Collapse of Simple Harmonic Universe · ⇒ universe in static state in the inﬁnite past 1 A. Borde, A.H. Guth and A. Vilenkin, Phys. Rev. Lett. 90 , 151301 (2003) A. Mithani Collapse

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


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)



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)



“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]



“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”



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



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




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



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



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



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)



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



Solutions to WDW

quantum harmonic oscillator

(Schrodinger)

- B.C.s ψ(±∞) → 0

- energy eigenvalues En


(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



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?



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



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)


Documents

Collapse of Simple Harmonic Universe · ⇒ universe in static state in the inﬁnite past 1 A. Borde, A.H. Guth and A. Vilenkin, Phys. Rev. Lett. 90 , 151301 (2003) A. Mithani Collapse