Dark Matter from Decaying Topological Defects · 2013. 9. 9. · Introduction TD models of dark matter production Dark Matter and Boltzmann equation with source Solutions Scenarios

IntroductionTD models of dark matter production

Dark Matter and Boltzmann equation with sourceSolutions

Scenarios and constraintsSummary

Dark Matter from Decaying TopologicalDefects

Mark Hindmarsh1,2 Russell Kirk3 Stephen West3,4

1Department of Physics & AstronomyUniversity of Sussex

2Helsinki Institute of PhysicsHelsinki University

3Department of Physics & AstronomyRoyal Holloway, University of London

4Rutherford Appleton Laboratory

COSMO 2013MH, Kirk, West (in prep.)

Mark Hindmarsh DM from Decaying TDs




Outline

Introduction

TD models of dark matter production

Dark Matter and Boltzmann equation with source

Solutions

Scenarios and constraints





Dark Matter

I Strong evidence from multiple sources for dark matter (DM)I Planck + ΛCDM:1 ΩDMh2 = 0.1186± 0.0031I A leading candidate: weakly interacting massive particle (WIMP)I Standard thermal freeze-out:2

relic abundance ∼ (total annihilation cross-section)−1I Refinements and other production mechanisms:

I co-annihilation, near-threshold/resonant annihilation,3

I Other production mechanismsI freeze-in4I gravitino decayI and ...

1Ade et al 20132Zel’dovich 1965; Lee, Weinberg 19773Greist, Seckel 19914Hall et al 2010





“Top-Down" production of particles

I BSM physics often also predicts extra symmetriesI spontaneous breaking at scale vd →

extra phase transitions at temperature T ' vdI phase transitions can produce topological defects:5

I cosmic stringsI textures, semilocal strings, monopoles, necklaces

I Decay of topological defects produces particlesI SM states (γ, e±, p, p̄, ν, ν̄)→ cosmic rays, γ-ray background6I ... and dark matter7

5Kibble 19766Review: Bhattacharjee, Sigl 20007Jeannerot, Zhang, Brandenberger 1999





TD = Topological Defect and Top-DownI TDs decay into a new sector of particles X (branch. frac. fX )I X particles decay into stable states including DMI Energy injection rate per unit volume Q(t) ∼ t4−pI Parameters of a TD model

I mass of DM particle mχI energy density injection rate at T = Tα = mχ: QαI exponent of power law pI average energy of X particles ĒXI average multiplicity of X decays Nχ

I DM number injection rate per unit volume:

j injχ =fX NχĒX

Q

I Important combination: qX = QαfX/ραHα(ρ - energy density, H - Hubble rate, evaluated at Tα)





Cosmic string TD models

I Strings decay into particles and gravitational radiationI Branching fractions uncertain and model-dependentI Strings parametrised by mass per unit length µ ' 2πv2dI Consider two string decay scenarios:

I A) Strings decay entirely into X particlesI B) Strings decay mostly into g-radiation, small fraction X particles

from string-antistring annihilation at cusps

c

X’=0

I X-particle decay scenarios:I X1) ĒX ∼ vd (X particle masses at symmetry-breaking scale)I X2) ĒX ∼ mχ (X particle masses at DM scale - e.g. Msusy)

I NB Third string scenario: particles from final string loop collapse8

- subdominant contribution to particle production.8Jeannerot, Zhang, Brandenberger 1999





Boltmann equation with source

Number density of dark matter states nχ obeys

ṅχ + 3Hnχ = −〈σχv〉(n2χ − n2χ,eq

)+

NχfX Q(t)ĒX

,

I 〈σχv〉: thermally-averaged dark matter annihilation cross sectionI ... weighted by v , relative speed of annihilating particlesI nχ,eq: equilibrium dark matter number density





Yield equation

I Definitions:I x = mχ/T (proportional to scale factor)I 〈σχv〉 = σ0x−n (s-wave: n = 0; p-wave: n = 1)I Dark matter yield Yχ = nχ/s (where s is entropy density)

I Equation for yield:

dYχdx

= − Axn+2

(Y 2χ − Y 2χ,eq

)+

Bx4−2p

,

where

A =√

π

45MPlmχσ0, B =

34

x2−2pα

(Nχmχ

ĒX

)(QαfXραHα

).

