Anisotropies
from Collapsing
Textures
Introduction
Textures
Analytical
approach
CMB Anisotropies from Collapsing Textures
Work in collaboration with J. Urrestilla
Numerical
methods
Results
Kepa Sousa
Conclusions
Departamento de Fı́sica Teórica e Historia de la Ciencia
UPV/EHU.
[email protected]
University of Sussex, 27/01/2014
1 / 21
Outline
Anisotropies
from Collapsing
Textures
Introduction
1
Introduction
2
Textures
3
Analytical approach
4
Numerical methods
5
Results
6
Conclusions
Textures
Analytical
approach
Numerical
methods
Results
Conclusions
2 / 21
Introduction
Anisotropies
from Collapsing
Textures
Textures are a class of topological defects.
Introduction
Textures
Analytical
approach
Numerical
methods
Results
Conclusions
They are formed generically in cosmological models where a
non-abelian global symmetry is spontaneously broken.
Should the ΛCDM be augmented with textures?:
Cruz et al. ’08,’09 showed through bayesian analysis that a
cosmic texture is consistent with the CMB anomaly know as the
“Cold Spot”.
Other possibilities: Sunyaev-Zeldovic effect or a large void.
Other works show discrepancies: Feeney et al. ’12.
3 / 21
Introduction
Anisotropies
from Collapsing
Textures
Textures are a class of topological defects.
Introduction
Textures
Analytical
approach
Numerical
methods
Results
Conclusions
They are formed generically in cosmological models where a
non-abelian global symmetry is spontaneously broken.
Should the ΛCDM be augmented with textures?:
Cruz et al. ’08,’09 showed through bayesian analysis that a
cosmic texture is consistent with the CMB anomaly know as the
“Cold Spot”.
Other possibilities: Sunyaev-Zeldovic effect or a large void.
Other works show discrepancies: Feeney et al. ’12.
3 / 21
Introduction
Anisotropies
from Collapsing
Textures
Textures are a class of topological defects.
Introduction
Textures
Analytical
approach
Numerical
methods
Results
Conclusions
They are formed generically in cosmological models where a
non-abelian global symmetry is spontaneously broken.
Should the ΛCDM be augmented with textures?:
Cruz et al. ’08,’09 showed through bayesian analysis that a
cosmic texture is consistent with the CMB anomaly know as the
“Cold Spot”.
Other possibilities: Sunyaev-Zeldovic effect or a large void.
Other works show discrepancies: Feeney et al. ’12.
All these analysis rely on the existing predictions on the
anisotropy pattern produced by global textures.
They use a very idealized analytical solution, (spherical symmetry
and self similar collapse) Turok et al. ’90
which is known to be unrealistic Borrill et al. ’92.
3 / 21
Topological Defects
Kibble mechanism
Anisotropies
from Collapsing
Textures
Introduction
Topological defects form generically during phase transitions.
Example: Domain Walls
Textures
V!Φ"
0.8
Analytical
approach
0.6
0.4
Numerical
methods
ϕ=-1
0.2
Results
-1.5
-1
-0.5
0.5
1
1.5
ϕ=1
Φ
Conclusions
L = 21 ∂µ φ ∂ µ φ − λ(φφ − η 2 )2 .
The vacuum manifold is given by M = {−1, 1}.
The topology of M determines the type of defects which can form:
π0 (M) 6=
→ Domain Walls
π2 (M) 6=
→ Monopoles
π1 (M) 6=
π3 (M) 6=
→ Cosmic Strings
→ Textures
4 / 21
Textures
The O(4)−model
Anisotropies
from Collapsing
Textures
Introduction
Textures appear in theories where a non-abelian global symmetry
group G is completely broken at low energies.
Global SU(2) symmetry rotating a Higgs doublet.
Family symmetry.
Textures
Analytical
approach
Numerical
methods
These symmetries are easily implemented in GUT’s.
There are mechanisms where a non-abelian gauge symmetry
leads to a global symmetry at low energies. Turok 90
Results
Conclusions
The simplest example: O(4)−model
L = 21 ∂µ φa ∂ µ φa −λ(φa φa −η 2 )2 ,
2
a = 1, . . . , 4.
-1
2
0 Φ2
φ̈a − ∇ φa = −4λ(φb φb − η ) φa .
1
The model depends on a single parameter:
√
−1
lφ ≡ mφ
= ( 8λη)−1
1.5
1
0
Φ1
-1
1
V!Φ1 , Φ2 "
0.5
0
The vacuum manifold is M ∼
= S 3.
5 / 21
Textures
Topology of the Vacuum Manifold
Anisotropies
from Collapsing
Textures
Introduction
Textures
A texture is a localized region of space where the field φa winds
around M.
