-
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
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
Outline
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Mark Hindmarsh DM from Decaying TDs
-
IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
“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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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)
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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 ).
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Increasing qX qX = 0
(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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
Numerical solutions: (n,p) = (0,1/2), (1,1/2)
Increasing qX qX = 0
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
Increasing qX qX = 0
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
Numerical solutions: summary
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
Increasing qX qX = 0
(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
Increasing qX qX = 0
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
Increasing qX qX = 0
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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 )
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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 -
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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 -
Mark Hindmarsh DM from Decaying TDs
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IntroductionTD models of dark matter production
Dark Matter and Boltzmann equation with sourceSolutions
Scenarios and constraintsSummary
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
Mark Hindmarsh DM from Decaying TDs
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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
)].
Mark Hindmarsh DM from Decaying TDs
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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,
Mark Hindmarsh DM from Decaying TDs
IntroductionTD models of dark matter productionDark Matter and
Boltzmann equation with sourceSolutionsScenarios and
constraintsSummaryAppendixAppendix