Download Gravity-anti-Gravity Symmetric Mini - Superspace Research proposal

Gravity-anti-Gravity Symmetric Mini Superspace
Research proposal
Submitted by: Tomer Ygael
Advisor: Prof. Aharon Davidson
Department of Physics
Faculty of Natural Sciences
Ben-Gurion University of the Negev
February 25, 2015
Contents
1 Abstract
2
2 Introduction
2
3 Preliminaries
3
3.1
The FLRW equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3.2
Dilaton Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
3.3
The Mini Superspace model . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
4 Research Plan and Preliminary results
7
4.1
Research Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
4.2
Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1
1
Abstract
A gravity-anti-gravity odd linear dilaton action offers an eternal inflation evolution governed
by the unified equation of state ρ − 3P = 4Λ.
At the mini super-space level, a ’two-particle’ variant of the no-boundary proposal is encountered allowing the energy of each ’particle’ to be different than zero. While a GaG - odd
wave function can only host a weak Big Bang boundary condition, albeit for any k, a strong
Big Bang boundary condition requires a GaG - even entangled wave function, and singles
out k = 0 flat space. At the second stage, one can add a Brans-Dicke kinetic, (ωBD 6= 0),
term for the dilaton.
To make contact with GR, a more general scalar potential is in order, such that the associated effective potential, which is relevant for the Klein-Gordon equation, becomes of the
double-well Higgs type.
2
Introduction
The idea that the Newton constant may have changed it’s sign during the very early universe is not new. Originally, it was presented as a possible consequence of the interaction of
geometry with electro/nuclear matter. With the early universe in mind, it has been recently
suggested that anti-gravity may have played a crucial role in taming the classical Big Bang
singularity by curing the geodesic incompleteness problem, in which a central ingredient of
this theory is the underlying conformal Weyl symmetry.
This work will start with a linear dilaton gravity Lagrangian, establish the classical mirror
cosmology and demonstrate how spatially symmetric mirror-gravity deflation can smoothly
2
evolve into eternal gravity inflation. Quantum mechanically we will work within the framework of the mini superspace model.
3
3.1
Preliminaries
The FLRW equations
The Einstein-Hilbert action
S=
√
R −gd4 x
(1)
can be varied with respect to gµν giving us the Einstein’s field equations with no sources
1
Rµν − gµν = 0
2
(2)
this combined with the FLRW metric

