Balkan Workshop BW2013 Beyond the Standard Models 25 – 29 April, 2013, Vrnjačka Banja, Serbia

1. Cosmological B−L Breaking:
(Dark) Matter & Gravitational Waves
Wilfried Buchm¨uller
DESY, Hamburg
with Valerie Domcke, Kai Schmitz & Kohei Kamada
1202.6679; 1203.0285, 1210.4105, 1304.XXX
BW2013, Vrnjaˇcka Banja, Serbia, April 2013

2. I. B−L breaking, inflation & dark matter
• Light neutrino masses can be explained by mixing with Majorana neutrinos
with GUT scale masses from B−L breaking (seesaw mechanism)
• Decays of heavy Majorana neutrinos natural source of baryon asymmetry
(leptogenesis; thermal (Fukugita, Yanagida ’86) or nonthermal (Lazarides, Shafi ’91) )
• In supersymmetric models with spontaneous B−L breaking, natural
connection with inflation (Copeland et al ’94; Dvali, Shafi, Schaefer ’94; ...)
• LSP (gravitino, higgsino,...) natural candidate for dark matter
• Consistent picture of inflation, baryogenesis and dark matter?
• Possible direct test: gravitational waves
1

3. Leptogenesis and gravitinos: for thermal leptogenesis and typical
superparticle masses, thermal production yields observed amount of DM,
Ω ˜Gh2
= C
TR
1010 GeV
100 GeV
m ˜G
m˜g
1 TeV
2
, C ∼ 0.5 ;
ΩDMh2
∼ 0.1 is natural value; but why TR ∼ TL ?
Starting point simple observation: heavy neutrino decay width
Γ0
N1
=
˜m1
8π
M1
vEW
2
∼ 103
GeV , m1 ∼ 0.01 eV , M1 ∼ 1010
GeV .
yields reheating temperature (for decaying gas of heavy neutrinos)
TR ∼ 0.2 · Γ0
N1
MP ∼ 1010
GeV ,
wanted for gravitino DM. Intriguing hint or misleading coincidence?
2

4. II. Spontaneous B−L breaking and false vacuum decay
Supersymmetric SM with right-handed neutrinos,
WM = hu
ij10i10jHu + hd
ij5∗
i 10jHd + hν
ij5∗
i nc
jHu + hn
i nc
inc
iS1 ,
in SU(5) notation: 10 = (q, uc
, ec
), 5∗
= (dc
, l); electroweak symmetry
breaking, Hu,d ∝ vEW , and B−L breaking,
WB−L =
√
λ
2
Φ v2
B−L − 2S1S2 ,
S1,2 = vB−L/
√
2 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!
3

5. Parameters of B−L breaking sector: mν =
√
m2m3 = 3 × 10−2
eV,
M1 M2,3 mS, m1 = (m†
DmD)11/M1, vB−L.
Spontaneous symmetry breaking: consider Abelian Higgs model in unitary
gauge (→ massive vector multiplet, no Wess-Zumino gauge!),
S1,2 =
1
√
2
S exp(±iT) , V = Z +
i
2g
(T − T∗
) .
Inflaton field Φ: slow motion (quantum corrections), changes mass of
‘waterfall’ field S, rapid change after critical point where mS = 0; basic
mechanism of hybrid inflation.
Shift around time-dependent background, s = 1√
2
(σ +iτ), σ →
√
2v(t)+σ
with v(t) = 1√
2
σ 2
(t, x)
1/2
x ; masses of fluctuations:
4

6. 0
1
2
3
1
0
1
0
0.5
Φ MP
H MP
V MP
4
S / vB-L
/
/
m2
σ =
1
2
λ(3v2
(t) − v2
B−L) , m2
τ =
1
2
λ(v2
B−L + v2
(t)) , m2
φ = λv2
(t) ,
m2
ψ = λv2
(t) , m2
Z = 8g2
v2
(t) , M2
i = (hn
i )2
v2
(t) ;
time-dependent masses of B−L Higgs, inflaton, vector boson, heavy
neutrinos, all supermultiplets!
5

