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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
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)
gauge coupling Yukawa coupling
+ 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
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)
moduli (SM singlets)
+ 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
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
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
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
Moduli at finite temperature Outline
Some basics
+ consider field–dependent couplings
g = g(S) and y = y(S)
+ 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
Michael Ratz, UC Irvine 07/August 15 2019 4/ 30
Moduli at finite temperature Outline
Some basics
+ consider field–dependent couplings
g = g(S) and y = y(S)
+ 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
Michael Ratz, UC Irvine 07/August 15 2019 4/ 30
Moduli at finite temperature Outline
Some basics
+ consider field–dependent couplings
g = g(S) and y = y(S)
+ S is weakly coupled: all couplings are suppressed by Λ (in stringsoften Λ ' MP)
+ S is light (in strings often mS ∼ m3/2?∼ TeV)
gravitino mass
+ 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
Michael Ratz, UC Irvine 07/August 15 2019 4/ 30
Moduli at finite temperature Outline
Some basics
+ consider field–dependent couplings
g = g(S) and y = y(S)
+ 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
Michael Ratz, UC Irvine 07/August 15 2019 4/ 30
Moduli at finite temperature Outline
Some basics
+ consider field–dependent couplings
g = g(S) and y = y(S)
+ 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
Michael Ratz, UC Irvine 07/August 15 2019 4/ 30
Moduli at finite temperature Outline
Some basics
+ consider field–dependent couplings
g = g(S) and y = y(S)
+ 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
Michael Ratz, UC Irvine 07/August 15 2019 4/ 30
Moduli at finite temperature Outline
Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .
ln a
ln ρ
ρS ∝ a −3
ρrad ∝ a −4
ρS ∝ a −3
ρrad ∝ a −4
ρrad ∝ a−3/2 dilution
ln T 4decay
S
VS
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
Michael Ratz, UC Irvine 07/August 15 2019 5/ 30
Moduli at finite temperature Outline
Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .
ln a
ln ρ
ρS ∝ a −3
ρrad ∝ a −4
ρS ∝ a −3
ρrad ∝ a −4
ρrad ∝ a−3/2 dilution
ln T 4decay
S
VS
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
Michael Ratz, UC Irvine 07/August 15 2019 5/ 30
Moduli at finite temperature Outline
Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .
ln a
ln ρ
ρS ∝ a −3
ρrad ∝ a −4
ρS ∝ a −3
ρrad ∝ a −4
ρrad ∝ a−3/2 dilution
ln T 4decay
S
VS
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
Michael Ratz, UC Irvine 07/August 15 2019 5/ 30
Moduli at finite temperature Outline
Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .
ln a
ln ρ
ρS ∝ a −3
ρrad ∝ a −4
ρS ∝ a −3
ρrad ∝ a −4
ρrad ∝ a−3/2 dilution
ln T 4decay
S
VS
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
Michael Ratz, UC Irvine 07/August 15 2019 5/ 30
Moduli at finite temperature Outline
Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .
ln a
ln ρ
ρS ∝ a −3
ρrad ∝ a −4
ρS ∝ a −3
ρrad ∝ a −4
ρrad ∝ a−3/2 dilution
ln T 4decay
S
VS
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
Michael Ratz, UC Irvine 07/August 15 2019 5/ 30
Moduli at finite temperature Outline
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 =√
Λ mS
“modulusdecay
constant”
mass+ briefly discuss constraints from late–decaying moduli
Michael Ratz, UC Irvine 07/August 15 2019 6/ 30
Moduli at finite temperature Outline
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
+ substantial change of couplings when
T ∼ Tcrit =√
Λ mS
“modulusdecay
constant”
mass
+ briefly discuss constraints from late–decaying moduli
Michael Ratz, UC Irvine 07/August 15 2019 6/ 30
Moduli at finite temperature Outline
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
+ substantial change of couplings when
T ∼ Tcrit =√
Λ mS
“modulusdecay
constant”
mass+ briefly discuss constraints from late–decaying moduli
Michael Ratz, UC Irvine 07/August 15 2019 6/ 30
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
(1)
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
Michael Ratz, UC Irvine 07/August 15 2019 7/ 30
Moduli at finite temperature Thermal potential for moduli
Thermal potential for moduli
+ main observation: Vthermal = F
free energyVthermal(S) = F(g(S), y(S),T
)temperature
+ leading contribution contributions to free energy from gauge andYukawa interactions
