Cosmological B L Breaking: (Dark) Matter & Gravitational Waves - - PowerPoint PPT Presentation

Cosmological BL Breaking: (Dark) Matter & Gravitational Waves

Wilfried Buchm¨ uller DESY, Hamburg with Valerie Domcke, Kai Schmitz & Kohei Kamada 1202.6679; 1203.0285, 1210.4105, 1305.3392

GGI, Florence, June 2013

• I. BL breaking, inflation & dark matter
• Light neutrino masses can be explained by mixing with Majorana neutrinos

with GUT scale masses from BL 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 BL 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

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 8⇡ ✓M1 vEW ◆2 ⇠ 103 GeV , e m1 ⇠ 0.01 eV , M1 ⇠ 1010 GeV . yields reheating temperature (for decaying gas of heavy neutrinos) TR ⇠ 0.2 · q Γ0

N1MP ⇠ 1010 GeV ,

wanted for gravitino DM. Intriguing hint or misleading coincidence?

2

• II. Spontaneous BL breaking and false vacuum decay

Supersymmetric SM with right-handed neutrinos, WM = hu

ij10i10jHu + hd ij5⇤ i 10jHd + h⌫ ij5⇤ i nc jHu + hn i nc inc iS1 ,

in SU(5) notation: 10 = (q, uc, ec), 5⇤ = (dc, l); electroweak symmetry breaking, hHu,di / vEW, and BL breaking, WBL = p

• 2 Φ
• v2

BL 2S1S2

• ,

hS1,2i = vBL/ p 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

Parameters of BL breaking sector: m⌫ = pm2m3 = 3 ⇥ 102 eV, M1 ⌧ M2,3 ' mS, e m1 = (m†

DmD)11/M1, vBL.

Spontaneous symmetry breaking: consider Abelian Higgs model in unitary gauge (! massive vector multiplet, no Wess-Zumino gauge!), S1,2 = 1 p 2 S0 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, s0 =

1 p 2(0+i⌧), 0 !

p 2v(t)+ with v(t) =

1 p 2h02(t, ~

x)i1/2

~ x ; masses of fluctuations:

4

        

     



   

m2

= 1

2(3v2(t) v2

BL) ,

m2

⌧ = 1

2(v2

BL + v2(t)) ,

m2

= v2(t) ,

m2

= v2(t) ,

m2

Z = 8g2v2(t) ,

M 2

i = (hn i )2v2(t) ;

time-dependent masses of BL Higgs, inflaton, vector boson, heavy neutrinos, all supermultiplets!

5

Constraints from cosmic strings and inflation: upper bound on string tension

(Planck Collaboration ’13)

Gµ < 3.2 ⇥ 107 , µ = 2⇡B()v2

BL ,

with = /(8 g2) and B() = 2.4 [ln(2/)]1 for < 102; further constraint from CMB (cf. Nakayama et al ’10), yields 3 ⇥ 1015 GeV . vBL . 7 ⇥ 1015 GeV , 104 . p . 101 . Final choice for range of parameters (analysis within FN flavour model): vBL = 5 ⇥ 1015 GeV , 105 eV  e m1  1 eV , 109 GeV  M1  3 ⇥ 1012 GeV . (range of e m1: uncertainty of O(1) parameters)

6

Tachyonic Preheating

Hybrid inflation ends at critical value Φc of inflaton field Φ by rapid growth

• f fluctuations of BL Higgs field S0 (‘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 Occupation number: nk k/m

Bosons: Lattice Tanh Fermions: Lattice Tanh

in addition, particles which couple to S0 are produced by rapid increase of ‘waterfall field’ (Garcia-Bellido, Morales ’02); no coherent oscillations!

7

Decay of false vacuum produces long wave-length -modes, true vacuum reached at time tPH (even faster decay with inflaton dynamics), h02i

• t=tPH

= 2v2

BL ,

tPH ' 1 2m ln ✓32⇡2

• ◆

. Initial state: nonrelativistic gas of -bosons, N2,3, ˜ N2,3, A, ˜ A, C (contained in superfield Z), ... ; energy fractions (↵ = mX/mS, ⇢0 = v4

BL/4):

⇢B/⇢0 ' 2 ⇥ 103 gs f(↵, 1.3) , ⇢F/⇢0 ' 1.5 ⇥ 103 gs f(↵, 0.8) . Time evolution: rapid N2,3, ˜ N2,3, A, ˜ A, C decays, yields initial radiation, thermal N1’s and gravitinos; decays produce nonthermal N1’s; N1 decays produce most of radiation and baryon asymmetry; details of evolution described by Boltzmann equations.