The Higgs lays on the vacuum manifold everywhere in space.
Analytical
approach
Numerical
methods
Results
Conclusions
φa φa = η 2 .
The texture configuration defines a mapping R3 −→ S 3 .
Since π3 (M) is non-trivial there are configurations which cannot
evolve into the homogeneous vacuum.
There is a conserved topological current:
µ
j =
µναβ abcd
1
φa
12π 2 η 4 Z
∂ν φb ∂α φd ∂β φd ,
Q=
d 3 x j 0 ∈ R.
The charge measures the fraction of M covered by the texture.
6 / 21
Textures
Spherically symmetric Q = 1 configuration
Anisotropies
from Collapsing
Textures
~ ≡ (φ1 , φ2 , φ3 ) = η sin χ(r ) rˆ,
φ
Introduction
Textures
χ(0) = 0,
Analytical
approach
Conclusions
χ(r > R) = π
=⇒
Q = 1.
!Φ1 ,Φ2 "
Numerical
methods
Results
φ4 = η cos χ(r ).
Φ4
1
12
10
0.5
8
0
6
4
-0.5
2
-1
2
4
6
8
10
12
0
2
4
6
8
10
12
14
7 / 21
Texture collapse and unwinding
Derrick’s theorem
Anisotropies
from Collapsing
Textures
Introduction
Textures
Analytical
approach
Numerical
methods
Results
Conclusions
Consider a localized texture configuration φa (~r ).
!Φ1 ,Φ2 "
!Φ1 ,Φ2 "
12
12
10
10
8
8
6
6
4
4
2
2
2
4
6
8
10
12
2
4
6
8
10
12
Transforming the configuration as φa (~r ) → φa (α~r ), with α > 1,
the energy decreases: textures are unstable to collapse.
E =
1
2
Z
~ a ∇φ
~ a → α−1 E
d 3 x ∇φ
When R . lφ the field gradients pull φa (~r ) over the potential barrier.
The topological charge is no longer conserved and the knot unwinds.
The energy escapes to infinity in an expanding shell of goldstone
bosons.
8 / 21
CMB Anisotropies by Collapsing Textures
Integrated Sachs-Wolfe effect
Anisotropies
from Collapsing
Textures
Introduction
Unwinding textures create a time-varying gravitational potential.
CMB photons crossing it receive a red- or a blue-shift.
Textures
Analytical
approach
Observer
Collapsing Texture
Last Scattering Surface
gµν
Numerical
methods
Θµν
Results
(0)
= gµν
+ hµν ,
= ∂µ φa ∂ν φa − gµν L
Conclusions
To first order in the perturbations
∆T
(n̂, x) = − 12
T
Z
λobs
dλ hij,0 (xγ (λ)) n̂i n̂j ,
xγ (λ) = x − λn̂
λem
Matter evolves in the unperturbed background,
photons travel along unperturbed trajectories.
9 / 21
CMB Anisotropies by Collapsing Textures
Small-angle approximation
Anisotropies
from Collapsing
Textures
Metric perturbations can be solved in terms of Θµν ,
the anisotropy can be calculated explicitely. Stebbins et al. 94
Introduction
Textures
Analytical
approach
Numerical
methods
Results
Conclusions
Small-angle approximation
When the geodesics are almost parallel to the line of sight n̂
Hindmarsh 94:
∆T
(x⊥ ) = −8πG ∇⊥ · U(x⊥ )
T
Z λobs
Θ0j (xγ (λ)) − n̂k Θjk (xγ (λ)) dλ
Ui (x⊥ ) = −(δij − n̂i n̂j )
∇2⊥
λem
Valid for fluctuations on small angular scales.
These results can be extended to the cosmological case provided
the impact parameter of the photons is small compared to H −1 ,
the unwinding time is small compared to H −1 .
10 / 21
CMB Anisotropies by Collapsing Textures
Cold Spot Formation
Anisotropies
from Collapsing
Textures
Photons crossing the texture before the unwinding are red-shifted.
Introduction
Textures
Analytical
approach
γ
They fall in the potential well of a
fraction of the texture,
tb
and climb out the potential well of
the full texture.
Numerical
methods
Results
Conclusions
t0
11 / 21
CMB Anisotropies by Collapsing Textures
Hot Spot Formation
Anisotropies
from Collapsing
Textures
Introduction
Photons crossing the texture after the unwinding are blue-shifted.
γ
Textures
Analytical
approach
Numerical
methods
t0
tb
They fall in the potential well of the
full texture,
Results
Conclusions
and climb out the potential well of a
fraction of the texture.