−1
0
0
0

2

a (t)
 0 1−kr
0
0
2

gµν = 
 0
0
a2 (t)r2
0

0
0
0
a2 (t)r2 sin2 θ








yield the ”equations of motion” for the cosmos:
2
ȧ
8πG
k
ρ
+ 2 =
a
a
3
4πG
ä
=−
(ρ + 3p)
a
3
(3)
(4)
(5)
Where ρ is the energy density, p is the pressure and generally neither are considered to be
constant. k describes the spatial curvature of space-time, where k = 0 is flat space. The
FLRW equations describe the development of the scale of the universe a(t) in time.
Notice that if we take flat space, i.e. k = 0, we can define
H=
ȧ
a
(6)
3
which is the Hubble parameter. Plugin this back into the Friedman equations we get one
differential equation for the Hubble parameter.
H2 =
3.2
8πG
ρ ;
3
Ḣ + H 2 = −
4πG
(ρ + 3p)
3
⇒
Ḣ = −4πG (ρ + p)
(7)
Dilaton Gravity
We will work with a scalar field (the dilaton) which is coupled to the Einstein-Hilbert action,
which is explained in section 3.1, we also add a general potential V (φ)
S=
√
(φR + V (φ)) −gd4 x
(8)
we now, once again, derive the equations of motion in accordance to gµν and φ. This leads,
respectively to the general set of the equations of motion
1
1
φ Rµν − gµν R + gµν 2φ − ∇µ ∇ν φ + gµν V (φ) = 0
2
2
R − V 0 (φ) = 0
(9)
(10)
It is quite interesting to find that even though we have not introduced a scalar kinetic the
Klein-Gordon equation does make it’s entrance as can be verified by tracing equation 9 and
plugin the constraint 10
−φR + 32φ + 2V (φ) = 0
⇒
2φ =
1
(φV 0 (φ) − 2V (φ))
3
(11)
Using the FLRW metric 3.1 yields the more general ”equations of motion” for the cosmos:
ȧ
ȧ2 + k
1
φ̇ +
φ =
V (φ)
2
a
a
6
1 0
ä ȧ2 + k
+
=
V (φ)
2
a
a
6
(12)
(13)
giving us the most general ”Friedman” equations for any given curvature k, scalar field φ
and potential V (φ).
4
3.3
The Mini Superspace model
Superspace is the infinite dimensional configuration of 3-geometries and matter field configurations on a spatial hyper-surface, Σ. It is on superspace that the wave function of the
universe is defined. In practice to work with the infinite dimensions of the full superspace
is not possible. One useful approximation therefore is to truncate the infinite degrees of
freedom to a finite number, thereby obtaining some particular mini superspace model.
An easy way to achieve this is by considering homogeneous and isotropic metrics which
instead of having a separate Wheeler-DeWitt equation for each point of Σ, simply have a
single WDW equation for all Σ.
Moreover we pick a metrics of the form
ds2 = −n2 (t)dt2 + hij (q α (t)) dxi dxj ,
α = 1, 2, . . . , n.
(14)
The results obtained by this approach are somewhat satisfying in that they do appear to
have some predictive power. However, the truncation to mini superspace has not as yet
been made part of a rigorous approximation scheme to full superspace quantum cosmology.
As they are currently formulated mini superspace models should therefore be viewed as toy
models, which nonetheless hopefully capture some of the essence of quantum cosmology.
Using the FLRW we have
dr2
i
j
2
2
2
2
2
hij dx dx = a (t)
+ r dθ + sin θdφ
,
1 − kr2
k = −1, 0, 1.
(15)
We now turn to the Einstein-Hilbert action writing it down explicitly
√
L −gd4 x =
r
1
6 (kn3 − aȧṅ + nȧ2 + naä)
2
3
=
r sin θ
drdθdφ a n
− 2Λ dt
kr2
a2 n3
S =
we call the second term Lmini , which is the mini superspace Lagrangian. Simplified,
Lmini = −2Λa3 n+
aȧ2
6a
dF (a, ȧ, n)
3
2
kn
−
a
ȧ
ṅ
+
n
ȧ
+
naä
=
+2Λa3 n−6akn+
(16)
2
n
n
dt
5
Ueff (a)
a
Figure 1: The effective potential with k = 1
Using the Legendre transform one is lead to the mini superspace Hamiltonian,
n
n 1 2
2
2
4
Hmini =
P + Uef f (a) , Uef f (a) = 12ka2 −4Λa4 .(17)
P + 24ka − 8Λa =
4a a
2a 2 a
The associated Hamiltonian constraint
Pn =
∂L
=0
∂ ṅ
⇒
Ṗn =
∂H
1
=
Pa2 + 24ka2 − 8Λa4 = 0
∂n
4a
(18)
gives rise to the WDW equation
Hmini Ψ = 0.
(19)
