Manuel KrämerInstitute of Physics

University of Szczecin

Singularity resolution inWheeler–DeWitt

quantum cosmology

Singularities of general relativity and their quantum fateBanach Center / IMPAN, Warsaw, June 29, 2016

• full Wheeler–DeWitt equation is mathematically difficult to handle

➡ quantization of a symmetry-reduced model of the universe

• consider a spatially flat homogeneous and isotropic universe

with a minimally coupled scalar field with potential

• infinitely many degrees of freedom of “superspace” are reduced to two:

➔ scale factor and scalar field ➔ minisuperspace

➡ Wheeler–DeWitt equation:

• clearly avoids mathematical problems of the functional WDW equation

The Wheeler–DeWitt equation in quantum cosmology2

� V (�)

2 := 8⇡G

a �

ds2 = � dt2 + a2(t) d⌦23

~22

✓2

6a2@

@a

✓a

@

@a

◆� 1

a3@2

@�2

◆ (a,�) + a3 V (�) (a,�) = 0

• to simplify the equation set:

• evaluate this equation for a cosmological model of your choice

‣ choose a potential

‣ give boundaryconditions

‣ example:cyclic model

‣ apply Born–Oppenheimer approximation:

➡ wave function of the Universe

Solving the Wheeler–DeWitt equation3

a ⌘ a0 exp(↵)

~22

✓2

6

@2

@↵2� @2

@�2

◆ (↵,�) + a60 e

6↵ V (�) (↵,�) = 0

19

The factor ordering has been chosen to be of the Laplace–Beltrami form,which has the advantage that it guarantees covariance in minisuperspace.

It is evident that equations such as (26) do not possess the mathematicalproblems of the full functional equation (7). One can thus focus attention onphysical applications. One important application is the imposition of bound-ary conditions. Popular proposals are the no-boundary condition [19] and thetunneling condition [54]. The no-boundary proposal makes essential use ofthe connection between covariant and canonical quantum gravity discussed inSection 2.6: it is defined conceptually by a Euclidean path integral, but alsorelies on solving a minisuperspace Wheeler–DeWitt equation such as (26).Other important applications include the discussion of wave packets, the va-lidity of the semiclassical approximation, the origin of classical behaviour andthe arrow of time, and the possible quantum avoidance of classical singular-ities [2,16].

Before picking out one particular model, I want to emphasize one im-portant conceptual point which is relevant for the problem of time discussedabove, see Figure 2.

Fig. 2 The classical and the quantum theory of gravity exhibit drastically differentnotions of determinism [2].

Consider a two-dimensional minisuperspace model with the variables aand φ as above. The figure on the left shows the classical trajectory in config-uration space for a universe which is expanding and recollapsing. Classically,one can give initial conditions, for example, on the left end of the trajectoryfor small a and then determine the whole trajectory. In this sense, the recol-lapsing part of the trajectory is the deterministic successor of the expandingpart. One could, of course, also start from the right end of the trajectorybecause there is no distinguished direction; but the important point is thata trajectory exists. Not so in the quantum theory where both the trajectoryand the time parameter t are absent! If one wants to find a solution of theWheeler–DeWitt equation which describes a wave packet following the clas-sical trajectory, one has to specify two packets at the would-be ends of theclassical trajectory, see the right figure. The reason is that (26) is a hyper-bolic equation with respect to intrinsic time a, and the natural formulation

(↵,�) = '(↵,�)C(↵)

• most prominent singularity: Big Bang

• depending on what kind of matter/energy is present in the universe,a variety of future singularities can occur:

‣ Type I – “Big Rip”: for :

‣ Type II – “Big Brake”: for :

‣ Type III – “Big Freeze”: for :

‣ Type IV – (“Big Separation”): for : , 2nd and higher time derivatives of divergeType V: divergence of the barotropic index

➡ Can these singularities be resolved in quantum cosmology?

Classical singularities in cosmology4

t ! tsing

t ! tsing

t ! tsing

t ! tsingH

a ! 1 , ⇢ ! 1 , |P | ! 1

a ! 0 , ⇢ ! 1 , |P | ! 1

energy density pressure

a ! asing , ⇢ ! ⇢sing , |P | ! 1

a ! asing , ⇢ ! 1 , |P | ! 1

a ! asing , ⇢ ! ⇢sing , |P | ! Psing

‣

Nojiri, Odintsov and Tsujikawa,Phys. Rev. D 71, 063004 (2005).

