the loop quantum cosmology bounce as a kasner transition · 2018. 7. 30. · there are two key...

The loop quantum cosmology bounceas a Kasner transition

Edward Wilson-Ewing

University of New Brunswick

Class. Quant. Grav. 35 (2018) 065005arXiv:1711.10943 [gr-qc]

Singularities in General Relativity and their Quantum Fate IIBanach Center

E. Wilson-Ewing (UNB) The LQC bounce as a Kasner transition May 23, 2018 1 / 28

Loop Quantum Cosmology

In loop quantum cosmology (LQC), the non-perturbativequantization techniques of loop quantum gravity are applied tocosmological models like the Friedmann and Bianchi space-times[Bojowald; Ashtekar, Bojowald, Lewandowski; Ashtekar, Paw lowski, Singh; . . . ].

There are two key inputs in loop quantum cosmology:

the basic operators encode areas and holonomies of theAshtekar-Barbero connection Ai

a = Γia + γK i

the predicted Planck-scale discreteness of geometry.

In particular, for the FLRW and Bianchi I space-times the space-timecurvature (which appears in the Hamiltonian constraint operator) isexpressed in terms of the holonomy of Ai

a around a loop with an areaof ∼ `2

Loop Quantum Cosmology

One of the most striking results in LQC is singularity resolution: inthese space-times the singularities of general relativity are genericallyresolved.

[Paw lowski, Pierini, WE]

For space-times where numericalstudies have been done, numericsshow the singularity is replaced by abounce, and in addition there exist‘effective equations’ that provide anexcellent approximation to the fullquantum dynamics.

Very similar quantum gravity corrections to cosmological dynamicsarise from the hydrodynamics of group field theory condensate states[Gielen, Oriti, Sindoni, WE, . . . ].

The BKL Conjecture

According to the Belinski-Khalatnikov-Lifshitz (BKL) conjecture, inthe approach to a space-like singularity, neighbouring points decoupleand spatial derivatives become negligible in comparison to time-likederivatives.

Then, the dynamics at each point is given by those of a Bianchispace-time, typically Bianchi IX.

Therefore, if the BKL conjecture is correct, the Bianchi space-timescan be expected to play a central role in quantum gravity and indetermining the fate of (at least some of) the singularities predictedby general relativity.

As a first step towards studying quantum gravity effects near genericsingularities, I want to understand the LQC dynamics of Bianchispace-times.

LQC Dynamics for Bianchi I

Numerical solutions to the LQC effective dynamics for the Bianchi Ispace-time show that the singularity is replaced by a bounce [Gupt, Singh].Studies of the full quantum dynamics confirm this [Paw lowski].

[Gupt, Singh][Paw lowski]

Away from the bounce, the dynamics are very well approximated bythe Kasner solution from general relativity.

The LQC Bounce as a Kasner Transition

Here I will focus on the effective equations since they are much easierto study than the full quantum dynamics.

Because the degrees of freedom in LQC are heavy degrees offreedom, quantum fluctuations in semi-classical states do not increasesufficiently to become important [Rovelli, WE]. For sharply-peaked statesthe effective equations are expected provide a good approximation tothe full quantum dynamics.

[Gupt, Singh]

Numerical solutions to the LQCeffective equations show a rapidbounce, with different classical(Kasner) solutions either side [Gupt, Singh].

Could the LQC bounce be viewed as atransition between Kasner solutions?

[Gupt, Singh]

Outline

1 Review of the Kasner Solution

2 The LQC Bounce as a Kasner Transition

3 Mixmaster Dynamics in LQC

4 Discussion

Bianchi I Variables

The line element is

ds2 = −N2dt2 +∑i

ai(t)2dx2i .

The basic variables I will use here are logarithmic scale factorsαi = ln ai and their conjugate momenta Πi ,

αi ,Πj = −8πGδij , αi , αj = Πi ,Πj = 0.

The Πi are related to the scale factors by, e.g.,

Π1 ∼ a1(a2a3 + a2a3).

Bianchi I Variables

The line element is

ds2 = −N2dt2 +∑i

ai(t)2dx2i .

The basic variables I will use here are logarithmic scale factorsαi = ln ai and their conjugate momenta Πi ,

αi ,Πj = −8πGδij , αi , αj = Πi ,Πj = 0.

The Πi are related to the scale factors by, e.g.,

Π1 ∼ a1(a2a3 + a2a3).

The Hamiltonian Constraint

For the lapse N = a1a2a3 = exp[∑

i αi ], the Hamiltonian constraintfor the vacuum Bianchi I space-time is

CH ∼ −Π1Π2 − Π1Π3 − Π2Π3 +1

1 + Π22 + Π2

Clearly, the Πi are constants of the motion and

α1 ∼ (Πj + Πk − Πi)τ + α(0)1 , etc.

This is the full solution for the vacuum Bianchi I space-time ingeneral relativity.

The Kasner Solution

The Kasner line element for the Bianchi I solution is

ds2 = −dt2 + (t − to)2k1 dx21 + (t − to)2k2 dx2

2 + (t − to)2k3 dx23 ,

where the ki are the Kasner exponents.

These can be related to the earlier solution by dt = exp[∑

i αi(τ)]dτ :

τ ∼ 1∑j Πj

ln(∑

Πi(t − to)−∑i

α(0)i

and then the Kasner exponents are given by, e.g.,

k1 =Π2 + Π3 − Π1

Π1 + Π2 + Π3.