Planck mass MPl = 1/√

G ' 1.22× 1019 GeV





Model parameters

I Recall that yield equation depends on two parameters

A =√

π

45MPlmχσ0, B =

34

x2−2pα

(Nχmχ

ĒX

)(QαfXραHα

).

I Define:I χ multiplicity parameter: νχ =

NχmχĒX

I X injection rate parameter: qX = Qα fXραHαI Scenario A: p = 1; Scenario B: p = 12 ;I Take νχ ' 1 (X particle decay scenario X2)I Derive constraints on qX for s-wave and p-wave annihilationI Gives 4 models: (n,p) = (0,1), (1,1), (0, 12 ), (1,

12 ).





Numerical solutions: (n,p) = (0,1), (1,1)

Increasing qX qX = 0

(0,1)

10 20 50 100 200 500 100010-14

10-13

10-12

10-11

10-10

10-9

10-8

x = m Χ T

Yie

ld


(1,1)

10 20 50 100 200 500 100010-14

10-13

10-12

10-11

10-10

10-9

10-8

x = m Χ TY

ield

mχ = 500 GeV, ĒX = 1 TeV, Nχ = 1 GeV (νχ = 0.5),(n,p) = (0,1): σ0 = 1.6× 10−26 cm3s−1(n,p) = (1,1): σ0 = 7.0× 10−25 cm3s−1Coloured lines: qX = 0,10−9,10−8,10−7

Solid black line: equilibrium yield





Numerical solutions: (n,p) = (0,1/2), (1,1/2)


H0, 1 2)

10 20 50 100 200 500 100010-14

10-13

10-12

10-11

10-10

10-9

10-8

x = m Χ T

Yie

ld


H1, 1 2)

10 20 50 100 200 500 100010-14

10-13

10-12

10-11

10-10

10-9

10-8

x = m Χ TY

ield

mχ = 500 GeV, νχ = 0.5,(n,p) = (0, 12 ): σ0 = 1.6× 10

−26 cm3s−1

(n,p) = (1, 12 ): σ0 = 7.0× 10−25 cm3s−1

qX = 0,10−9,10−8,10−7 are plotted.Solid black line: equilibrium yield





Numerical solutions: summary


(0,1)

10 20 50 100 200 500 100010-14

10-13

10-12

10-11

10-10

10-9

10-8

x = m Χ T

Yie

ld


(1,1)

10 20 50 100 200 500 100010-14

10-13

10-12

10-11

10-10

10-9

10-8

x = m Χ T

Yie

ld


H0, 1 2)

10 20 50 100 200 500 100010-14

10-13

10-12

10-11

10-10

10-9

10-8

x = m Χ T

Yie

ld


H1, 1 2)

10 20 50 100 200 500 100010-14

10-13

10-12

10-11

10-10

10-9

10-8

x = m Χ T

Yie

ld





Comments on numerical solutions

I Increasing in asymptotic yield with increasing qX (expected)I Recall yield eqn: dYχdx = −

Axn+2

(Y 2χ − Y 2χ,eq

)+ Bx4−2p ,

post freeze-out behaviour depends on sign of (n + 2)− (4− 2p)I ( n + 2 > 4− 2p ) source drops less quickly than annihilation term

– relic density dominated by source decays after freeze-oute.g. (n, p) = (1, 1)

I ( n + 2 < 4− 2p ) source drops more quickly than annihilation term– relic density close to ordinary freeze-oute.g. (n, p) = (0, 12 )

I ( n + 2 = 4− 2p ) source and annihilation terms drop at same rate– rapid asymptote to Yχ(∞) =