• universe filled with ideal fluid, equation of state:

‣ conventional: , Chaplygin gas:

• convert this to a scalar field with potential

➡ Wheeler–DeWitt equation:

• use Born–Oppenheimer approximation:

• require to satisfy:

➡ gravitational part:

Construction of a quantum-cosmological model 5

P = f(⇢)

P = w⇢ P = � A

⇢

⇢!=

1

2�2 + V (�) P

!=

1

2�2 � V (�)

� V (�)

~22

✓2

6

@2

@↵2� @2

@�2

◆ (↵,�) + a60 e

6↵ V (�) (↵,�) = 0

(↵,�) = '(↵,�)C(↵)

' �~22

@2'

@�2+ a60 e

6↵V (�)' = E(↵)'

@C

@↵

@'

@↵+ C

@2'

@↵2+

✓2

6

@2C

@↵2+ 2E(↵)C

◆' = 0

Normalizable solutions of the Wheeler–DeWitt equation vanishat the classical singularity

-15

-10

-5

0

5

10

15

0 2 4 6 8 10a

φ

2 4

6 8

10 12 -10

-5 0

5 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2 4

6 8

10 12 -10

-5 0

5 10

ττ

φφ

Ψ(τ,φ)Ψ(τ,φ)

similar result for the corresponding loop quantum cosmology

(Kamenshchik, C. K., Sandhofer 2007)

• expansion of the universe comes to a halt in the future, :

‣ can be described by an anti-Chaplygin gas:

‣ potential for :

‣ WDW equation:

• can be solved analytically after using Born–Oppenheimer approximation

➡ normalizable solutions vanish at the classical singularities (Big Brake and Big Bang)

Example 1: Big Brake6

~22

✓2

6

@2

@↵2� @2

@�2

◆ (↵,�)� V0

|�| e6↵ (↵,�) = 0

V (�) /hsinh

⇣p3|�|

⌘� sinh�1

⇣p3|�|

⌘iP = A/⇢

�

Kamenshchik, Kiefer and Sandhöfer, Phys. Rev. D 76, 064032 (2007).�e6↵

(↵,�)

a ! asing , ⇢ ! ⇢sing , |P | ! 1 , H ! 1t ! tsing

Normalizable solutions of the Wheeler–DeWitt equation vanishat the classical singularity

-15

-10

-5

0

5

10

15

0 2 4 6 8 10a

φ

2 4

6 8

10 12 -10

-5 0

5 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2 4

6 8

10 12 -10

-5 0

5 10

ττ

φφ

Ψ(τ,φ)Ψ(τ,φ)

similar result for the corresponding loop quantum cosmology

(Kamenshchik, C. K., Sandhofer 2007)

Example 1: Big Brake7

Kamenshchik, Kiefer and Sandhöfer,Phys. Rev. D 76, 064032 (2007).�

-15

-10

-5

0

5

10

15

0 2 4 6 8 10a

φ

Figure: Classical trajectory inconfiguration space.

a6

classical trajectory

Big Bang

Big Brake

Big Brake

(a,�)

• from observations (SN Type Ia): expansion of universe is accelerating

• one way to model this acceleration: dark energy ➔ negative pressure

• observationally even cannot be excluded

‣ phantom field, violates the null energy condition

‣ induces Big Rip singularity: for

‣ WDW equation becomes elliptic

‣ wave-packet solutions disperse at the classical singularity

➡ time & classical evolution come to an end, stationary quantum state left

Example 2: Big Rip8

Dąbrowski, Kiefer and Sandhöfer, Phys. Rev. D 74, 044022 (2006).

P = w⇢ w < �1

3w . �1

⇢+ P > 0

t ! tsing

~22

✓2

6

@2

@↵2+

@2

@�2

◆ (↵,�) + e

6↵

✓V0 cosh

2

✓�

F

◆+

⇤

6

◆ (↵,�) = 0

a ! 1 , ⇢ ! 1 , |P | ! 1

• expansion of the universe comes to a halt in the future, :

‣ can be described by a generalized anti-Chaplygin gas:

‣ type IV singularity for: for , where

• double-well potential (➔ tunnelling?)