7. Constraints from cosmic strings and inflation: upper bound on string tension
(Planck Collaboration ’13)
Gµ < 3.2 × 10−7
, µ = 2πB(β)v2
B−L ,
with β = λ/(8 g2
) and B(β) = 2.4 [ln(2/β)]
−1
for β < 10−2
; further
constraint from CMB (cf. Nakayama et al ’10), yields
3 × 1015
GeV vB−L 7 × 1015
GeV ,
10−4
√
λ 10−1
.
Final choice for range of parameters (analysis within FN flavour model):
vB−L = 5 × 1015
GeV , 10−5
eV ≤ m1 ≤ 1 eV ,
109
GeV ≤ M1 ≤ 3 × 1012
GeV .
(range of m1: uncertainty of O(1) parameters)
6

8. Tachyonic Preheating
Hybrid inflation ends at critical value Φc of inflaton field Φ by rapid growth
of fluctuations of B−L Higgs field S (‘spinodal decomposition’):
0.001
0.01
0.1
1
5 10 15 20 25 30
<φ2(t)>1/2/v,nB(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
Occupationnumber:nk
k/m
Bosons: Lattice
Tanh
Fermions: Lattice
Tanh
in addition, particles which couple to S are produced by rapid increase of
‘waterfall field’ (Garcia-Bellido, Morales ’02); no coherent oscillations!
7

9. Decay of false vacuum produces long wave-length σ-modes, true vacuum
reached at time tPH (even faster decay with inflaton dynamics),
σ
2
t=tPH
= 2v2
B−L , tPH
1
2mσ
ln
32π2
λ
.
Initial state: nonrelativistic gas of σ-bosons, N2,3, ˜N2,3, A, ˜A, C (contained
in superfield Z), ... ; energy fractions (α = mX/mS, ρ0 = λv4
B−L/4):
ρB/ρ0 2 × 10−3
gs λ f(α, 1.3) , ρF /ρ0 1.5 × 10−3
gs λ f(α, 0.8) .
Time evolution: rapid N2,3, ˜N2,3, A, ˜A, C decays, yields initial radiation,
thermal N1’s and gravitinos; σ decays produce nonthermal N1’s; N1 decays
produce most of radiation and baryon asymmetry; details of evolution
described by Boltzmann equations.
8

10. Reheating Process
Major work: solve network of Boltzmann equations for all (super)particles;
treat nonthermal and thermal contributions differently, varying equation
of state; result: detailed time resolved description of reheating process,
prediction of baryon asymmetry and gravitino density (possibly dark matter).
Illustrative example for parameter choice
M1 = 5.4 × 1010
GeV , m1 = 4.0 × 10−2
eV ,
meG = 100 GeV , m˜g = 1 TeV ; Gµ = 2.0 × 10−7
fixes (within FN flavour model) all other masses, CP asymmetries etc.
Note: emergence of temperature plateau at intermediate times; final result:
ηB 3.7 × 10−9
ηnt
B , ΩeGh2
0.11 ,
i.e., dynamical realization of original conjecture.
9

11. Thermal and nonthermal number densities
aRH
i aRH aRH
f
Σ Ψ Φ
N2,3 N2,3
N1
nt
N1
nt
N1
th
N1
th
2N1
eq
R
B L
G
100
101
102
103
104
105
106
107
108
1025
1030
1035
1040
1045
1050
10 1
100
101
102
103
Scale factor a
absNa
Inverse temperature M1 T
Comoving number densites of thermal and nonthermla N1s,..., B−L,
gravitinos and radiation as functions of scale factor a.
10

12. Time evolution of temperature: intermediate plateau
aRH
i aRH aRH
f
100
101
102
103
104
105
106
107
108
107
108
109
1010
1011
1012
10 1
100
101
102
103
Scale factor a
TaGeV
Inverse temperature M1 T
Gravitino abundance can be understood from ‘standard formula’ and
effective ‘reheating temperature’ (determined by neutrino masses).
11