8

Reheating Process

Major work: solve network of Boltzmann equations for all (super)particles; treat nonthermal and thermal contributions differently, varying equation

• f 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 , e m1 = 4.0 ⇥ 102 eV , m e

G = 100 GeV ,

m˜

g = 1 TeV ;

Gµ = 2.0 ⇥ 107 fixes (within FN flavour model) all other masses, CP asymmetries etc. Note: emergence of temperature plateau at intermediate times; final result: ⌘B ' 3.7 ⇥ 109 ' ⌘nt

B ,

Ω e

Gh2 ' 0.11 ,

i.e., dynamical realization of original conjecture.

9

Thermal and nonthermal number densities

aRH

i

aRH aRH

f

s+y+f N2,3+N é

2,3

N1

nt+N

é

1 nt

N1

th+N

é

1 th

2N1

eq

R B - L G é

100 101 102 103 104 105 106 107 108 1025 1030 1035 1040 1045 1050 10-1 100 101 102 103 Scale factor a abs NHaL Inverse temperature M1 êT

Comoving number densites of thermal and nonthermla N 0

1s,..., BL,

gravitinos and radiation as functions of scale factor a.

10

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 THaL @GeVD Inverse temperature M1 êT

Gravitino abundance can be understood from ‘standard formula’ and effective ‘reheating temperature’ (determined by neutrino masses).

11

• III. Gravitinos & Dark Matter

Thermal production of gravitinos is origin of DM; depending on pattern

• f SUSY breaking, gravitino DM or higgsino/wino DM. Mass spectrum of

superparticles motivated ‘large’ Higgs mass measured at the LHC, mLSP ⌧ msquark,slepton ⌧ m e

G .

LSP is typically ‘pure’ wino or higgsino (bino disfavoured, overproduction in thermal freeze-out), almost mass degenerate with chargino. Thermal abundance of wino ( e w) or higgsino (e 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

e w,e hh2 = c e w,e h

✓ m e

w,e h

1 TeV ◆2 , c e

w = 0.014 ,

ce

h = 0.10 ,

Heavy gravitinos (10 TeV . . . 103 TeV) consistent with BBN, ⌧ e

G ' 24 ⇥

12

(10 TeV/m e

G)3sec. Total higgsino/wino abundance

Ω e

w,e hh2 = Ω e G e w,e hh2 + Ωth e w,e hh2 ,

Ω

e G LSPh2 = mLSP

m e

G

Ω e

Gh2 ' 2.7 ⇥ 102 ⇣ mLSP

100 GeV ⌘ ✓TRH(M1, e m1) 1010 GeV ◆ , with ‘reheating temperature’ determined by neutino masses (takes reheating process into account), TRH ' 1.3 ⇥ 1010 GeV ✓ e 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 e m1).

13

4He

D WLSP > WDM

• bs

10-4 10-2 100 m é

1 @eVD

101 102 103 109 1010 1011 mG

é @TeVD

TRH @GeVD

Wh

é > WDM

• bs

Ww

é > WDM

• bs

w é h é G é

10-5 10-4 10-3 10-2 10-1 100 500 1000 1500 2000 2500 3000 m é

1 @eVD

mLSP @GeVD 100 ¥ mG

é @TeVD

For each ‘reheating temperature’, i.e. pair (M1, e 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

• n e

m1, ‘reheating temperature’ (right panel); more stringent for higgsino mass, since freeze-out contribution larger. E.g., m1 = 0.05 eV implies me

h <

⇠ 900 GeV, m e

G & 10 TeV.

14

• 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 BL breaking: prediction

• f GW spectrum with all contributions!