• quantum analysis:

➡ not exactly a type IV singularity, but “close enough” (form of potential does not change)

Example 3: Type IV singularity9

Bouhmadi-López, Kiefer and M.K., Phys. Rev. D 89, 064016 (2014).

t ! tsing

P = A/⇢�

�1

2< � < 0 � 6= 1

2p� 1

2p 2 N

V (�) /

2

4sinh2

1+�

p3

2|1 + �||�|

!� 1

sinh2�

1+�

⇣p32 |1 + �||�|

⌘

3

5

...H

4

0.5 1.0 1.5 2.0

500

1000

1500

−3/2(1 + β)α

φ2 |A|−

11+

β

0.5 1.0 1.5 2.0

!1.5

!1.0

!0.5

0.5

1.0

1.5

−3/2(1 + β)α(√

3/2)κ(1

+β)φ

FIG. 1. Kinetic energy of the scalar field (left) and dependence of the field on the logarithmic scale factor α = ln(a/amax)(right). In the left Figure, the value β = −

√

2/3 is chosen. The singularity is at φ = 0, where a = amax.

!1.0 !0.5 0.5 1.0

!0.2

0.2

0.4

(√3/2)κ(1 + β)φ

V(φ)/V1

✸

Contracting phase Expanding phase

➠➠

➠➠

FIG. 2. The potential defined in Eq. (11) as a function of the scalar field for the value β = −

√

2/3 (we have chosen this valuefor β to make sure that it cannot be written as 1/(2p)− 1/2, where p is an integer). It has the form of a double-well potentialwell known from quantum mechanics.

where V1 = |A|1

1+β /2, cf. [21]. The potential is displayed in Fig. 2 for a typical value of β.Notice that near amax (φ = 0) the potential is negative and finite. This is not surprising, since in a type IV

singularity both the energy density and the pressure are finite. We emphasize that the potential (11) is of the formof a double-well potential and is regular everywhere. This is in stark contrast to the cases discussed in [18, 20, 21]and is connected with the soft nature of the type IV singularity. It will have direct consequences for the study of thequantum theory below.Close to the type IV singularity, the potential can be approximated as

V (φ) ≃ −V1

!√3

2κ|1 + β||φ|

"− 2β1+β

, (12)

cf. Eq. (16) in [21]. In the limiting case β = −1/2, this corresponds to an inverted harmonic oscillator.At small scale factor (or large value of the scalar field), the potential can be approximated by the exponential form

V (φ) ≃ 2−2

1+β V1 exp#√

3κ|φ|$

. (13)

Such a potential occurs also in the cases of the big rip with a phantom field [18] and the big bang with an anti-Chaplygin gas [20]. In the latter case, it was shown that the big-bang singularity is avoided in the quantum theory

� = �1

2

a ! asing , ⇢ ! ⇢sing , P ! Psing , H, , . . . ! 1

• after Born–Oppenheimer decomposition ,matter part of the WDW equation reads

➡ can be solved in terms of confluent Heun functions

• regular at the origin , increase as power for large

• left side symmetric; right side antisymmetric ➔ vanishes at

• gravitational part does not spoil this result

➡ only a subset of wave functions vanishes at

➡ from the structure of the double-well potential, one can conclude thatthis result is also valid for , i.e. in the case of a type IV sing.

Example 3: Type IV singularity10

�1 + x

2�@

2'

@x

2+ x

@'

@x

� ⇠x

2(x2 � 1)' = �✏'

'(x) = c1 e�

p⇠

2 x

2 Hc

✓·,�1

2, ·, ·, ·;�x

2

◆+ c2 x e

�p

⇠2 x

2 Hc

✓·, 12, ·, ·, ·;�x

2

◆

Hc(·, ·, ·, ·, ·; 0) = 1x

x = � = 0

✏ :=32Ek(↵)

32

⇠ :=32V1a60e

6↵

32x := sin

✓p3

4�

◆ (↵,�) = '(↵,�)C(↵)

C(↵)

� = � 12 + "

(↵,�) � = 0

• analysis can be repeated for a phantom field

➡ leads to a periodic potential:

‣ evolution of the universe goes through a type IV singularity

‣ asymptotically de Sitter

➡ again, singularity resolution only for a subset of the wave functions

Example 3bis: Type IV singularity – phantom field11

5

value of the kinetic energy for the scalar field and thechange of the field in terms of the scale factor. The sin-gularity is located at φ = 0 where we have set φmin = 0for simplicity.

0.5 1.0 1.5 2.0

0.05

0.10

0.15

0.20

0.25

3/2(1 + β)α

φ2 A

−1

1+β

0.5 1.0 1.5 2.0

!1.0

!0.5

0.5

1.0

3/2(1 + β)α√3/2(1+β)κφ

FIG. 3. The kinetic energy of the scalar field (top) and thedependence of the field on the logarithmic scale factor α =ln(a/aamin) (bottom). In the upper Figure, the value β =−

√

2/3 is used. The singularity is at φ = 0, where a = amin.

