Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Dark Matter from Decaying Topological
Defects
Mark Hindmarsh1,2
1
Russell Kirk3
Stephen West3,4
Department of Physics & Astronomy
University of Sussex
2
Helsinki Institute of Physics
Helsinki University
3
Department of Physics & Astronomy
Royal Holloway, University of London
4
Rutherford Appleton Laboratory
COSMO 2013
MH, Kirk, West (in prep.)
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
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
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Dark Matter
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Strong evidence from multiple sources for dark matter (DM)
Planck + ΛCDM:1 ΩDM h2 = 0.1186 ± 0.0031
A leading candidate: weakly interacting massive particle (WIMP)
Standard thermal freeze-out:2
relic abundance ∼ (total annihilation cross-section)−1
Refinements and other production mechanisms:
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co-annihilation, near-threshold/resonant annihilation,3
Other production mechanisms
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freeze-in4
gravitino decay
and ...
1 Ade
et al 2013
1965; Lee, Weinberg 1977
3 Greist, Seckel 1991
4 Hall et al 2010
2 Zel’dovich
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
“Top-Down" production of particles
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BSM physics often also predicts extra symmetries
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spontaneous breaking at scale vd →
extra phase transitions at temperature T ' vd
phase transitions can produce topological defects:5
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cosmic strings
textures, semilocal strings, monopoles, necklaces
Decay of topological defects produces particles
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SM states (γ, e± , p, p̄, ν, ν̄) → cosmic rays, γ-ray background6
... and dark matter7
5 Kibble
1976
Bhattacharjee, Sigl 2000
7 Jeannerot, Zhang, Brandenberger 1999
6 Review:
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
TD = Topological Defect and Top-Down
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TDs decay into a new sector of particles X (branch. frac. fX )
X particles decay into stable states including DM
Energy injection rate per unit volume Q(t) ∼ t 4−p
Parameters of a TD model
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mass of DM particle mχ
energy density injection rate at T = Tα = mχ : Qα
exponent of power law p
average energy of X particles ĒX
average multiplicity of X decays Nχ
DM number injection rate per unit volume:
jχinj =
I
fX Nχ
Q
ĒX
Important combination: qX = Qα fX /ρα Hα
(ρ - energy density, H - Hubble rate, evaluated at Tα )
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Cosmic string TD models
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Strings decay into particles and gravitational radiation
Branching fractions uncertain and model-dependent
Strings parametrised by mass per unit length µ ' 2πvd2
Consider two string decay scenarios:
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A) Strings decay entirely into X particles
B) Strings decay mostly into g-radiation, small fraction X particles
c
X’=0
from string-antistring annihilation at cusps
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X-particle decay scenarios:
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X1) ĒX ∼ vd (X particle masses at symmetry-breaking scale)
X2) ĒX ∼ mχ (X particle masses at DM scale - e.g. Msusy )
NB Third string scenario: particles from final string loop collapse8
- subdominant contribution to particle production.
8 Jeannerot,
Zhang, Brandenberger 1999
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Boltmann equation with source
Number density of dark matter states nχ obeys
Nχ fX Q(t)
2
ṅχ + 3Hnχ = −hσχ v i nχ2 − nχ,eq
+
,
ĒX
I
hσχ v i: thermally-averaged dark matter annihilation cross section
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... weighted by v , relative speed of annihilating particles
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nχ,eq : equilibrium dark matter number density
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Yield equation
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Definitions:
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x = mχ /T (proportional to scale factor)
hσχ v i = σ0 x −n (s-wave: n = 0; p-wave: n = 1)
Dark matter yield Yχ = nχ /s (where s is entropy density)
Equation for yield:
dYχ
A
B
2
= − n+2 Yχ2 − Yχ,eq
+ 4−2p ,
dx
x
x
where
r
3
Nχ m χ
π
Qα f X
.
MPl mχ σ0 , B = xα2−2p
45
4
ρ α Hα
ĒX
√
Planck mass MPl = 1/ G ' 1.22 × 1019 GeV
A=
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Model parameters
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Recall that yield equation depends on two parameters
r
π
3
Nχ m χ
Qα f X
A=
MPl mχ σ0 , B = xα2−2p
.