F = Fnon−interacting + ∆F(1)gauge + ∆F
(1)
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
Michael Ratz, UC Irvine 07/August 15 2019 7/ 30
Moduli at finite temperature Thermal potential for moduli
Thermal potential for moduli
+ main observation: Vthermal = F
free energyVthermal(S) = F
(g(S), y(S),T
)temperature
+ leading contribution contributions to free energy from gauge andYukawa interactions
F = Fnon−interacting + ∆F(1)gauge + ∆F
(1)
Yukawa + O(g3, y3, g2y2)
gauge Yukawa
∆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
Michael Ratz, UC Irvine 07/August 15 2019 7/ 30
Moduli at finite temperature Thermal potential for moduli
Thermal potential for moduli
+ main observation: Vthermal = F
free energyVthermal(S) = F
(g(S), y(S),T
)temperature
+ leading contribution contributions to free energy from gauge andYukawa interactions
F = Fnon−interacting + ∆F(1)gauge + ∆F
(1)
Yukawa + O(g3, y3, g2y2)
gauge Yukawa
∆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
Michael Ratz, UC Irvine 07/August 15 2019 7/ 30
Moduli at finite temperature Thermal potential for moduli
Thermal potential for moduli
+ main observation: Vthermal = F
free energyVthermal(S) = F
(g(S), y(S),T
)temperature
+ leading contribution contributions to free energy from gauge andYukawa interactions
F = Fnon−interacting + ∆F(1)gauge + ∆F
(1)
Yukawa + O(g3, y3, g2y2)
gauge Yukawa
∆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
Michael Ratz, UC Irvine 07/August 15 2019 7/ 30
Moduli at finite temperature Thermal potential for moduli
Thermal potential for moduli
+ main observation: Vthermal = F
free energyVthermal(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
(1)
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
Michael Ratz, UC Irvine 07/August 15 2019 7/ 30
Moduli at finite temperature Thermal potential for moduli
Thermal potential for moduli
+ main observation: Vthermal = F
free energyVthermal(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
(1)
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
Michael Ratz, UC Irvine 07/August 15 2019 7/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
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)
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
+ switch on thermal corrections ∝ 1/(Re S)
Michael Ratz, UC Irvine 07/August 15 2019 9/ 30
Moduli at finite temperature Dilaton destabilization at high temperature
Critical temperature
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
O(m2S M2
P)
O(MP)
slope ∼ m2S MP y T 4/MP ∼ m2
S MP
bottom–line:
critical temperature T∗ ∼√
mS MP
Michael Ratz, UC Irvine 07/August 15 2019 10/ 30
Moduli at finite temperature Dilaton destabilization at high temperature
Critical temperature
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
O(m2S M2
P)
O(MP)
slope ∼ m2S MP y T 4/MP ∼ m2
S MP
bottom–line:
critical temperature T∗ ∼√
mS MP
Michael Ratz, UC Irvine 07/August 15 2019 10/ 30
Moduli at finite temperature Dilaton destabilization at high temperature
Critical temperature
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
O(m2S M2
P)
O(MP)
slope ∼ m2S MP y T 4/MP ∼ m2
S MP
bottom–line:
critical temperature T∗ ∼√
mS MP
Michael Ratz, UC Irvine 07/August 15 2019 10/ 30
Moduli at finite temperature Dilaton destabilization at high temperature
Critical temperature
V
Re S
vacuumw/ realistic
gauge coupling
run–awayvacuum
O(m2S M2
P)
O(MP)
slope ∼ m2S MP y T 4/MP ∼ m2
S MP
bottom–line:
critical temperature T∗ ∼√
mS MP
Michael Ratz, UC Irvine 07/August 15 2019 10/ 30
Moduli at finite temperature Dilaton destabilization at high temperature
Discussion
+ if the dilaton has been destabilized, it will run away and cannot comeback
model–independent contraint:
TR . T∗ ∼√
mS MP
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)
Michael Ratz, UC Irvine 07/August 15 2019 11/ 30
Moduli at finite temperature Dilaton destabilization at high temperature
Discussion
+ if the dilaton has been destabilized, it will run away and cannot comeback
model–independent contraint:
TR . T∗ ∼√
mS MP
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)
Michael Ratz, UC Irvine 07/August 15 2019 11/ 30
Moduli at finite temperature Dilaton destabilization at high temperature
Discussion
+ if the dilaton has been destabilized, it will run away and cannot comeback
model–independent contraint:
TR . T∗ ∼√
mS MP
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)
Michael Ratz, UC Irvine 07/August 15 2019 11/ 30
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
yuij
(SΛ
)nuij
Q i Φ uj +3∑
i ,j=1
ydij
(SΛ
)ndij
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
Λ+
m2σ(T )
2σ2 +
κ
3!σ3 +
λS
4σ4 + . . .