√B/A

e.g. (n, p) = (0, 1), (1, 12 )





Fitting to Planck dark matter abundance

(0, 1/2)

(0, 1)

(1, 1/2)

(1, 1)

-26 -25 -24 -23 -22 -21 -20

-12

-10

-8

-6

-4

Log@Σ0 cm3 s-1D

Log

@q XD





Comments on fit to Planck dark matter abundance

I Large qX : power-lawrelationship – final yield stilldepends on DM annihilationcross-sectiona

I Small qX : yield asymptotes toordinary freeze-out value andbecomes independent of source

I Slope of curve depends on(n,p)

aIncorrect to integrate source fromfreeze-out

(0, 1/2)

(0, 1)

(1, 1/2)

(1, 1)

-26 -25 -24 -23 -22 -21 -20

-12

-10

-8

-6

-4

Log@Σ0 cm3 s-1DLog

@q XD





Analytic solution: Ricatti equation

I As x gets large, Yχ,eq → 0 and Boltzmann equation can beapproximated as

dYχdx

= − Axn+2

Y 2χ +B

x4−2p.

I Ricatti equation form - exact solution available.

I In large qX limit: Yχ(∞) ≈ (α + β)β−αα+β

Bαα+β Γ

(β

α+β

)A

βα+β Γ

(α

α+β

)where α = n + 1 and β = 3− 2p.

I e.g. n + 2 = 4− 2p gives Yχ(∞) '√

B/A as above





Comparison: analytic and numerical (n,p) = (1,1)

-25 -23 -21 -19 -17

-13

-12

-11

-10

-9

-8

-7

Log@Σ0 cm3s-1D

Log

@q XD





Unitarity limit

I Annihilation cross-section constrained:9

〈σvrel〉 ≤4(2n + 1)

√πxd

m2χI Sourced freeze-out temperature xd defined by

Yχ(xd )− Yχ,eq(xd ) ≈ cYχ,eq(xd ) with c = O(1).

9Griest, Kamionkowski 1990Mark Hindmarsh DM from Decaying TDs




Indirect Fermi-LAT limit (model-dependent)

I Searches for γ continuum in dwarf galaxies givemodel-dependent limits to DM density10

I Assumptions in representative model:I s-wave annihilation (n = 0)11I χχ→ WW

10Fermi-LAT 2011, Drlica-Wagner (talk) 201211Constraints on p-wave annihilation (n = 1) much weaker due to v -dependence of

annihilationMark Hindmarsh DM from Decaying TDs




Diffuse γ-ray background (model-dependent)

I X particles may also decay into SM particlesI Cascade decays to γ, e±, p, p̄, ν, ν̄I Interaction with cosmic backgrounds, magnetic fieldsI Result: cosmic rays, γ-ray background (GRB)12

I Observed GRB limits energy injection rate into EM cascadetoday13

Q0 < 2.2× 10−23(3p − 1)h eV cm−3s−1

I No significant constraints for p < 1 (Q decays too quickly)

12Review: Bhattacharjee, Sigl 200013Sigl, Lee, Bhattacharjee, Yoshida 1998, using EGRET data





Constraints

(0, 1)

-28 -26 -24 -22 -20

-14

-12

-10

-8

-6

-4

Log@Σ0 cm3s-1D

Log

@q XD

(1, 1)

-25 -23 -21 -19 -17

-13

-12

-11

-10

-9

-8

-7

Log@Σ0 cm3s-1DLog

@q XD

(n,p) Unitarity Fermi-LAT EGRET(0,1) qX . 4.6× 10−6 qX . 2.3× 10−9 qX . 2.4× 10−9(1,1) qX . 2.0× 10−8 - qX . 2.4× 10−9

(0,1/2) qX . 19 qX . 6.1× 10−6(1,1/2) qX . 3.8× 10−4 -





Cosmic string models

I String mass per unit length µ ' 2πv2d .I String density ρd, average equation of state wd, density

parameter Ωd = ρd/ρ.I Numerical simulations: wd ' 0I Total energy injection rate into (particles) + (gravitational

radiation): QI Conservation of energy: QρH = 3(w − wd)Ωd '

