Possible effect of negative shear
stress on cosmological
perturbations
Matej Škovran
Supervisor: Vladimír Balek
Comenius University in Bratislava
A BOUT THE W ORK
N UMERICAL R ESULTS3
R ELASTICITY – II.
We have computed cosmological perturbations in a
universe containing a solid component with w = 1/3
and negative shear stress. Unlike in the case previously
considered in the literature [Bucher and Spergel, 1999],
with the solid component having negative w and positive
shear stress, the perturbations are most strongly influenced
by the presence of the solid matter in the first period after it
appears.
Theory of relasticity in a cosmological setting:
1. scalar perturbations (longitudinal waves)
2
1 a ε + σ 3 ȧ
Ψk = 2 3
y01 + y11
2k a
2a
Scalar perturbations
1
1
Φk = Ψk
0.75
0.75
0.5
0.5
0
2. vector perturbations (transversal waves)
hk
0.5
0.5
kη∗ = 10−2
kη∗
kη∗
kη∗
kη∗
0
-0.25
kη∗ = 10−2
0.25
= 10−1
=1
= 10
= 102
kη∗
kη∗
kη∗
kη∗
0
-0.25
−0.5
40
= 10−1
=1
= 10
= 102
0
−0.5
1.5
1.5
40
C OSMOLOGICAL P ERTURBATIONS
1
1
0.25
1 a2 µs
Φk = Ψk + 2 3 y11
2k a
Tensor perturbations
kη∗ = 10−1
kη∗ = 1
kη∗ = 10
Φk
30
20
hk
1
1
20
• Scalar perturbations – gauge invariant variables Φ, Ψ.
Without shear stress they satisfy Φ = Ψ and in the
Newtonian gauge they coincide with the gravitational
potential.
• Vector perturbations – gauge invariant variable Vi.
Usually ignored because of their quick decay, as well
as the fact that there are just a few known physical
processes in which they can be generated.
• Tensor perturbations – gauge invariant variable hTijT .
Otherwise called gravitational waves, described by the
transversal traceless part of the metric. They have
two polarizations whose evolution is given by the same
equation.
3. tensor perturbations (gravitational waves)
kη∗ = 10−2
ȧ
ḧk + 3 ḣk + k 2a−2 + µsa−3 hk = 0
a
0
0
-10
−20
20
Ψk
10
0
0
This corresponds to:
kη∗
kη∗
kη∗
kη∗
−20
-20
-30
−40
10−3
10−2
10−1
10−2
0
10−1
1
η/η∗
−0.5
10
The figures in the first row show the case
without shear stress. The rest depicts
perturbations (relative to the perturbation
at the end of the inflation) influenced by
radiation-like shear stress (wµ = 1/3) with
the parameters set to ξ = −10−2, η̃s = 10−2
and ∆η̃ = 10−3.
-10
Let us consider an elastic continuum with the shear stress
of the form:
a 3wµ 1 η − ηs
0
µs(a) = µs,0
1 + tanh
a
2
∆η
= 10−1
=1
= 10
= 102
10−3
20
P ROPOSED M ODEL
kη∗
kη∗
kη∗
kη∗
0
= 10−1
=1
= 10
= 102
η/η∗ 10
1
Figure 1: Evolution of perturbations
2
2
1.5
1.5
Φk (η∗)
hk (η∗)
1
1
1
1
• time scale of the phase transition ∆η
0
Relasticity is a theory describing elastic continuum within
the framework of general relativity.