α = γ ∂TY∂ε∼ 10−2
Michael Ratz, UC Irvine 07/August 15 2019 13/ 30
Moduli at finite temperature Constraints on flavons
Field–dependent fermion masses
+ e.g. Froggatt–Nielsen mechanismFroggatt & Nielsen (1979)
LFN =3∑
i ,j=1
yuij
(SΛ
)nuij
Q i Φ uj +3∑
i ,j=1
ydij
(SΛ
)ndij
Q i Φ dj + h.c.
+ 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
Λ+
m2σ(T )
2σ2 +
κ
3!σ3 +
λS
4σ4 + . . .
α = γ ∂TY∂ε∼ 10−2
Michael Ratz, UC Irvine 07/August 15 2019 13/ 30
Moduli at finite temperature Constraints on flavons
Field–dependent fermion masses
+ e.g. Froggatt–Nielsen mechanismFroggatt & Nielsen (1979)
LFN =3∑
i ,j=1
yuij
(SΛ
)nuij
Q i Φ uj +3∑
i ,j=1
ydij
(SΛ
)ndij
Q i Φ dj + h.c.
+ 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
Λ+
m2σ(T )
2σ2 +
κ
3!σ3 +
λS
4σ4 + . . .
α = γ ∂TY∂ε∼ 10−2
Michael Ratz, UC Irvine 07/August 15 2019 13/ 30
Moduli at finite temperature Constraints on flavons
Field–dependent fermion masses
+ e.g. Froggatt–Nielsen mechanismFroggatt & Nielsen (1979)
LFN =3∑
i ,j=1
yuij
(SΛ
)nuij
Q i Φ uj +3∑
i ,j=1
ydij
(SΛ
)ndij
Q i Φ dj + h.c.
+ 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
Λ+
m2σ(T )
2σ2 +
κ
3!σ3 +
λS
4σ4 + . . .
α = γ ∂TY∂ε∼ 10−2
Michael Ratz, UC Irvine 07/August 15 2019 13/ 30
Moduli at finite temperature Constraints on flavons
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
σ
+ as the temperature decreases, the flavon undergoes oscillationsaround the T = 0 minimum, which behave like nonrelativistic matter
Michael Ratz, UC Irvine 07/August 15 2019 14/ 30
Moduli at finite temperature Constraints on flavons
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
σ
+ as the temperature decreases, the flavon undergoes oscillationsaround the T = 0 minimum, which behave like nonrelativistic matter
Michael Ratz, UC Irvine 07/August 15 2019 14/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Thermal moduli potential (cartoon)
V
S〈S〉T =0
〈S〉T
Michael Ratz, UC Irvine 07/August 15 2019 15/ 30
Moduli at finite temperature Constraints on flavons
Flavon oscillations
ΩHuL
-u4 ±u
32 DΩ
0.1 0.2 0.5 1.0 2.0 5.0-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
Temperature HuL
ΩHuL
ω =Λσ
αM2H
w/
MH =MP
1.66√
g∗' 1.4 · 1017 GeV
u = T/T∗ T∗ =√
mσ MH
Michael Ratz, UC Irvine 07/August 15 2019 16/ 30
Moduli at finite temperature Constraints on flavons
Flavon oscillations
ΩHuL
-u4 ±u
32 DΩ
0.1 0.2 0.5 1.0 2.0 5.0-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
Temperature HuL
ΩHuL
ω =Λσ
αM2H
w/
MH =MP
1.66√
g∗' 1.4 · 1017 GeV
u = T/T∗ T∗ =√
mσ MH
Michael Ratz, UC Irvine 07/August 15 2019 16/ 30
Moduli at finite temperature Constraints on flavons
Flavon oscillations
ΩHuL-u
4 ±u32 DΩ
0.1 0.2 0.5 1.0 2.0 5.0-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
Temperature HuL
ΩHuL
Michael Ratz, UC Irvine 07/August 15 2019 17/ 30
Moduli at finite temperature Constraints on flavons
Flavon oscillations
ΩHuL-u
4
-u4 ±u
32 DΩ
0.1000.050 0.2000.030 0.1500.070-0.003
-0.002
-0.001
0.000
0.001
Temperature HuL
ΩHuL
Michael Ratz, UC Irvine 07/August 15 2019 18/ 30
Moduli at finite temperature Constraints on flavons
Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .
ln a
ln ρ
ρS ∝ a −3
ρrad ∝ a −4
ρS ∝ a −3
ρrad ∝ a −4
ρrad ∝ a−3/2 dilution
ln T 4decay
S
VS
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
Michael Ratz, UC Irvine 07/August 15 2019 19/ 30
Moduli at finite temperature Constraints on flavons
Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .
ln a
ln ρ
ρS ∝ a −3
ρrad ∝ a −4
ρS ∝ a −3
ρrad ∝ a −4
ρrad ∝ a−3/2 dilution
ln T 4decay
S
VS
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
Michael Ratz, UC Irvine 07/August 15 2019 19/ 30
Moduli at finite temperature Constraints on flavons
Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .
