Collapse of Simple Harmonic Universe · ⇒ universe in static state in the infinite past 1 A....
Transcript of Collapse of Simple Harmonic Universe · ⇒ universe in static state in the infinite past 1 A....
Collapse of Simple Harmonic Universe
Audrey T. Mithani
Alexander Vilenkin
Miami 2011
December 19, 2011
Intro Collapse through tunneling Wave Function Conclusions
Introduction
What is the “simple harmonic universe” and does it collapse?
did the universe have a beginning?
- avoiding the initial singularity
simple harmonic universe
-classically stable
quantum instabilities
- semiclassical tunneling probability
- tunneling from nothing
- wave function of the universe
conclusions
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Did the Universe have a beginning?
Is it possible to avoid the initial singularity?
Singularity theorems showing that geodesics must be
past-incomplete require Havg > 0
- where H = a
a, averaged along the geodesic 1
However, if
Havg = 0,
can have inflation without an initial singularity
⇒ universe in static state in the infinite past
1A. Borde, A.H. Guth and A. Vilenkin, Phys. Rev. Lett. 90, 151301 (2003)
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Did the Universe have a beginning?
emergent universe must
- exist for an infinite amount of time
- stability with respect to perturbations
mechanism to trigger inflation
- example: massless
scalar field φ in potential
V (φ)
2 3
2G.F.R. Ellis and R. Maartens, Class. Quant. Grav. 21: 223 (2004)
3D.J. Mulryne, R. Tavakol, J.E. Lidsey and G.F.R. Ellis, Phys. Rev. D71,
123512 (2005)
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
“Simple Harmonic Universe”
Graham et al 4 proposed an emergent universe scenario which
is stable to linear perturbations
closed FRW universe k = +1
negative cosmological constant Λ < 0
matter source with equation of state p = wρ
The universe is stable for −1 < w < −1/3 and c2s > 0
- one example is a network of domain walls (w = −2/3)
4P.W. Graham, B. Horn, S. Kachru, S. Rajendran, and G. Torroba,
arXiv:1109.0282 [hep-th]
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
“Simple Harmonic Universe”
With w = −2/3, the Friedmann equation
a2 + 1 =
8πG
3
�Λ+ ρ0a
−1
�a
2
has oscillatory solutions
a(t) = ω−1(γ −
�γ2 − 1 cos(ωt))
where ω =�
8π3
G|Λ| and γ =
�2πGρ2
0
3|Λ|
“simple harmonic universe”
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Fate of the simple harmonic universe
Graham et al showed that the simple harmonic universe is
classically stable
We will check for quantum mechanical instabilities
semiclassical tunneling
tunneling from “nothing”
quantum cosmology with Wheeler-DeWitt
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Classical Hamiltonian dynamics
Hamiltonian describes dynamics of the system
H = −G
3πa
�p
2a + U(a)
�
with momentum
pa = −3π
2Gaa
and our potential is dependent on k ,Λ, ρ0
U(a) =
�3π
2G
�2
a2
�1 −
8πG
3(ρ0a + Λa
2)
�
with the Hamiltonian constraint H = 0, this produces the
simple harmonic universe
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Semiclassical Tunneling
U(x) = λ−2x
2(1 − 2γx + x2)
where x = ωa, λ = 16G2|Λ|9
, γ =
�2πGρ2
0
3|Λ|
turning points at x± = γ ±�γ2 − 1
0.5 1.0 1.5x
-200
200
400
U HxL
Figure: λ = .05 and γ = 1.3
0.5 1.0 1.5x
50
100
150
200
250
300
350
U �x�
Figure: λ = .05 and γ = 1
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Semiclassical tunneling
Tunneling probability from the WKB action P ∼ e−2SWKB
SWKB =
�x−
0
�U(x)dx = λ−1
�γ2
2+
γ
4
�γ2
− 1
�ln
�γ − 1
γ + 1
�−
1
3
�
probability for tunneling each time universe bounces at x−
For a static universe (γ = 1, so x− = x+ = 1),
Sγ=1 =1
6λ
⇒ simple harmonic universe cannot last forever
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Tunneling from nothing
Universe can also be created from “nothing” via tunneling from
x = 0 to x = x−
Euclideanized Friedmann equation
x2 = ω2(x+ − x)(x− − x)
instanton solution:
x(τ) = γ −
�γ2 − 1 cosh[ω(τ − τ0/2)]
τ is Euclidean time and τ0 = ω−1 ln
�γ+1
γ−1
�
solution starts at x(0) = 0 and reaches maximum at
x(τ0/2) = x−, then returns to x(τ0) = 0
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Quantum Cosmology
Canonical quantization of the theory 5
conjugate momentum is replaced with differential operator
pa → −id
da
wave function of the universe satisfies the Wheeler-DeWitt
equation
Hψ = 0
in the minisuperspace where ψ = ψ(a),
�−
d2
da2+ U(a)
�ψ(a) = 0
5B.S. DeWitt, Phys. Rev. 160, 1113 (1967)
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Quantum Cosmology for Simple Harmonic Universe
For the simple harmonic universe,
WDW �−
d2
dx2+ U(x)
�ψ(x) = 0
with potential
U(x) = λ−2x
2(1−2γx+x2)
where x = ωa,
λ = 16G2|Λ|9
, γ =
�2πGρ2
0
3|Λ|
0.5 1.0 1.5x
-200
200
400
U HxL
turning points at x± = γ ±�
γ2 − 1
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Solutions to WDW
quantum harmonic oscillator
(Schrodinger)
- B.C.s ψ(±∞) → 0
- energy eigenvalues En
simple harmonic universe
(WDW)
0.5 1.0 1.5x
-200
200
400
U HxL
- fixed energy eigenvalue
(Hψ = 0)
→ apply ψ(∞) = 0 to determine wave function as x → 0
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Solutions to WDW
Numerical solutions for oscillating and static (γ = 1) universe
0.5 1.0 1.5 2.0x
-600
-400
-200
200
U HxL , y HxL
λ = .05, γ = 1.3
0.5 1.0 1.5x
20
40
60
80
U HxL , y HxL
λ = 0.1, γ = 1 (static
universe)
the wave function is non-zero at x = 0, indicating a
non-zero probability for collapse
- is it possible to arrange for ψ(0) = 0?
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Solutions to WDW
Can quantum collapse be avoided?
λ determines the depth of the well and the number of
oscillations
for a given γ, changing λ causes ψ(0) to oscillate between
positive and negative values
0.5 1.0 1.5 2.0x
�600
�400
�200
200
400
600
U �x� , Ψ �x�
Figure: λ = .0473 and γ = 1.3
A. Mithani Collapse of Simple Harmonic Universe
Intro Collapse through tunneling Wave Function Conclusions
Conclusions
simple harmonic universe → escape the initial singularity
classically stable
quantum cosmology: tunneling to a = 0
semiclassical tunneling
creation from “nothing”
generic solutions to WDW: ψ(0) �= 0
beyond the minisuperspace? ψ(a,φ)
A. Mithani Collapse of Simple Harmonic Universe