The scalar field potential can be written as

V (φ) = V−1

!

sin−2β

1+β

"√3

2κ|1 + β||φ|

#

+ sin2

1+β

"√3

2κ|1 + β||φ|

#$

, (23)

where V−1 = A1

1+β /2 and 0 < (√3/2)κ|1 + β||φ| ≤ π/2,

cf. Eq. (21) in [21]. Notice that near amin (φ = 0), thepotential is positive and finite, in contrast to the casesdiscussed in [21]. This is, again, not surprising, as ina type IV singularity both the energy density and thepressure are finite. The potential (23) is, in contrast tothe case of the standard field, periodic in φ. The shape ofthe potential in terms of the scalar field is shown in Fig. 4.In the expanding branch, the evolution starts from thesingularity located at φ = 0, then the scalar field rollsup the potential and asymptotically reaches the top ofthe potential, which is located at

√3(β + 1)κφ/2 = π/2,

while a → ∞. Classically, the various parts (extensions

of the part shown in Fig. 4; i.e., for example, outside themaxima of the potential marked with two vertical lines)correspond to different classical solutions. This may haveconsequences in the quantum theory.Close to the singularity, the potential can be approxi-

mated by

V (φ) ≃ V−1

"√3

2κ|1 + β||φ|

#− 2β1+β

. (24)

We now turn to the quantum versions of these models.

!3 !2 !1 1 2 3

0.5

1.0

1.5

2.0

✸

Contracting phase

➞Expanding phase➞

➞

❙❙

➠ ➠

√3/2κ(1 + β)φ

V(φ)/V−1

FIG. 4. The potential defined in Eq. (23) as a function ofthe scalar field for the value β = −

√

2/3 (we have chosenthis value for β to make sure that it cannot be written as1/(2p)−1/2, where p is an integer). The potential is periodic,but the model we have discussed here corresponds to the rangeof values of the scalar field that cover two consecutive maxima(blue vertical lines).

III. QUANTUM ANALYSIS

In this section, we investigate the question whetherthe classical type IV singularity can be avoided in thequantum theory or not. In treating the Wheeler–DeWittequation, we apply the methods used in the earlier papers[18, 20, 21]. We also want to emphasize that the quantumcosmology of a GCG was first discussed in [33]. In ourcase, we have for the wave function Ψ (α,φ) a Wheeler–DeWitt equation of the form

!2

2

%

κ2

6

∂2

∂α2− ℓ

∂2

∂φ2

&

Ψ (α,φ) + a60e6αV (φ)Ψ (α,φ) = 0,

(25)where a0 corresponds to the location of the singular-ity, which is a0 = amax for the model of Sec. II.A anda0 = amin for the model in Sec. II.B. Here, we have used

V (�) = V�1

"sin�

2�1+�

p3

2|1 + �||�|

!+ sin

21+�

p3

2|1 + �||�|

!#

Bouhmadi-López, Kiefer and M.K.,Phys. Rev. D 89, 064016 (2014).

‣ How are classical singularities resolved in quantum-cosmological models?

‣ Big Rip (I): wave packets disperse at the classical singularity

‣ Big Brake (II): normalizable solutions vanish at Big Brake and Big Bang

‣ Big Freeze (III): boundary condition: let wave function vanish in classically forbidden region ➔ wave function vanishes also at the singularity

‣ Type IV: only a subset of solutions vanishes at the singularity

‣ Little Rip: wave function vanishes for late timesBig Rip for

➡ Is a generalization of these results possible?12

Summary

a, ⇢, P ! 1

⇢, P ! 1

P ! 1

H,...H, ... ! 1

t ! 1

Kamenshchik, Kiefer, Sandhöfer, PRD 76, 064032 (’07), 0705.1688.

Dąbrowski, Kiefer, Sandhöfer, PRD 74, 044022 (’06), hep-th/0605229.

Bouhmadi-López, Kiefer, Sandhöfer, Moniz, PRD 79, 124035 (’09), 0905.2421.

Bouhmadi-López, Kiefer, M.K., PRD 89, 064016 (’14), 1312.5976.

Albarran, Bouhmadi-López, Kiefer, Marto, Moniz, 1604.08365.