Perturbations in flat FRW background, ds2 = a2(⌧)(⌘µ⌫ + hµ⌫)dxµdx⌫ , ¯ hµ⌫ = hµ⌫ 1 2⌘µ⌫h⇢

⇢ ,

determined by linearized Einstein equations, ¯ h00

µ⌫(x, ⌧) + 2a0

a ¯ h0

µ⌫(x, ⌧) r2 x¯

hµ⌫(x, ⌧) = 16⇡GTµ⌫(x, ⌧) . Spectrum of GW background, ΩGW(k, ⌧) = 1 ⇢c @⇢GW(k, ⌧) @ ln k ,

15

Z 1

1

d ln k@⇢GW(k, ⌧) @ ln k = 1 32⇡G D ˙ hij (x, ⌧) ˙ hij (x, ⌧) E ; 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, ⌧) = ∆2

t

12 k2 a2

0H2

T 2

k(⌧)

= ∆2

t

12 Ωr gk

⇤

g0

⇤

g0

⇤,s

gk

⇤,s

!4/3 8 > < > :

1 2 (keq/k)2 ,

k0 ⌧ k ⌧ keq 1 , keq ⌧ k ⌧ kRH

1 2 C3 RH (kRH/k)2

kRH ⌧ k ⌧ kPH

16

with boundary frequences (f = k/(2⇡a0)), f0 = 3.58 ⇥ 1019 Hz ✓ h 0.70 ◆ , feq = 1.57 ⇥ 1017 Hz ✓Ωmh2 0.14 ◆ , fRH = 4.25 ⇥ 101 Hz ✓ TRH 107 GeV ◆ , fPH = 1.99 ⇥ 104 Hz ✓ 104 ◆1/6 ✓ vBL 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

with characteristic scalar and vector scales ⇣ R(s)

PH

⌘1 = ( vBL | ˙ c|)1/3 , ⇣ R(v)

PH

⌘1 ⇠ mZ = 2 p 2 g vBL . Contribution from cosmic strings (Abelian Higgs): ΩGW(k) ' 1 6⇡2F r ✓vBL MPl ◆4 Ωrh2 8 < : (keq/k)2, k0 ⌧ k ⌧ keq 1, keq ⌧ k ⌧ kRH (kRH/k)2, kRH ⌧ k constant F r recently determined in numerical simulation (cf. Figueroa, Hindmarsh,

Urrestilla ’12). Result similar to contribution from inflation, but very different

normalization!

18

inflation cosmic strings f0 feq fRH fPH fPH

HsL

fPH

HvL

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 WGW h2 k@Mpc-1D

19

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) = ↵ti ΓGµ(t ti) ; rate for amplitude h and frequency f (cf. Kuroyanagi et al ’12), d2R dzdh(f, h, z) ' 3 4 ✓2

m

(1 + z)(↵ + ΓGµ)h 1 ↵2t4(z) dV (z) dz Θ(1 ✓m) ; 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

Abelian-Higgs Strings vs. Nambu-Goto Strings

a =10-6 a =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 10 20 10 25

f@HzD WGW h2 k@Mpc-1D

21

Observational Prospects

H7L H8L H3L H4L H5L H6L H1L H2L H9L 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 10 20 10 25

f@HzD WGW h2 k@Mpc-1D

22

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

• f

relic gravitational wave background, may provide determination of reheating temperature

23

(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 m é

1 @eVD

M1 @GeVD

hBHm é

1,M1L

vB-L = 5.0 ¥ 1015 GeV Inflation & strings hB

nt = hB th

hB

th > hB

• bs

hB

nt > hB

• bs

hB < hB

• bs

Upper bound on M1 from inflation; lower bound from baryogenesis

24

Gravitino Dark Matter vs. leptogenesis

2¥1010 3¥1010 5¥1010 7¥1010 7¥1010 1¥1011 2¥1011

10-5 10-4 10-3 10-2 10-1 100 5 10 20 50 100 200 500 m é

1 @eVD

mG

é @GeVD

M1 @GeVD such that WG

é h2 = 0.11

vB-L = 5.0 ¥ 1015 GeV mg

é = 1 TeV

M1 @GeVD hB

nt > hB

• bs

hB < hB

• bs

Gravitino mass range: 10 GeV < ⇠ m e

G <

⇠ 700 GeV; heavy neutrino mass range: 2 ⇥ 1010 GeV < ⇠ M1 < ⇠ 2 ⇥ 1011 GeV (more stringent than inflation)

25