ln a
ln ρ
ρS ∝ a −3
ρrad ∝ a −4
ρS ∝ a −3
ρrad ∝ a −4
ρrad ∝ a−3/2 dilution
ln T 4decay
S
VS
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
Michael Ratz, UC Irvine 07/August 15 2019 19/ 30
Moduli at finite temperature Constraints on flavons
Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .
ln a
ln ρ
ρS ∝ a −3
ρrad ∝ a −4
ρS ∝ a −3
ρrad ∝ a −4
ρrad ∝ a−3/2 dilution
ln T 4decay
S
VS
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
Michael Ratz, UC Irvine 07/August 15 2019 19/ 30
Moduli at finite temperature Constraints on flavons
Moduli problemsCoughlan, Fischler, Kolb, Raby & Ross (1983); de Carlos, Casas, Quevedo & Roulet (1993) ,. . .
ln a
ln ρ
ρS ∝ a −3
ρrad ∝ a −4
ρS ∝ a −3
ρrad ∝ a −4
ρrad ∝ a−3/2 dilution
ln T 4decay
S
VS
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
Michael Ratz, UC Irvine 07/August 15 2019 19/ 30
Moduli at finite temperature Constraints on flavons
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
ener
getic
Michael Ratz, UC Irvine 07/August 15 2019 20/ 30
Moduli at finite temperature Constraints on flavons
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
ener
getic
Michael Ratz, UC Irvine 07/August 15 2019 20/ 30
Moduli at finite temperature Constraints on flavons
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
ener
getic
Michael Ratz, UC Irvine 07/August 15 2019 20/ 30
Moduli at finite temperature Constraints on flavons
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
ener
getic
Michael Ratz, UC Irvine 07/August 15 2019 20/ 30
Moduli at finite temperature Constraints on flavons
BBN constraints
1012
1013
1014
1015
1016
1017
1018
1019
10 102
103
104
105
TR = 109 GeV
TR = 107 GeV
TR = 105 GeV
γ=
10 −7
γ=
10 −5
γ=
10 −3
mσ [GeV]
Λ[G
eV]
dilution factor
Michael Ratz, UC Irvine 07/August 15 2019 21/ 30
Moduli at finite temperature Constraints on flavons
BBN constraints
1012
1013
1014
1015
1016
1017
1018
1019
10 102
103
104
105
TR = 109 GeV
TR = 107 GeV
TR = 105 GeV
γ=
10 −7
γ=
10 −5
γ=
10 −3
mσ [GeV]
Λ[G
eV]
dilution factor
Michael Ratz, UC Irvine 07/August 15 2019 21/ 30
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
Michael Ratz, UC Irvine 07/August 15 2019 23/ 30
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
Michael Ratz, UC Irvine 07/August 15 2019 23/ 30
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
Michael Ratz, UC Irvine 07/August 15 2019 23/ 30
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
Michael Ratz, UC Irvine 07/August 15 2019 23/ 30
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
Michael Ratz, UC Irvine 07/August 15 2019 23/ 30
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
Michael Ratz, UC Irvine 07/August 15 2019 25/ 30
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 dilaton
fixing: TR .√
mS MP
+ cosmological constraints on models with “light” flavons
moduli problem more severe than generally appreciated
Michael Ratz, UC Irvine 07/August 15 2019 25/ 30
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 dilaton
fixing: TR .√
mS MP
+ cosmological constraints on models with “light” flavons
moduli problem more severe than generally appreciated
Michael Ratz, UC Irvine 07/August 15 2019 25/ 30
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 dilaton
fixing: TR .√
mS MP
+ cosmological constraints on models with “light” flavons
moduli problem more severe than generally appreciated
Michael Ratz, UC Irvine 07/August 15 2019 25/ 30
Moduli at finite temperature Summary
Summary
inflation ends
baryogenesis
nucleosynthesist = 1sec
today (August 15 2019)
creation ofdark mattermany
openquestions
physics(more or less)
understood
Michael Ratz, UC Irvine 07/August 15 2019 26/ 30
Moduli at finite temperature Summary
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)
Michael Ratz, UC Irvine 07/August 15 2019 27/ 30
Moduli at finite temperature Summary
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)
Michael Ratz, UC Irvine 07/August 15 2019 27/ 30
Moduli at finite temperature Summary
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)
Michael Ratz, UC Irvine 07/August 15 2019 27/ 30
Moduli at finite temperature Summary
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)
Michael Ratz, UC Irvine 07/August 15 2019 27/ 30
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.
Michael Ratz, UC Irvine 07/August 15 2019 29/ 30
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.
Michael Ratz, UC Irvine 07/August 15 2019 30/ 30