12 / 21
Analytical approach
Non-linear sigma model
Anisotropies
from Collapsing
Textures
At low energies the massive d.o.f. are not excited, thus φa (x) is
constrained to stay in M.
φa φa = η 2 .
Introduction
Textures
Analytical
approach
Numerical
methods
We use an effective action with a Lagrange multiplier β:
L = 21 ∂µ φa ∂ µ φa − β(φa φa − η 2 ).
Results
Conclusions
∇2 φa + η −2 (∂µ φb ∂ µ φb )φa = 0.
The effective e.o.m. admit an analytic solution describing a
spherically symmetric texture collapse Turok et al. 90
~ ≡ (φ1 , φ2 , φ3 ) = η sin χ rˆ,
φ
φ4 = η cos χ.
χ(r , t < 0) = 2 arctan(−r /t),
χ(r , t > 0) =
2 arctan(r /t) + π,
2 arctan(t/r ) + π,
r <t
r > t.
13 / 21
Analytical approach
Non-linear sigma model
Anisotropies
from Collapsing
Textures
Introduction
Textures
Analytical
approach
Numerical
methods
Results
Note that:
Turok’s solution extends all the way to spatial infinity,
and it has a linearly divergent mass:
M(r < Λ) ∼ 8πη 2 Λ
Conclusions
The small angle approximation is used implicitly:
The photon emission point,
the unwinding site,
and the observer
are at an infinite distance from each other.
14 / 21
Analytical approach
Problems of the Self-Similar solution
Anisotropies
from Collapsing
Textures
!T
"""""""
T
"T
#######
T
0.5
Introduction
!20
!10
"T
#######
T
1
1
10
!0.5
20
t0
10
0.8
!0.2
0.6
!0.4
0.4
Numerical
methods
Results
Conclusions
40
r
!0.8
Textures
Analytical
approach
30
!0.6
0.2
!1
20
10
20
30
40
r
!1
Template used to compare the CMB data with the texture model:
∆T
t0
(r , t0 ) = ,
T
(2r 2 + t02 )1/2
≡ 8π 2 G η 2 .
Turok et al. 90, Stebbins et al. 94
Known problems
The sigma-model approximation breaks at the unwinding.
All photons crossing the unwinding site receive the same amount
of red/blue-shift, independently of the time of passing.
The anisotropy profile decays very slowly ∼ r −1 .
15 / 21
Analytical approach
Problems of the Self-Similar solution
Anisotropies
from Collapsing
Textures
!T
"""""""
T
"T
#######
T
0.5
Introduction
!20
!10
"T
#######
T
1
1
10
20
t0
!0.5
10
0.8
!0.2
0.6
!0.4
0.4
30
40
r
!0.6
0.2
!0.8
Textures
!1
20
10
20
30
40
r
!1
Analytical
approach
Numerical
methods
Results
Conclusions
Template used to compare the CMB data with the texture model:
t0
∆T
,
(r , t0 ) = T
(2r 2 + t02 )1/2
≡ 8π 2 G η 2 .
Turok et al. 90, Stebbins et al. 94
The self-similar solution requires very special initial conditions
Q ∈ Z, ( in a cosmological context fractional charges are generic.)
Spherical symmetry.
A coherent velocity is needed to preserve the self-similar state.
Borrill et al. 92,94
15 / 21
Numerical Methods
Discrete equations of motion
Anisotropies
from Collapsing
Textures
We evolve a discrete version of the O(4)−model in a cubic lattice.
Fields are evolved in discrete time-steps:
Introduction
Textures
φa (x) −→ φi,n
a .
ϕai,n
Analytical
approach
Derivatives are represented by finite
differences:
Numerical
methods
Results
Conclusions
3
N = 96 ,
i,n+
∆x = 5∆t = 4lφ .
φ̇a (x) −→ φ̇a
1
2
=
φi,n+1
− φi,n
a
a
.
∆t
At the edges of the lattice we use periodic boundary conditions,
we run the simulation for 96 grid units.
The discrete e.o.m. in a Minkowski background:
1
i,n+ 2
φ̇a
φai,n+1
1
i,n− 2
i,n i,n
2
i,n
+ ∇2 φi,n
∆t
a − 4λ(φb φb − η ) φa
=
φ̇a
=
φi,n
a + φ̇a
1
i,n+ 2
∆t
16 / 21
Numerical Methods
Calculation of the anisotropy
Anisotropies
from Collapsing
Textures
Introduction
We calculate the anisotropy on a sequence of 48 photon planes
Planes are separated by one lattice spacing.
Analytical
approach
They travel along one of the mayor axis of the lattice.