13. III. Gravitinos & Dark Matter
Thermal production of gravitinos is origin of DM; depending on pattern
of SUSY breaking, gravitino DM or higgsino/wino DM. Mass spectrum of
superparticles motivated ‘large’ Higgs mass measured at the LHC,
mLSP msquark,slepton meG .
LSP is typically ‘pure’ wino or higgsino (bino disfavoured, overproduction
in thermal freeze-out), almost mass degenerate with chargino. Thermal
abundance of wino (w) or higgsino (h) LSP significant for masses above
1 TeV, well approximated by (Arkani-Hamed et al ’06; Hisano et al ’07, Cirelli et al ’07)
Ωth
ew,eh
h2
= cew,eh
mew,eh
1 TeV
2
, cew = 0.014 , ceh = 0.10 ,
Heavy gravitinos (10 TeV . . . 103
TeV) consistent with BBN, τeG 24 ×
12

14. (10 TeV/meG)3
sec. Total higgsino/wino abundance
Ωew,ehh2
= Ω
eG
ew,eh
h2
+ Ωth
ew,eh
h2
,
Ω
eG
LSPh2
=
mLSP
meG
ΩeGh2
2.7 × 10−2 mLSP
100 GeV
TRH(M1, m1)
1010 GeV
,
with ‘reheating temperature’ determined by neutino masses (takes reheating
process into account),
TRH 1.3 × 1010
GeV
m1
0.04 eV
1/4
M1
1011 GeV
5/4
.
Requirement of LSP dark matter, i.e. ΩLSPh2
= ΩDMh2
0.11, yields
upper bound on the reheating temperature, TRH < 4.2 × 1010
GeV; lower
bound on TRH from successfull leptogenesis (depends on m1).
13

15. 4
He
D
LSP DM
obs
10 4
10 2
100
m1 eV
101
102
103
109
1010
1011
mG
TeV
TRHGeV
h DM
obs
w DM
obs
w
h
G
10 5
10 4
10 3
10 2
10 1
100
500
1000
1500
2000
2500
3000
m1 eV
mLSPGeV
100mG
TeV
For each ‘reheating temperature’, i.e. pair (M1, m1), lower bound on
gravitino mass (taken from Kawasaki et al ’08) (left panel). Requirement of
higgsino/wino dark matter puts upper bound on LSP mass, dependent
on m1, ‘reheating temperature’ (right panel); more stringent for higgsino
mass, since freeze-out contribution larger. E.g., m1 = 0.05 eV implies
meh
<∼ 900 GeV, meG 10 TeV.
14

16. IV. Gravitational Waves
Relic gravitational waves are window to very early universe; contributions
from inflation, preheating and cosmic strings (Rubakov et al ’82; Garcia-Bellido and
Figueroa ’07; Vilenkin ’81; Hindmarsh et al ’12); cosmological B−L breaking: prediction
of GW spectrum with all contributions!
Perturbations in flat FRW background,
ds2
= a2
(τ)(ηµν + hµν)dxµ
dxν
, ¯hµν = hµν −
1
2
ηµνhρ
ρ ,
determined by linearized Einstein equations,
¯hµν(x, τ) + 2
a
a
¯hµν(x, τ) − 2
x
¯hµν(x, τ) = 16πGTµν(x, τ) .
Spectrum of GW background,
ΩGW (k, τ) =
1
ρc
∂ρGW (k, τ)
∂ ln k
,
15

17. ∞
−∞
d ln k
∂ρGW (k, τ)
∂ ln k
=
1
32πG
˙hij (x, τ) ˙hij
(x, τ) ;
use initial conditions for super-horizon modes from inflation, calculate
correlation function of stress energy tensor.
Contribution from inflation (primordial spectrum, transfer function; cf.
Nakayama et al ’08):
ΩGW(k, τ) =
∆2
t
12
k2
a2
0H2
0
T2
k (τ)
=
∆2
t
12
Ωr
gk
∗
g0
∗
g0
∗,s
gk
∗,s
4/3



