Gravitational waves from cosmic string - domain wall
networks
Ken’ichi SaikawaICRR, The University of Tokyo
Collaborate with T. Hiramatsu (YITP), M. Kawasaki (ICRR) and T. Sekiguchi (Nagoya U.)
28 September 2011, JGRG21 (Tohoku U.) 1/132011年9月30日金曜日
• Cosmic string ex.) U(1) → I
• Domain wall ex.) Z2 → I
Cosmic strings and domain walls
complex scalar field
real scalar field
field space
field space
coordinate space
coordinate space
28 September 2011, JGRG21 (Tohoku U.) 2/132011年9月30日金曜日
• Sequence of symmetry breaking
• ex.) U(1) → ZN → I
• Domain walls bounded by strings
• Naturally arise in some particle physics models (such as axions)
Hybrid defects
string
N discrete vacua at
28 September 2011, JGRG21 (Tohoku U.) 3/132011年9月30日金曜日
Axion cosmology and topological defects
• Axion : Pseudo-Nambu-Goldstone boson from spontaneous breaking of U(1)PQ
• Peccei-Quinn (PQ) field
•
• Spontaneous breaking of U(1)PQ
• Formation of cosmic strings
•
• Axion acquires a mass (QCD effect)
• ( )
• Spontaneous breaking of ZNDW
• Formation of domain walls28 September 2011, JGRG21 (Tohoku U.) 4/13
2011年9月30日金曜日
Axionic domain wall problem
• Domain wall number NDW
• If NDW>1, string-wall networks are stable
• come to overclose the universe (domain wall problem)
• However, it can be avoided if we introduce a bias (unstable domain walls)
• Existence of string-wall networks in the early universe
• Source of the stochastic gravitational wave backgrounds observed today
: depend on models
28 September 2011, JGRG21 (Tohoku U.) 5/132011年9月30日金曜日
• Solve the classical field equations for on 3D lattice
• Input parameters
Numerical simulations
Scheme Leap frog
Number of grids 256×256×256
Era Radiation dom.
Initial time 2 ( in unit of )
Final time 25
Time resolution 0.01
Box size 60
0.1
0.1
( : conformal time )
28 September 2011, JGRG21 (Tohoku U.) 6/132011年9月30日金曜日
• Solve the classical field equations for on 3D lattice
• Input parameters
Numerical simulations
Scheme Leap frog
Number of grids 256×256×256
Era Radiation dom.
Initial time 2 ( in unit of )
Final time 25
Time resolution 0.01
Box size 60
0.1
0.1
( : conformal time )
28 September 2011, JGRG21 (Tohoku U.) 6/132011年9月30日金曜日
• Solve the classical field equations for on 3D lattice
• Input parameters
Numerical simulations
Scheme Leap frog
Number of grids 256×256×256
Era Radiation dom.
Initial time 2 ( in unit of )
Final time 25
Time resolution 0.01
Box size 60
0.1
0.1
( : conformal time )
28 September 2011, JGRG21 (Tohoku U.) 6/132011年9月30日金曜日
Calculation of GW spectrum
• Linearized theory
• “Green function” method J. Dufaux, et al., PRD76, 123517 (2007)
28 September 2011, JGRG21 (Tohoku U.) 7/132011年9月30日金曜日
107
108
109
1010
10-1 100
S k
k [in unit of ]
NDW = 2NDW = 3NDW = 4NDW = 5NDW = 6
GW spectrum
• NDW>2 : spectrum similar to that of simple Z2 model with real scalar field
• Nearly flat spectrum extends between two relevant scales (Hubble radius and wall width)
• NDW = 2 : different from others
see M. Kawasaki and KS, JCAP09(2011)008
28 September 2011, JGRG21 (Tohoku U.) 8/132011年9月30日金曜日
Field configuration in NDW = 2
• Vstring distributs like a wall rather than string
• Width is determined by the size of strings
• Produced only if and
28 September 2011, JGRG21 (Tohoku U.) 9/13
( axion model : )
spectrum extends to higher frequency
NDW = 2, = 0.1, m = 0.1
0
0.005
0.01
0.015
0.02
0.025NDW = 3, = 0.1, m = 0.1
0
0.005
0.01
0.015
0.02
0.025
2011年9月30日金曜日
Termination of GW production• GW production is terminated when string-wall
networks collapse
• The collapse is caused by a bias
• Lifts degenerate vacua
• Phenomenologically parameterized as
• Affects as a pressure on walls annihilate them
• Life time of the networks
• The present GW energy density is obtained as
P. Sikivie, PRL48, 1156 (1982)
28 September 2011, JGRG21 (Tohoku U.) 10/13
: the surface energydensity of domain walls
2011年9月30日金曜日
Constraints for bias parameter
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Overproduction of axionsfrom domain wall decay
r : fraction of the wall energy which goes into axion radiations
The white region is still allowed