Half cross the unwinding site before the event,
Numerical
methods
half cross the unwinding site afterwards.
Textures
Results
Conclusions
The anisotropy is calculated from
the small-angle formula:
∇2⊥
∆T
(x⊥ ) = −8πG ∇⊥ · U(x⊥ ),
T
which is solved using a F.F.T.
The integration constant is fixed
requiring < ∆T
T >= 0. Borril et al. 94
17 / 21
Numerical methods
Initial configuration
Anisotropies
from Collapsing
Textures
Introduction
Textures
Analytical
approach
Numerical
methods
The initial correlation lenght was set to 16 grid points
The initial configuration is set assigning random points from M
on a 33 sublattice.
Intermediate points are filled using an algorithm which
minimizes the total energy.
Only a 4% of the initial conditions lead to a texture unwinding.
φ2 < η 2 /4,
Results
Conclusions
Borril et al. 93, 94
We obtained an ensemble of random initial conditions leading to
unwinding events.
We evolved 1300 initial conditions,
ϕai,n
obtaining 50 texture unwindings.
Only 33 were selected for the ensemble,
discarding multiple unwinding events and
those close to the begining and end of the
simulation.
18 / 21
Numerical methods
Limitations of this approach
Anisotropies
from Collapsing
Textures
Introduction
Textures
Analytical
approach
The evolution of cosmic textures involves physics both at
microscopical and cosmological length-scales.
Numerical
methods
The microscopical length-scale: lφ = mφ−1
Results
Cosmological length-scale: ξc ∼ H −1
Conclusions
R = H −1 /lφ ∼ 1050
The asymptotic regime is reached when R ∼ 200, Borril et al. 92.
19 / 21
Time evolution and radial profile of the anisotropy
Anisotropies
from Collapsing
Textures
Averaged anisotropy profiles:
Introduction
Textures
"T
#######
T
!T
"""""""
T
1
1
0.5
0.8
"T
#######
T
10
Numerical
methods
!40
!20
20
!0.5
!1
40
t0
30
40
t0
!0.4
0.6
Analytical
approach
20
!0.2
!0.6
0.4
!0.8
0.2
10
20
30
40
t0
!1
Results
Conclusions
Textures become spherical close to the collapse. Turok et al .90
Maximum of the hot spot profile
minimum of the cold spot profile
∆T
T |max
∆T
T |min
= (+0.77 ± 0.21),
= (−0.49 ± 0.13),
in agreement with Borrill et al. 94
The unwinding event is resolved, δt ∼ 40lφ .
The hot and cold spot profile functions are more localized than
in the analytic solution.
20 / 21
Time evolution and radial profile of the anisotropy
Anisotropies
from Collapsing
Textures
Averaged anisotropy profiles:
"T
#######
T
!T
"""""""
T
1
1
0.5
0.8
Introduction
Textures
Analytical
approach
Numerical
methods
"T
#######
T
10
!20
20
!0.5
!1
40
t0
30
40
t0
!0.4
0.6
!40
20
!0.2
!0.6
0.4
!0.8
0.2
10
20
30
40
t0
!1
Results
Conclusions
The brightness of the spot decays at early and late times.
The CMB analyses depend on the expected number of spots due
to textures.
dNs
Ns (θ > θc )
=⇒
∼ θ−3
dθc
This estimate is partially based on numerical simulations which
do not consider how photons interact with the texture.
Our results also indicate a strong dependence on the photon
emission and detection times.
20 / 21
Conclusions
Anisotropies
from Collapsing
Textures
Introduction
Textures
Analytical
approach
Numerical
methods
Results
A library of 33 random initial conditions leading to single unwinding
events.
Calculation the corresponding anisotropy profiles using the small-angle
approximation.
The averaged anisotropy pattern is significantly more localized, and
has a smaller peak than the Turok solution (20 − 50%).
Results
Conclusions
Textures are only observable during a finite interval around the
unwinding event.
21 / 21
Conclusions
Anisotropies
from Collapsing
Textures
Introduction
Textures
Analytical
approach
Numerical
methods
Results
A library of 33 random initial conditions leading to single unwinding
events.
Calculation the corresponding anisotropy profiles using the small-angle
approximation.
The averaged anisotropy pattern is significantly more localized, and
has a smaller peak than the Turok solution (20 − 50%).
Results
Conclusions
Textures are only observable during a finite interval around the
unwinding event.
To be done:
Repeat the simulations in a FRW background.
Resimulate in a larger lattice to increase precision and diminish
boundary effects.
Characterize the small scale anisotropies due to random fluctuations
of the fields.
21 / 21