1
2 (keq/k)
2
, k0 k keq
1 , keq k kRH
1
2 C3
RH (kRH/k)
2
kRH k kPH
16

18. with boundary frequences (f = k/(2πa0)),
f0 = 3.58 × 10−19
Hz
h
0.70
,
feq = 1.57 × 10−17
Hz
Ωmh2
0.14
,
fRH = 4.25 × 10−1
Hz
TRH
107 GeV
,
fPH = 1.99 × 104
Hz
λ
10−4
1/6
vB−L
5 × 1015 GeV
2/3
TRH
107 GeV
1/3
.
Contribution from preheating (cf. Dufaux et al ’07):
ΩGW (kPH) h2
cPH (RPHHPH)2 aPH
aRH
Ωrh2 gRH
∗
g0
∗
g0
∗,s
gRH
∗,s
4/3
,
17

19. with characteristic scalar and vector scales
R
(s)
PH
−1
= (λ vB−L | ˙φc|)1/3
, R
(v)
PH
−1
∼ mZ = 2
√
2 g vB−L .
Contribution from cosmic strings (Abelian Higgs):
ΩGW (k)
1
6π2
Fr vB−L
MPl
4
Ωrh2



(keq/k)2
, k0 k keq
1, keq k kRH
(kRH/k)2
, kRH k
constant Fr
recently determined in numerical simulation (cf. Figueroa, Hindmarsh,
Urrestilla ’12). Result similar to contribution from inflation, but very different
normalization!
18

20. inflation
cosmic strings
f0
feq fRH fPH
fPH
s
fPH
v
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 1020 1025
f Hz
GWh2
k Mpc 1
19

21. Are macroscopically long cosmic strings Nambu-Goto strings? Energy loss of
string network by ‘massive radiation’ or gravitational waves? GW radiation
from NG strings, radiated by loops of length
l(t, ti) = αti − ΓGµ(t − ti) ;
rate for amplitude h and frequency f (cf. Kuroyanagi et al ’12),
d2
R
dzdh
(f, h, z)
3
4
θ2
m
(1 + z)(α + ΓGµ)h
1
αγ2t4(z)
dV (z)
dz
Θ(1 − θm) ;
can be approximately integrated analytically over z and h; results
differs qualitatively from Abelian-Higgs prediction; five orders of magnitude
difference in normalization! Truth somewhere inbetween?
20

22. Abelian-Higgs Strings vs. Nambu-Goto Strings
Α 10 6
Α 10 12
Nambu Goto
Abelian Higgs
10 20 10 15 10 10 10 5 100 105 1010
10 15
10 10
10 5
10 5 100 105 1010 1015 1020 1025
f Hz
GWh2
k Mpc 1
21

23. Observational Prospects
78
3
4
5
6
1
2
9
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
100
10 5 100 105 1010 1015 1020 1025
f Hz
GWh2
k Mpc 1
22

24. Summary and Outlook
• Decay of false vacuum of unbroken B-L symmetry leads to
consistent picture of inflation, baryogenesis and dark matter
(everything from heavy neutrino decays)
• Prediction: relations between neutrino and superparticle
masses for gravitino or higgsino/wino dark matter
• Possible direct test: detection of relic gravitational
wave background, may provide determination of reheating
temperature
23

25. (Non)thermal leptogenesis in M1 − ˜m1 plane
10 11
10 10
10 9
10 8
10 7
10 6
10 5
10 4
10 3
10 2
10 1
100
108.5
109
109.5
1010
1010.5
1011
1011.5
1012
1012.5
1013
m1 eV
M1GeV
ΗB m1,M1
vBL5.01015
GeV
Inflation
& strings
ΗB
nt
ΗB
th
ΗB
th
ΗB
obs
ΗB
nt
ΗB
obs
ΗB ΗB
obs
Upper bound on M1 from inflation; lower bound from baryogenesis
24

26. Gravitino Dark Matter vs. leptogenesis
2 1010
3 1010
5 1010
7 1010
7 10101 1011
2 1011
10 5
10 4
10 3
10 2
10 1
100
5
10
20
50
100
200
500
m1 eV
mG
GeV
M1 GeV such that G h2
0.11
vBL5.01015
GeVmg1TeV
M1 GeV
ΗB
nt
ΗB
obs
ΗB ΗB
obs
Gravitino mass range: 10 GeV <∼ meG
<∼ 700 GeV; heavy neutrino mass
range: 2 × 1010
GeV <∼ M1 <∼ 2 × 1011
GeV (more stringent than inflation)
25