In [Polák and Balek, 2008] a universe filled by an elastic
continuum is studied1:
• Friedmann equations remain unchanged
2
ȧ
1 −3
∂ε
=
εa + ρ0 ,
= −3a−1σ
a
6
∂a
• material characteristics of the continuum satisfy
∂σ
= −a−1 (2σ + 3λ + 2µ)
∂a
• proportionality to the power of the scale parameter a after
the phase transition
• amplitude of the shear stress at present µs,0 ≡ 2ξ ΩRH02
ξ=0
ξ
ξ
ξ
ξ
−1
-1
1. longitudinal acoustic waves
2µ + 3λ + 5σ ȧ a ε + σ
2µ + λ + 3σ 1
ẏ01 =
+
y01 −
y11
3
ε+σ
a 4ȧ a
ε+σ
2
!
k2 3 ε + σ
a ε+σ
ẏ11 = 2 2 +
y01 −
y11
3
3
2 a
4ȧ a
a
10−3
2
1
0
0
An analytic solution does exist in the following case:
• deep within radiation dominated era – a(η) ∝ η
• for a radiation-like continuum – wµ = 1/3
ξ
ξ
ξ
ξ
0
1
ξ
ξ
ξ
ξ
ξ
−1
-1
−2
10−3
10−2
10−1
1
=0
= −10−2
= −10−3
= −10−4
= −10−5
kη∗
10
= −10−2
= −10−3
= −10−4
= −10−5
10−2
0
10−1
1
kη∗
1. for scalar perturbations
R EFERENCES
i
1h 2
Φk = 2 n (1−n)(Ajn +Byn)+(2+n+n2)φ (Ajn+1 +Byn+1)
η
1
Ψk = 2 [−n(1 − n) (Ajn + Byn) + 2φ (Ajn+1 + Byn+1)]
η
2. for tensor perturbations
hk = Cjn(kη) + Dyn(kη)
2
The solutions are linear combinations of spherical Bessel functions jn and yn, where
the parameter n is defined by n(n + 1)p≡ −µs,0/(ΩR H02) ≡ −2ξ. Scalar perturbations
are functions of the variable φ ≡ kη (1 + ξ)/3. Coefficient in front of the Bessel
functions are constants dependent of the model parameters and initial conditions
imposed on the perturbations at the end of the inflation.
9th Vienna Central European Seminar
Dark Matter, Dark Energy, Black Holes and Quantum Aspects of the Universe
November 30 - December 02, 2012, Vienna, Austria
[Bucher and Spergel, 1999] Bucher, M. and Spergel, D.
(1999). Is the dark matter a solid?
Phys. Rev., D
60:043505. [arXiv:astro-ph/9812022v3].
[Polák and Balek, 2008] Polák, V. and Balek, V. (2008).
Plane waves in a relativistic, homogeneous and isotropic
elastic continuum. Class. Quantum Grav., 25:045007.
[arXiv:gr-qc/0701055].
3
−0.5
10
Perturbations at the recombination offer
us a comparison of the observations and
the theory. Observations imply that a
flat spectrum (believed to be produced at
the inflation) should be preserved untill
the recombination.
The shear stress
prevents this. However, for sufficiently
small amplitude ξ, perturbations up to the
observable scales (kη∗ ∼ 10−2) retain flat
spectrum.
Figure 2: Perturbations at the time of recombination
The solution after the phase transition is2:
2. transversal acoustic waves
3. gravitational waves
Calculations are performed in units in which 16πκ = 1 and c = 1. Energy per
particle, pressure per particle and Lame coefficients (which are defined analogically
as in nonrelativistic relasticity) are denoted as ε, σ and µ, λ respectively.
= −10−2
= −10−3
= −10−4
= −10−5
−2
2
• for an instantaneous phase transition – ∆η → 0+
• the shear stress of such continuum is µs = µ + σ
• there are three modes of wave propagation:
ξ=0
Ψk (η∗)
A NALYTICAL S OLUTIONS
0.5
0.5
0
1
0.5
0.5
10
• phase transition at the conformal time ηs
R ELASTICITY – I.
kη∗ = 102
Calculations are performed for wµ = 1/3 and a two-component (matter-radiation)
universe. In such universe the scale parameter can be written as a = aeq (η̃ 2 + 2η̃),
where η̃ ≡ η/η∗ and η∗ is approximately the time of the recombination.
The numerical results correspond to a fixed time of the phase transition η̃s = 10−2 on
the scale ∆η̃ = 10−3. Although non-physically close to the time of the recombination,
it offers a good qualitative image of the modification. The numerical calculations
become unstable for much earlier times of the phase transition, however then an
analytic solution can be taken.