45
4
ρ α Hα
ĒX
Define:
Nχ mχ
ĒX
qX = ρQααHfXα
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χ multiplicity parameter: νχ =
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X injection rate parameter:
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Scenario A: p = 1; Scenario B: p = 12 ;
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Take νχ ' 1 (X particle decay scenario X2)
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Derive constraints on qX for s-wave and p-wave annihilation
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Gives 4 models: (n, p) = (0, 1), (1, 1), (0, 12 ), (1, 12 ).
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Numerical solutions: (n, p) = (0, 1), (1, 1)
10 -8
(0,1)
10 -9
10 -9
10 -10
10 -10
10
-11
10
-12
Yield
Yield
10 -8
Increasing q X
qX = 0
10 -13
10 -14
(1,1)
10 -11
10 -12
Increasing q X
qX = 0
10 -13
10
20
50
100
x = m Χ T
200
500
1000
10 -14
10
20
50
100
x = m Χ T
mχ = 500 GeV, ĒX = 1 TeV, Nχ = 1 GeV (νχ = 0.5),
(n, p) = (0, 1): σ0 = 1.6 × 10−26 cm3 s−1
(n, p) = (1, 1): σ0 = 7.0 × 10−25 cm3 s−1
Coloured lines: qX = 0, 10−9 , 10−8 , 10−7
Solid black line: equilibrium yield
Mark Hindmarsh
DM from Decaying TDs
200
500
1000
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Numerical solutions: (n, p) = (0, 1/2), (1, 1/2)
H0, 1  2)
10 -8
10 -9
10 -10
10 -10
Yield
Yield
10
10 -11
10
10
-12
Increasing q X
qX = 0
10 -11
10 -12
qX = 0
Increasing q X
10 -13
-13
10 -14
H1, 1  2)
10 -8
-9
10
20
50
100
x = m Χ T
200
500
1000
10 -14
10
20
50
mχ = 500 GeV, νχ = 0.5,
(n, p) = (0, 12 ): σ0 = 1.6 × 10−26 cm3 s−1
(n, p) = (1, 12 ): σ0 = 7.0 × 10−25 cm3 s−1
qX = 0, 10−9 , 10−8 , 10−7 are plotted.
Solid black line: equilibrium yield
Mark Hindmarsh
DM from Decaying TDs
100
x = m Χ T
200
500
1000
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Numerical solutions: summary
10 -8
(0,1)
10 -9
10 -9
10 -10
10 -10
10
Yield
Yield
10 -8
-11
10 -12
Increasing q X
10
20
50
100
x = m Χ T
200
500
10 -14
1000
H0, 1  2)
10 -10
10 -10
Yield
10 -9
10 -11
10 -12
Increasing q X
qX = 0
qX = 0
20
50
100
x = m Χ T
200
500
1000
H1, 1  2)
10 -11
10 -12
qX = 0
Increasing q X
10 -13
-13
10 -14
10
10 -8
10 -9
10
Increasing q X
10 -13
10 -8
Yield
10 -11
10 -12
qX = 0
10 -13
10 -14
(1,1)
10
20
50
100
x = m Χ T
200
500
1000
Mark Hindmarsh
10 -14
10
20
50
100
DM from Decaying TDs x = m Χ  T
200
500
1000
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Comments on numerical solutions
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Increasing in asymptotic yield with increasing qX (expected)
dY
A
B
2
+ x 4−2p
Yχ2 − Yχ,eq
,
Recall yield eqn: dxχ = − x n+2
post freeze-out behaviour depends on sign of (n + 2) − (4 − 2p)
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( n + 2 > 4 − 2p ) source drops less quickly than annihilation term
– relic density dominated by source decays after freeze-out
e.g. (n, p) = (1, 1)
( n + 2 < 4 − 2p ) source drops more quickly than annihilation term
– relic density close to ordinary freeze-out
e.g. (n, p) = (0, 21 )
( n + 2 = 4 − 2p ) source and annihilation
terms drop at same rate
p
– rapid asymptote to Yχ (∞) = B/A
e.g. (n, p) = (0, 1), (1, 21 )
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Fitting to Planck dark matter abundance
-4
(0, 1/2)
Log@q X D
-6
-8
(1, 1)
(0, 1)
(1, 1/2)
-10
-12
-26
-25
-24
Log@Σ 0  cm 3 s -1 D
Mark Hindmarsh
-23
-22
-21
DM from Decaying TDs
-20
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Comments on fit to Planck dark matter abundance
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Large qX : power-law
relationship – final yield still
depends on DM annihilation
cross-sectiona
-4
Slope of curve depends on
(n, p)
a Incorrect
(0, 1/2)
-6
Small qX : yield asymptotes to
ordinary freeze-out value and
becomes independent of source
Log@q X D
I
-8
(1, 1/2)
-10
-12
-26
-25
-24
to integrate source from
freeze-out
Mark Hindmarsh
(1, 1)
(0, 1)
DM from Decaying TDs
Log@Σ 0  cm 3 s -1 D
-23
-22
-21
-20
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Analytic solution: Ricatti equation
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As x gets large, Yχ,eq → 0 and Boltzmann equation can be
approximated as
A
B
dYχ
= − n+2 Yχ2 + 4−2p .
