An analysis of the symmetries of Cosmological Billiards; Talk presented at Fourteenth Marcel Grossmann Meeting - MG14, University of Rome "La Sapienza" , Rome, July 12-18, 2015, Parallel Session Exact Solutions (Physical Aspects) on 14 July 2015

1. An analysis of the symmetries
of
Cosmological Billiards
O.M. Lecian2
July 18, 2015
2
Sapienza University of Rome, Physics Department and ICRA,
Piazzale Aldo Moro,5- 00185 Rome, Italy
O.M. Lecian3
An analysis of the symmetries of Cosmological Billiards

2. Summary
• Cosmological Billiards
• Integral invariants
• Stochastization of the Dynamics
O.M. Lecian4
An analysis of the symmetries of Cosmological Billiards

3. Bianchi IX Universes
dl2 = a2(t)dx2
1 + b2(t)dx2
2 + c2(t)dx2
3
a(t) = tp1 , b(t) = tp2 , c(t) = tp3
i pi = i (pi )2 = 1
the Kasner solution can be considered valid as long as the rhs of
Einstein Eq.’s can be neglected
dl2 = gαβdxαdxβ, α, β = 1, 2, 3,
gαβ = hab(t)ea
α(x)eb
β(x) =
a2(t)lαlβ + b2(t)mαmβ + c2(t)nαnβ, a, b = 1, 2, 3
habea
αeb
β = a2(t)lαlβ + b2(t)mαmβ + c2(t)nαnβ
gij = d
a=1 e−2βa
Na
i Na
j
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An analysis of the symmetries of Cosmological Billiards

4. p1(u) = −u
u2+u+1
, p2(u) = 1+u
u2+u+1
, p3(u) = u(u+1)
u2+u+1
, 1 < u < ∞
for which the following Kasner Transformations TK are defined
0 < u < 1 u′ = 1
u
p1(u′) = p1(u) p2(u′) = p3(u) p3(u′) = p2(u)
− 1
2
< u < 0 u′ = − 1+u
u
p1(u′) = p3(u) p2(u′) = p1(u) p3(u′) = p2(u)
−1 < u < − 1
2
u′ = − u
u+1
p1(u′) = p3(u) p2(u′) = p2(u) p3(u′) = p1(u)
−2 < u < −1 u′ = − 1
u+1
p1(u′) = p2(u) p2(u′) = p3(u) p3(u′) = p1(u)
−∞ < u < −2 u′ = −u − 1 p1(u′) = p2(u) p2(u′) = p1(u) p3(u′) = p3(u)
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An analysis of the symmetries of Cosmological Billiards

5. Cosmological Billiards
Billiards can be visualized as a closed domains, where the billiard
ball moves.
The dynamics (the trajectories of the billiard ball)
is then characterized by
• free evolution (geodesic motion)
• bounces against the the billiard (≡ potential) walls
The statistical analysis is based on
• the symmetries of the billiard, that allow for the construction
of maps
• the existence of conserved quantities, that allow one to
establish probabilitieskeep track of the orientation
O.M. Lecian7
An analysis of the symmetries of Cosmological Billiards

6. O.M. Lecian8
An analysis of the symmetries of Cosmological Billiards

7. Group domains
O.M. Lecian9
An analysis of the symmetries of Cosmological Billiards

8. PGL(2, Z)
Its fundamental domain consists coincides with the small-billiard
table. It is generated by the transformations against the sides
R1(z) = −¯z,
R2(z) = −¯z + 1,
R3(z) = 1
¯z
The domain is not symmetric, no identification between the side is
possible
T = R2R1, T−1
= R1R2, (2a)
S = R1R3, S−1
= S, (2b)
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An analysis of the symmetries of Cosmological Billiards

9. Γ2 (PGL(2, Z))
No identification of the sides is required, as the big billiard map T
allows one to follow the dynamics throughout the corners, and to
keep track of the orientation.
A : z → −¯z, A ≡ R1,
B : z → −¯z − 2, B ≡ R1(R2R1)(R2R1)
C : z → − ¯z
2¯z+1, C ≡ (R1R3)(R2R1)(R2R1)R3,
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An analysis of the symmetries of Cosmological Billiards

10. Integral invariants
Poincar´e-Cartan 2-form
ω
(2)
PC := −dσ
(1)
PC = dqi
∧ dpi − dt ∧ dH(q, p, t)
H = E-reduced energy-shell hypersurface:
phase-space dim 3
Hγ(u, v, pu, pv ) = 1
2v2 p2
u + p2
v + V∞(γ(u, v))
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An analysis of the symmetries of Cosmological Billiards

11. Poincar´e return map
ωreduced
on ∂B = [du ∧ dpu + dv ∧ dpv ]v=const = du∧dpu = 2
du+ ∧ du−
(u+ − u−)2
Liouville 3-form
Ω
(3)
L = ω(2)
∧ ds
• reduced phase-space for the definition of epochs and eras
• probabilities for the reduced phase-space as areas, according
to the ω measure
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An analysis of the symmetries of Cosmological Billiards

12. Epochs
but D ω(2) = ∞
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An analysis of the symmetries of Cosmological Billiards

13. Eras
first epoch → next first epoch
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An analysis of the symmetries of Cosmological Billiards

14. CB-LKSKS maps
Tz =
1
¯z − n + 1
− 1 ≡ T−1
SR1T−n+1
z, (4)
defined on a domain with boundary identification
t1,2z = T−1SR1T−n+1z, for(u+, u−) ∈ S1
baand(u+, u−) ∈ S2
ba
t2′,3,3′
z = T−1SR1T−n+1R3z, for(u+, u−) ∈ S2′
ba, (u+, u−) ∈
S3
ba, and(u+, u−) ∈ S3′
ba
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An analysis of the symmetries of Cosmological Billiards

15. O.M. Lecian17
An analysis of the symmetries of Cosmological Billiards

16. Generalized BKL probabilities
invariant measure ω(2) = 2 du+∧du−
(u+−u−)2 normalized density of the
invariant measure for the billiard map for different phenomena
W (u+) ≡ 1
A(D) D(u+)
du−
(u+−u−)2
P(n1, n2, ..., nk) = du+ω(2) = du+du−ω(2)
PBKL
symm(k) ≡ P(n1,n2,...,nk )
P(n1,n2,...,nk ) symm
P(n1, n2, ..., nk) symm ≡ symm{ k} PBKL(k)
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An analysis of the symmetries of Cosmological Billiards

17. the integration domain D in the
reduced phase space depends on
the symmetry quotienting
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An analysis of the symmetries of Cosmological Billiards

18. A Stochastizing BKL Dynamics
limS symm(k) PBKL(k)
symm{ k}
limS symm(k) PBKL(k)
symm{ k} ≃ nsymm(k) i=k
i=1 P(ni )
limS PBKL(n1, n2, ..., nk) ≃ Ck
i=k
i=1
1
n2
i
symm(k) limS PBKL(k)
symm{ k} ≃ Cknsymm(k) i=k
i=1
1
n2
i
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An analysis of the symmetries of Cosmological Billiards

19. References
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