1. COSMO - CBPF Nelson Pinto Neto CBPF, Rio de Janeiro, July 2017
XVI BSCG
CBPF, Rio de Janeiro, July 2017
BOUNCING MODELS:
ACHIEVEMENTS AND CHALLENGES.
Nelson Pinto Neto
COSMO - CBPF

2. The problem: the initial singularity
THE STANDARD COSMOLOGICAL MODEL
(FRIEDMANN-1922)
The problem: the initial singularity
- All Friedmann models contain one.
- A point where no physics is possible.
- General Relativity indicates its own limits:
what really happens when we approach the singularity?
- New physics!
The puzzle: Why the Universe was so much homogeneous and isotropic in the past?
n 3 initial constants equal to zero!

3. Big-Bang

4. Inflation is very important!
More puzzles
a) Existence of particle horizons makes things worst!
b) Why the spatial hypersurface is so flat?
Ω T ≡ ρT / ρc ≈1
(ΩT – 1)N < 10-18
c) Origin of structures in the universe.
Inflation is very important!

5. Bouncing models

6. What bouncing models can say about the standard
cosmological puzzles?

7. 2) Flatness problem: if the contraction phase is much longer
1) No horizon problem.
2) Flatness problem: if the contraction phase is much longer
then the expansion phase, then the Universe is almost flat because
it has not expanded enough!
3) Origin of perturbations: quantum vacuum fluctuations. RH
lphys
Ω T ≡ ρT / ρc ≈1
INFLATION IS NOT NECESSARY IF THE UNIVERSE DOES NOT HAVE A BEGINNING

8. BUT CONCLUSION: QUANTUM COSMOLOGY MAY HAVE AN
HOWEVER:
- NEITHER SOLVE THE HOMOGENEITY PROBLEM.
ONLY A THEORY OF INITIAL CONDITIONS MAY SOLVE THIS
PROBLEM (QUANTUM COSMOLOGY): Penrose, Goldwirth-Piran.
CONCLUSION:
QUANTUM COSMOLOGY MAY HAVE AN
IMPORTANT ROLE FOR SOLVING BOTH
THE SINGULARITY PROBLEM AND THE
HOMOGENEITY PUZZLE
BUT

9. Does quantum cosmology makes sense?
Answer of the Copenhaguen interpretation: NO
-- Need of a classical domain with classical observers:
no classical domain if the whole Universe is quantized.
Within the Copenhaguen interpretation we are stuck:
no further developments.
Contemporary quantum theory … constitutes an optimum formulation
of [certain] connections … [but] offers no useful point of departure
for future developments.
Albert Einstein.
Fortunately, there are alternative quantum theories!
Many Worlds
Consistent Histories
Spontaneous collapse
de Broglie-Bohm

10. THE DE BROGLIE-BOHM THEORY
“The kinematics of the world, in this ortodox picture, is given by a
wave function for the quantum part, and classical variables
variables which have values - for the classical part:
(Ψ(t,q ...), X(t) ...). The Xs are somehow macroscopic. This is not
spelled out very explicitly. The dynamics is not very precisely
formulated either. It includes a Schrödinger equation for the
quantum part, and some sort of classical mechanics for the
classical part, and `collapse’ recipes for their interaction.
It seems to me that the only hope of precision with the dual (Ψ,x)
kinematics is to omit completely the shifty split, and let both Ψ and x
refer to the world as a whole. Then the xs must not be confined to
some vague macroscopic scale, but must extend to all scales.”
John Stewart Bell.

11. The de Broglie-Bohm quantum theory
The guidance relation allows the
determination of the trajectories
(different from the classical)
If P(x,t=0) = A2 (x, t=0), all the statistical
predictions of quantum mechanics are recovered.
However, P(x,t=0) ≠ A2 (x, t=0), relaxes rapidly to P(x,t) = A2 (x, t)
(quantum H theorem -- Valentini)
Born rule may be obtained, not postulated

12. Q is highly non-local and context dependent!
Some remarks
Q is highly non-local and context dependent!
(Bell´s inequalities are violated, like in usual QM)
It offers a simple characterization of the classical limit:
Q=0
b) Probabilities are not fundamental in this theory.
The unknown variable is the initial position.
c) With objective reality but with the same statistical
predictions of standard quantum theory.
d) One postulate more (existence of a particle trajectory)
and two postulates less (collapse and Born rule) than
standard quantum theory: 1-2 = -1 postulate