32 Ωd





Constraints on cosmic string scenarios

I A: constant fraction fX ∼ 1 into X particles, p = 1I Numerical simulations: Ωd ' 5.3(8πGµ)I qX '

( vd1016 GeV

)2 10−3(n, p) Unitarity Fermi-LAT EGRET(0, 1) vd < 7.1 · 1014 GeV vd < 1.6 · 1013 GeV vd < 1.6 · 1013 GeV(1, 1) vd < 4.7 · 1013 GeV - vd < 1.6 · 1013 GeV

I B: subdominant X emission from cusps on string loops, p = 12I Main loop decay channel gravitational waves, power Pg = ΓGµ2I Lower µ→ higher loop density→ more cusps→ more particlesI qX = E

(1016 GeV

vd

) 52 (mχ

TeV

)10−11 (E = O(1) parameter combination)

(n, p) Unitarity Fermi-LAT EGRET(0, 1/2) vd > 2.1 · 1010E

25 GeV vd > 8.3 · 1012E

25 GeV

(1, 1/2) vd > 1.6 · 1012E25 GeV -





Summary

I Dark matter produced “top-down” by decaying topological defectsI Analytic formula for DM yield in TD scenariosI Depends on

I DM particle mass mχ, annihilation cross-section parameter σ0I DM multiplicity parameter: νχ = Nχmχ/ĒXI X injection rate parameter: qX = QαfX/ραHα

I (qX , σ0) parameter space consistent with Planck relic densityI Constraints on cosmic strings from unitarity, indirect detection

(c.f. GRB)I Scenario A: upper bounds on vdI Scenario B: lower bounds on vd

I Outlook: specific modelsI Combine direct detection, collider limits, cosmic rays, g-wavesI New predictions for indirect detectionI New limits for TDs


Appendix

Back-up slide A.1

I Ricatti equation dYχdx = −A

xn+2 Y2χ +

Bx4−2p ,

I Exact asymptotic solution

Yχ(∞) ≈ (α + β)β−αα+β

Bαα+β Γ

(β

α+β

)I −αα+β

(2√

AB(α+β)x (α+β)/2d

)A

βα+β Γ

(α

α+β

)I αα+β

(2√

AB(α+β)x (α+β)/2d

) ,where α = n + 1 and β = 3− 2p,

I xd defined as sourced freeze-out temperature:Yχ(xd )− Yχ,eq(xd ) = cYχ,eq(xd ), with c = O(1) chosen to fitnumerical solutions.

I Iterative solution: xd ≈ log[Ac(c + 2)k ]−(n + 12

)log[Ac(c + 2)k ]−

log[

12

(1 +

√1 + 4Ac(c+2)B

(log[Ac(c+2)k ])6+n−2p

)].


Appendix

Back-up slide A.2

I Loop number density distribution: n(`, t) = νt

32 (`+βt)

52

I ν = O(1) constantI β = ΓGµ, with Γ ∼ 102 (gravitational radiation efficiency)

I Cusp emission power: Pc = βcµ/√

vd`

I Energy injection rate: Qc =∫∞

0 d`βcµ√

1vd`

n(`, t)

I qX = QcρH∣∣∣Tα' βcν

β2µ

m2P

(π2g90

) 14(

T 2αmPvd

) 12.

I qX ∼ βcνΓ2100

(1016 GeV

vd

) 52 ( Tα

TeV

)10−11,


IntroductionTD models of dark matter productionDark Matter and Boltzmann equation with sourceSolutionsScenarios and constraintsSummaryAppendixAppendix

Documents

Dark Matter from Decaying Topological Defects · 2013. 9. 9. · Introduction TD models of dark matter production Dark Matter and Boltzmann equation with source Solutions Scenarios