 2019)

creation ofdark mattermany

openquestions

physics(more or less)

understood

Outlook

+ there are many conceivable implications of thetemperature–dependence of couplings:• affect relic abundance of DM and other particles

• out–of–equilibrium condition for baryogenesis easier to satisfy• phase transitions change• flatness of inflaton potential may be easier to accomplish if couplings

switch off (interplay between moduli and inflaton)

Outlook

+ there are many conceivable implications of thetemperature–dependence of couplings:• affect relic abundance of DM and other particles• out–of–equilibrium condition for baryogenesis easier to satisfy

• phase transitions change• flatness of inflaton potential may be easier to accomplish if couplings

Outlook

+ there are many conceivable implications of thetemperature–dependence of couplings:• affect relic abundance of DM and other particles• out–of–equilibrium condition for baryogenesis easier to satisfy• phase transitions change

• flatness of inflaton potential may be easier to accomplish if couplingsswitch off (interplay between moduli and inflaton)

Outlook

+ there are many conceivable implications of thetemperature–dependence of couplings:• affect relic abundance of DM and other particles• out–of–equilibrium condition for baryogenesis easier to satisfy• phase transitions change• flatness of inflaton potential may be easier to accomplish if couplings

Thanks a lot!Thanks a lot!

Moduli at finite temperature References

References I

Martin Bauer, Torben Schell & Tilman Plehn. Hunting the Flavon. Phys.Rev., D94(5):056003, 2016. doi: 10.1103/PhysRevD.94.056003.

Wilfried Buchmüller, Koichi Hamaguchi, Oleg Lebedev & Michael Ratz.Dilaton destabilization at high temperature. Nucl. Phys., B699:292–308, 2004.

G. D. Coughlan, W. Fischler, Edward W. Kolb, S. Raby & Graham G.Ross. Cosmological Problems for the Polonyi Potential. Phys. Lett.,B131:59, 1983. doi: 10.1016/0370-2693(83)91091-2.

B. de Carlos, J.A. Casas, F. Quevedo & E. Roulet. Model independentproperties & cosmological implications of the dilaton & moduli sectorsof 4-d strings. Phys. Lett., B318:447–456, 1993. doi:10.1016/0370-2693(93)91538-X.

C.D. Froggatt & Holger Bech Nielsen. Hierarchy of Quark Masses,Cabibbo Angles & CP Violation. Nucl. Phys., B147:277, 1979. doi:10.1016/0550-3213(79)90316-X.

Moduli at finite temperature References

References II

Renata Kallosh & Andrei D. Linde. Landscape, the scale of SUSYbreaking & inflation. JHEP, 0412:004, 2004. doi:10.1088/1126-6708/2004/12/004.

Gordon Kane & Martin Wolfgang Winkler. Deriving the Inflaton inCompactified M-theory with a De Sitter Vacuum. 2019.

M. Yu. Khlopov & Andrei D. Linde. Is It Easy to Save the Gravitino? Phys.Lett., 138B:265–268, 1984. doi: 10.1016/0370-2693(84)91656-3.

Lev Kofman, Andrei D. Linde, Xiao Liu, Alexander Maloney, LiamMcAllister & Eva Silverstein. Beauty is attractive: Moduli trapping atenhanced symmetry points. JHEP, 05:030, 2004. doi:10.1088/1126-6708/2004/05/030.

Benjamin Lillard, Michael Ratz, M. P. Tait, Tim & Sebastian Trojanowski.The Flavor of Cosmology. JCAP, 1807(07):056, 2018. doi:10.1088/1475-7516/2018/07/056.

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