dx
x
x
Ricatti equation form - exact solution available.
α
β
B α+β Γ α+β
β−α
In large qX limit: Yχ (∞) ≈ (α + β) α+β
β
α
A α+β Γ α+β
where α = n + 1 and β = 3 − 2p.
p
e.g. n + 2 = 4 − 2p gives Yχ (∞) ' B/A as above
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Comparison: analytic and numerical (n, p) = (1, 1)
-7
-8
Log@q X D
-9
-10
-11
-12
-13
-25
-23
Log@Σ 0  cm3 s -1D
Mark Hindmarsh
-21
-19
DM from Decaying TDs
-17
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Unitarity limit
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9
Annihilation cross-section
√ constrained:
4(2n + 1) πxd
hσvrel i ≤
mχ2
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Sourced freeze-out temperature xd defined by
Yχ (xd ) − Yχ,eq (xd ) ≈ cYχ,eq (xd ) with c = O(1).
9 Griest,
Kamionkowski 1990
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Indirect Fermi-LAT limit (model-dependent)
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Searches for γ continuum in dwarf galaxies give
model-dependent limits to DM density10
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Assumptions in representative model:
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s-wave annihilation (n = 0)11
χχ → WW
10 Fermi-LAT
2011, Drlica-Wagner (talk) 2012
on p-wave annihilation (n = 1) much weaker due to v -dependence of
11 Constraints
annihilation
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Diffuse γ-ray background (model-dependent)
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X particles may also decay into SM particles
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Cascade decays to γ, e± , p, p̄, ν, ν̄
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Interaction with cosmic backgrounds, magnetic fields
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Result: cosmic rays, γ-ray background (GRB)12
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Observed GRB limits energy injection rate into EM cascade
today13
Q0 < 2.2 × 10−23 (3p − 1)h eV cm−3 s−1
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No significant constraints for p < 1 (Q decays too quickly)
12 Review:
13 Sigl,
Bhattacharjee, Sigl 2000
Lee, Bhattacharjee, Yoshida 1998, using EGRET data
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Constraints
-4
-7
(0, 1)
(1, 1)
-8
-6
Log@q X D
Log@q X D
-9
-8
-10
-10
-11
-12
-12
-14
-13
-28
(n, p)
(0, 1)
(1, 1)
(0, 1/2)
(1, 1/2)
-26
Log@Σ 0  cm 3 s -1 D
-24
-22
Unitarity
qX . 4.6 × 10−6
qX . 2.0 × 10−8
qX . 19
qX . 3.8 × 10−4
-20
-25
-23
Fermi-LAT
qX . 2.3 × 10−9
qX . 6.1 × 10−6
-
Mark Hindmarsh
DM from Decaying TDs
Log@Σ 0  cm 3 s -1 D
-21
-19
-17
EGRET
qX . 2.4 × 10−9
qX . 2.4 × 10−9
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Cosmic string models
I
String mass per unit length µ ' 2πvd2 .