14. Bell in Speakable and unspeakable in quantum mechanics
“In 1952 I saw the impossible done. It was in papers by David Bohm.
… the subjectivity of the orthodox version, the
necessary reference to the ‘observer,’ could be eliminated But why
then had Born not told me of this ‘pilot wave’? If only to point out
what was wrong with it? Why did von Neumann not consider it? . . .
Why is the pilot wave picture ignored in text books? Should it not be
taught, not as the only way, but as an antidote to the prevailing complacency?
To show us that vagueness, subjectivity, and indeterminism,
are not forced on us by experimental facts, but by deliberate theoretical
choice?” (Bell, page 160)

15. a(η) = ab [(/0)2 + 1]1/2 Bohmian quantum trajectory:
A first simple application to Cosmology: radiation (p=ρ/3).
Classical solution for the background: like a free particle
when written in conformal time: a = η
Wheeler-DeWitt equation:
Initial condition: gaussian
Bohmian quantum trajectory:
pa = 2 da/dη = ∂S/∂a
a(η) = ab [(/0)2 + 1]1/2

16. J. A.de Barros, M. A. Saggioro-Leal, NPN,
Phys. Lett. A 241, 229 (1998)

17. II) Quantum cosmological perturbations in quantum backgrounds.
Scalar perturbations:
Φ(x) is the inhomogeneous perturbation, related to the Newtonian potential in the nonrelativistic limit. dt = a dη. There is also δφ(x).
NOW PERTURBATIONS AND BACKGROUND
SHOULD BE QUANTIZED!
MORE GENERAL THAN MINISUPERSPACE AND SEMICLASSICAL
THEORY OF COSMOLOGICAL PERTURBATIONS

18. w = p/ρ QUANTUM EQUATIONS FOR PERTURBATIONS For the modes we have:
For the modes we have:
where now a is the quantum trajectory.
For a canonical scalar field cs2 =1

19. Before the bounce, in the contracting phase After the bounce,
Before the bounce,
in the contracting
phase
After the bounce,
In the expanding
phase.

20. Spectral indices for large wavelengths: (k is the wavenumber)
Primordial universe models: yield initial conditions for the evolution of the perturbations.
Spectral indices for large wavelengths:
(k is the wavenumber)
Scalar perturbations:
Tensor perturbations:

21. THE POWER SPECTRUM w = p/ρ Non relativistic fluid (dark matter?), w~0:
Non relativistic fluid (dark matter?), w~0:
scale invariant.
It is not necessary to have ordinary matter dominating all
along; just at the moment when perturbation scale becomes
comparable with the sound horizon.

22. Challenges:
i) With a single scalar field within General Relativity, the ratio r=T/S > 0.1, but observations (Planck satellite) give r < 0.1.
ii) Dark energy may be important in the contracting phase. However, they make it difficult to impose vacuum initial conditions in the far past of the contracting phase.

23. III) Scalar field with exponential potential
There is an attractor (repeller)

24. The critical points: The critical points: Dark energy: w=-1, x=0
The critical points:
The critical points:
Dark energy:
w=-1, x=0
Φ = const.

25. Either there is DE in the expanding phase or in the contracting phase, never in both!
Necessarily asymmetric.
We chose DE in the expanding phase.

26. More convenient variables:

27. Quantum bounce solution
S. Colin and NPN
arxiv:

28. PERTURBATIONS

30. One fluid: in a single shot we could accomodate the matter contraction necessary in bouncing models, and a dark energy phase which does not contrive quantum vacuum initial conditions.
The quantum bounce was fundamental for that, not only through the connection between the contracting and expanding phases, but also because the quantum effects can yield large scalar perturbations with respect to tensor perturbations, which does not happen in classical models with a single fluid.
Achievements:
- ns close to one: almost scale invariant.
reasonable amplitudes, for bounces between nucleosynthesis
and Planck scale.
dark energy naturally accomodated.
Challenges:
non gaussianities
more than one fluid: entropy perturbations
particle production in the bounce

31. FEATURES OF THE MODEL No singularity.
Perturbations of quantum mechanical origin.
Enhancement of perturbations at the bounce.
No horizon problem.
Flatness problem: if the contraction phase is much longer
then the expansion phase, then the Universe is almost flat because
it has not expanded enough!
6) Features of perturbations in accordance with observations.
7) Reasonable values for the curvature radius L at the
bounce: L > 103 lpl.
8) Dark energy appears naturally.