The corresponding Schrödinger energy is thus E = 0.
In our case we have a single scalar field which is coupled to the Ricci scalar
S=
√
−g (φR + V (φ)) d4 x
(20)
following the same procedure we will find that our mini superspace has only two coordinates
a(t) and φ(t).
6
4
Research Plan and Preliminary results
4.1
Research Plan
Within the framework of the mini superspace model, our reserach plan is to:
• Focus on dilaton cosmology, with a(t), n(t) and φ(t) as canonical variables.
• Discuss the simplest scalar potential is V (φ) = 4Λφ,
Λ > 0, with the positive pref-
actor interpreted as the positive cosmological constant
• Introduce a non-zero Brans-Dicke kinetic term
ωBD α
∇ φ∇α φ.
φ
• Generalize V (φ) to make contact with general relativity (as the vacuum of the theory).
4.2
Preliminary results
Our starting point is the particular dilation gravity theory characterized by a linear scalar
potential, namely
V (φ) = 4Λφ,
Λ > 0.
(21)
In a similar manner to 3.3 we integrate out the maximally symmetric spatial sub-manifold,
and up to a total derivative, the mini Lagrangian is given by
6
aφȧ2 + a2 ȧφ̇ + nφ 6ka − 4a3 Λ
Lmini = −
n
(22)
By supplementing φ → −φ by a → −a GaG has been elevated to a full symmetry at the
mini-superspace. A new pair of GaG parity conjugate variables, x± =
a
2
(a ± φ), is invoked
to diagonalize the quadratic kinetic term. Consequently, the GaG-odd mini Lagrangian, gets
decomposed into L = L+ − L− .
Using the Legendre transform one is led to
Hmini = n (H (x+ , p+ ) − H (x− , p− )) .
(23)
7
The upside-down harmonic oscillator building block
H (x, p) =
p2
+ U (x),
12
U (x) = 6kx − 4Λx2
(24)
resembles the Hartle-Hawking Hamiltonian 3.3.
The two almost decoupled ’one-particle’ physical systems communicate with each other solely
by means of the overall Hamiltonian constraint H = 0. On the quantum mechanics level
the Hamiltonian constraint kills the explicit time dependence of the universe wave function. A time independent wave eigenfunction ΨE (x+ , x− ) = ψE (x+ )ψE (x− ) then obeys the
bifurcated WDW equation
~ ∂2
+ U (x) − E ψE (x) = 0,
−
12 ∂x2
for x = x± .
(25)
While a GaG-odd wave function can only host a weak Big Bang boundary condition, albeit for
any k, a strong Big Bang boundary condition requires a GaG-even entangled wave function,
and singles out k = 0 flat space.
References
[1] Aharon Davidson and Tomer Ygael,
Gravity-anti-Gravity Symmetric Mini-Superspace: Quantum Entanglement and Cosmological Scale Factor Grid, arXiv:1410.3399v2 [gr-qc]
[2] David L. Wiltshire,
An Introduction to Quantum Cosmology, arXiv:gr-qc/0101003v2
[3] Ljubisa Nesic,
Minisuperspace Modles in Quantum Cosmology, Department of Physics, University of
Nis, Serbia.
[4] A.D. Linde,
Chaotic inflation with constrained fields, Phys. Lett. B93, 394 (1930).
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[5] M. Cooke,
An Introduciton to Quantum Cosmology, Department of Physics, Imperial College London, United Kingdom.
[6] I. Bars, P. Steinhardt and N. Turok,
Traversing Cosmological Singularities, Complete Journeys Through Spacetime Including
Antigravity, Phys. Rev. D89, 043515 (2014);
[7] M.P. Hertzberg,
Inflation, Symmetry, and B-Modes, arXiv:1403.5253 [hep-th];
[8] I. Quiros,
Scale invariance: fake appearances, arXiv:1405.6668 [gr-qc];
[9] R. Jackiw and S.Y. Pi,
Fake Conformal Symmetry in Conformal Cosmological Models, arXiv:1407.8545 [gr-qc];
[10] C. Brans and R.H. Dicke,
Mach’s Principle and a Relativistic Theory of Gravitation Phys. Rev. 124, 925 (1961).
[11] Spacetime and Geometry An Introduction to General Relativity,
Sean Carroll, Pearson Education 2004, San Francisco.
[12] J.B. Hartle and S W. Hawking,
Wave function of the Universe, Phys. Rev. D28, 2960 (1983);
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