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String density ρd , average equation of state wd , density
parameter Ωd = ρd /ρ.
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Numerical simulations: wd ' 0
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Total energy injection rate into (particles) + (gravitational
radiation): Q
Q
Conservation of energy: ρH
= 3(w − wd )Ωd ' 32 Ωd
I
Mark Hindmarsh
DM from Decaying TDs
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Constraints on cosmic string scenarios
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A: constant fraction fX ∼ 1 into X particles, p = 1
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Numerical simulations: Ωd ' 5.3(8πGµ)
2
qX ' 1016vdGeV 10−3
(n, p)
(0, 1)
(1, 1)
I
Unitarity
vd < 7.1 · 1014 GeV
vd < 4.7 · 1013 GeV
Fermi-LAT
vd < 1.6 · 1013 GeV
-
EGRET
vd < 1.6 · 1013 GeV
vd < 1.6 · 1013 GeV
B: subdominant X emission from cusps on string loops, p =
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1
2
Main loop decay channel gravitational waves, power Pg = ΓGµ2
Lower µ → higher loop density → more cusps → more particles
16
5
2
mχ qX = E 10 v GeV
10−11 (E = O(1) parameter combination)
TeV
d
(n, p)
(0, 1/2)
(1, 1/2)
Unitarity
2
vd > 2.1 · 1010 E 5 GeV
2
vd > 1.6 · 1012 E 5 GeV
Mark Hindmarsh
Fermi-LAT
2
vd > 8.3 · 1012 E 5 GeV
DM from Decaying TDs
EGRET
Introduction
TD models of dark matter production
Dark Matter and Boltzmann equation with source
Solutions
Scenarios and constraints
Summary
Summary
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Dark matter produced “top-down” by decaying topological defects
Analytic formula for DM yield in TD scenarios
Depends on
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(qX , σ0 ) parameter space consistent with Planck relic density
Constraints on cosmic strings from unitarity, indirect detection
(c.f. GRB)
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DM particle mass mχ , annihilation cross-section parameter σ0
DM multiplicity parameter: νχ = Nχ mχ /ĒX
X injection rate parameter: qX = Qα fX /ρα Hα
Scenario A: upper bounds on vd
Scenario B: lower bounds on vd
Outlook: specific models
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Combine direct detection, collider limits, cosmic rays, g-waves
New predictions for indirect detection
New limits for TDs
Mark Hindmarsh
DM from Decaying TDs
Appendix
Back-up slide A.1
dYχ
dx
A
= − x n+2
Yχ2 +
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Ricatti equation
I
Exact asymptotic solution
α
β
α+β
B α+β Γ
β−α
Yχ (∞) ≈ (α + β) α+β
A
Γ
B
,
x 4−2p
β
α+β
α
α+β
√
2 AB
(α+β)/2
(α+β)xd
√
2 AB
(α+β)/2
(α+β)xd
I −α
α+β
α
I α+β
,
where α = n + 1 and β = 3 − 2p,
I
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xd defined as sourced freeze-out temperature:
Yχ (xd ) − Yχ,eq (xd ) = cYχ,eq (xd ), with c = O(1) chosen to fit
numerical solutions.
Iterative solution: xd ≈ log[Ac(c + 2)k ] − n + 21 log[Ac(c + 2)k ] −
q
h i
4Ac(c+2)B
log 21 1 + 1 + (log[Ac(c+2)k
.
])6+n−2p
Mark Hindmarsh
DM from Decaying TDs
Appendix
Back-up slide A.2
I
Loop number density distribution: n(`, t) =
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5
ν = O(1) constant
β = ΓGµ, with Γ ∼ 102 (gravitational radiation efficiency)
√
Cusp emission power: Pc = βc µ/ vd `
q
R∞
Energy injection rate: Qc = 0 d`βc µ v1d ` n(`, t)
2 41 2 12
Tα
π g
Qc qX = ρH
.
' ββc2ν mµ2 90
m P vd
Tα
I
ν
3
t 2 (`+βt) 2
qX ∼
βc ν
Γ2100
10
P
16
GeV
vd
52
Tα
TeV
10−11 ,
Mark Hindmarsh
DM from Decaying TDs