32. IV - CONCLUSION
There are no observational reasons for a beginning of the Universe, so why not exploring the consequences of bouncing models?
Allows calculations of potentially observational effects.
Basic General Relativity and de Broglie-Bohm QuantumTheory yield a sensible bouncing model which can explain the origin of cosmological perturbations in accordance with observations.
In such models inflation can be present but it is not necessary: another perspective concerning initial conditions.
P. Peter, E. Pinho and NPN, JCAP 07, 014 (2005).
P. Peter, E. Pinho and NPN, Phys. Rev. D 73, (2006).
P. Peter, E. Pinho and NPN, Phys.Rev. D 75, (2007).
E. Pinho and NPN, Phys. Rev. D 76, (2007).
F. T. Falciano and NPN, Phys. Rev. D 79, (2009).
R. Maier, S. Pereira, N. Pinto-Neto, and B. B. Siffert, Phys. Rev. D85, (2012).
S. D. P.Vitenti and N. Pinto-Neto, Phys. Rev. D85, (2012).
S. D. P.Vitenti, F. T. Falciano and NPN, Phys. Rev. D87, (2013).
D. Celani, S. D. P.Vitenti and N. Pinto-Neto, Phys. Rev. D95, (2017).
A. Scardua and N. Pinto-Neto, Phys. Rev. D95, (2017).

33. Louis de Broglie "To try to stop all attempts to pass beyond the
present viewpoint of quantum physics could be
very dangerous for the progress of science and
would furthermore be contrary to the lessons
we may learn from the history of science.
This teaches us, in effect, that the actual state
of our knowledge is always provisional
and that there must be, beyond what is actually
known, immense new regions to discover."
Louis de Broglie

34. HAMILTONIAN FORMULATION: TWO IMPORTANT EXAMPLES 1) Scalar field .
ds2 = N2(t) dt2 – a2(t) (dx2 + dy2 + dz2)
(K=0)
 = ln a
H = NH0 = N[ - (P)2 + (P)2 + V() ]
Hamilton equations:
dα/dt = {α,H} = -2NP , dφ/dt = {φ,H} = 2NPφ , dP /dt = {P ,H}, dPφ /dt = {Pφ ,H},
- (P)2 + (P)2 + V() = 0  Friedmann equation

35. Hamilton-Jacobi formulation:
- (S/α)2 + (S/)2 + V(φ) = 0
P = S/α
Pφ = S/φ
dα/dt = -2NP
dφ/dt = 2NPφ
If V=0, α ± φ = cte.

36. PT is a constant of motion identified with the total amount
2) Radiation: (p=ρ/3).
PT is a constant of motion identified with the total amount
of radiation in the Universe.
It appears linearly  a prefered time gauge choice: N = a e T=t
 T=η (conformal time)
Hamiltonian of a free particle generating η translations
Hamilton-Jacobi: S/η+ (S/a)2 = 0, p=S/a
Classical solution for the background: like a free particle
when written in conformal time: a = η

37.  = ln a V = 0 TWO IMPORTANT EXAMPLES a) canonical scalar field .
H = NH0 = N[ - (P)2 + (P)2 + V() ]
Solution  phase  a Bohmian trajectory
α() and () through: p = ’ = S/ and p = ’ = S/
Ĥ0 Ψ = 0
V = 0
R. Colistete Jr.,
J. C. Fabris, NPN
Phys. Rev. D62,
83507 (2000)

38. b) K-essence representing radiation: (p=ρ/3).
L = (α φ .α φ)2
J. A.de Barros, M. A. Saggioro-Leal, NPN, Phys. Lett. A 241, 229 (1998)
pa = 2 da/dη = ∂S/∂a
a(η) = ab [(/0)2 + 1]1/2

39. x=-1
contraction
x=1
expansion
x=-1
expansion
x=1
contraction