T. Hiramatsu, M. Kawasaki and KS, JCAP08(2011)030
28 September 2011, JGRG21 (Tohoku U.) 11/132011年9月30日金曜日
NDW = 1
• NDW =1 ; walls quickly disappear (no domain wall problem)
• Strings decay due to the domain wall tension
• Axions produced by the decay contribute to the cold dark matter (CDM) component of the universe
• Estimation of CDM abundance [work in progress]28 September 2011, JGRG21 (Tohoku U.) 12/13
2011年9月30日金曜日
• Calculate GW spectrum produced by string-domain wall networks
• Based on 3D lattice simulations
• Spectrum is similar to that calculated in simple model in which discrete Z2 symmetry is spontaneously broken(except the model with NDW=2)
• For axion models, the signal can be observed in future GW interferometers
• Estimation of axion CDM abundance produced by the decay of networks is left as future works
Summary
28 September 2011, JGRG21 (Tohoku U.) 13/132011年9月30日金曜日
Appendix
28 September 2011, JGRG21 (Tohoku U.)2011年9月30日金曜日
• Treat and as two independent real scalar fields with correlation function
• No correlation in the k space
• Generate as Gaussian with
• Fourier transform to obtain and
Initial Conditions
: volume of the simulation box
28 September 2011, JGRG21 (Tohoku U.)2011年9月30日金曜日
Comments on the numerical study
• One must consider three extremely different length scales
• Core of the string
• Width of the wall
• Hubble radius
• At the final time of the simulation, the core of the string is marginally resolvable
at: lattice spacing
28 September 2011, JGRG21 (Tohoku U.)2011年9月30日金曜日
Effect on Big Bang Nucleosynthesis (BBN)
• Domain walls dominates the energy density of the universe at the temperature
• During the BBN epoch, domain walls contributes as an extra particle d.o.f.
• Observations indicate
• However, the contribution from domain walls is negligible
The wall domination occurs after the BBN epoch
28 September 2011, JGRG21 (Tohoku U.)2011年9月30日金曜日
Magnitude of GW• Suppose that gravitational waves are generated at
some length scale
• The energy of gravitational waves
• The energy density of gravitational wave
• define the efficiency parameter
where : the mass energy of domain walls
: the surface energy density of domain walls
: energy density of gravitational wavescalculated in numerical simulations
does not depend on the scale
28 September 2011, JGRG21 (Tohoku U.)2011年9月30日金曜日
Efficiency and area density
•
• Scaling solution :
• The amplitude of GWs is enhanced if NDW is large
10-3
10-2
10-1
10
Area
/ Vo
lum
e
conformal time
NDW=2NDW=3NDW=4NDW=5NDW=6
28 September 2011, JGRG21 (Tohoku U.)
10-1
100
101
102
103
5 10 15 20 25
effic
ienc
y gw
conformal time
NDW = 2NDW = 3NDW = 4NDW = 5NDW = 6
2011年9月30日金曜日
Observable?
• Emission of GWs is terminated at
• Future experiments can detect signals probe axion models
Intensity
Spectrum extends from
to
cf. DECIGO
28 September 2011, JGRG21 (Tohoku U.)2011年9月30日金曜日
Schematics
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!"#!'
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!"#'
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)*+$
,-./0
Axionic domain wall (?)
28 September 2011, JGRG21 (Tohoku U.)2011年9月30日金曜日
Bias
•
• Two forces acting on domain walls
• Tension (straightens the wall)
• Pressure (collapses the wall)
• Pressure dominates when
• Decay time of walls
28 September 2011, JGRG21 (Tohoku U.)2011年9月30日金曜日
Bounds for
• Neutron Electric Dipole Moment (NEDM)
• The bias term spoils the PQ solution to the strong CP problem
• Shifts the value from zero
• Observation of NEDM Upper bound for
• Overclosure bound
• Wall dominates when
• Require Lower bound for
28 September 2011, JGRG21 (Tohoku U.)2011年9月30日金曜日
Cold axions from domain walls
• Decay of domain walls production of axions
• The fraction of the wall energy goes into axion radiations
• Radiated axions are barely relativistic with Lorents factor
• Abundance of cold axions from domain walls at the time of equality between matter and radiation
• Another lower bound on
• must be small
S. Chang, C. Hagmann and P. Sikivie, PRD59, 023505 (1999)
become CDM component of the universe
28 September 2011, JGRG21 (Tohoku U.)2011年9月30日金曜日