By inspection∑

i ki = 1, and from CH = 0 it follows that∑

i k2i = 1.

The Kasner Solution

The Kasner line element for the Bianchi I solution is

ds2 = −dt2 + (t − to)2k1 dx21 + (t − to)2k2 dx2

2 + (t − to)2k3 dx23 ,

where the ki are the Kasner exponents.

These can be related to the earlier solution by dt = exp[∑

i αi(τ)]dτ :

τ ∼ 1∑j Πj

ln(∑

Πi(t − to)−∑i

α(0)i

and then the Kasner exponents are given by, e.g.,

k1 =Π2 + Π3 − Π1

Π1 + Π2 + Π3.

By inspection∑

i ki = 1, and from CH = 0 it follows that∑

i k2i = 1.

Bianchi II Solution

The Hamiltonian constraint for the Bianchi II space-time is the sameas for the Bianchi I space-time, with an additional exponentialpotential in the α1 direction:

CH ∼ C(B.I )H + e4α1 .

The dynamics can be viewed as a Bianchi I solution bouncing onceoff the potential and transitioning to a different Kasner solution.

Away from the Kasner transition, the potential is negligible and sothe Πi are constant. During the transition only Π1 changes since thepotential only depends on α1.

During the Kasner transition, Πi → Πi , with Π2 = Π2 and Π3 = Π3,while

Π1 = Π1 + ∆Π1.

It turns out that ∆Π1 can be found quite easily.

Bianchi II Solution

The Hamiltonian constraint for the Bianchi II space-time is the sameas for the Bianchi I space-time, with an additional exponentialpotential in the α1 direction:

CH ∼ C(B.I )H + e4α1 .

The dynamics can be viewed as a Bianchi I solution bouncing onceoff the potential and transitioning to a different Kasner solution.

Away from the Kasner transition, the potential is negligible and sothe Πi are constant. During the transition only Π1 changes since thepotential only depends on α1.

During the Kasner transition, Πi → Πi , with Π2 = Π2 and Π3 = Π3,while

Π1 = Π1 + ∆Π1.

It turns out that ∆Π1 can be found quite easily.

The Transition Rule

Away from the Kasner transition, the potential is negligible.Therefore, both Kasner solutions Πi and Πi must each satisfyC(B.I )H = 0. This gives the condition

∆Π1

[∆Π1 − 2(Π2 + Π3 − Π1)

which is satisfied by ∆Π1 = 0 before the transition, and by

∆Π1 = 2(Π2 + Π3 − Π1)

after the transition.

Putting this result into the relation between the Kasner exponentsand the Πi gives the well-known Kasner transition map for theKasner exponents [Belinski, Khalatnikov, Lifshitz]:

k1 =−k1

1 + 2k1, k2 =

k2 + 2k1

1 + 2k1, k3 =

k3 + 2k1

1 + 2k1.

The Transition Rule

Away from the Kasner transition, the potential is negligible.Therefore, both Kasner solutions Πi and Πi must each satisfyC(B.I )H = 0. This gives the condition

∆Π1

[∆Π1 − 2(Π2 + Π3 − Π1)

which is satisfied by ∆Π1 = 0 before the transition, and by

∆Π1 = 2(Π2 + Π3 − Π1)

after the transition.

Putting this result into the relation between the Kasner exponentsand the Πi gives the well-known Kasner transition map for theKasner exponents [Belinski, Khalatnikov, Lifshitz]:

k1 =−k1

1 + 2k1, k2 =

k2 + 2k1

1 + 2k1, k3 =

k3 + 2k1

1 + 2k1.

Bianchi I LQC Effective Dynamics

The LQC effective dynamics for the vacuum Bianchi I space-time aregenerated by the Hamiltonian constraint (for N = exp[

∑i αi ] ∼ V )

[Chiou, Vandersloot; Ashtekar, WE]

CH ∼ −V 2

∑i 6=j

sin µiγKi sin µjγKj

∼ − V 2

∑i 6=j 6=k

sinγ√

2V(Πj + Πk − Πi) sin

2V(Πi + Πk − Πj),

with µ1 =√

p1∆/p2p3 =√

∆ p1/V , while γ is the Barbero-Immirziparameter and ∆ is the minimal non-zero eigenvalue of the LQG areaoperator.

An important point here is that αi enters CH only in terms of thecombination V ∼ exp[

∑i αi ].

Dynamics for Πi in LQC

Note that from αi ,Πj = −8πGδij , it follows that

V ∼ exp[∑i

αi ], Πi ,V = 8πGV ,

and as a result the equation of motion for the Πi are identical sinceCH depends on αi only through the combination V :

dτ= Πi , CH = 8πGV

Well before the LQC bounce the Πi are constant (since LQC effectsare negligible), and after the bounce the new Πi are also constant.Since their equations of motion are identical, they are all shifted bythe same amount ∆Π by the LQC bounce:

Πi = Πi + ∆Π.

Dynamics for Πi in LQC

Note that from αi ,Πj = −8πGδij , it follows that

V ∼ exp[∑i

αi ], Πi ,V = 8πGV ,

and as a result the equation of motion for the Πi are identical sinceCH depends on αi only through the combination V :

dτ= Πi , CH = 8πGV

Well before the LQC bounce the Πi are constant (since LQC effectsare negligible), and after the bounce the new Πi are also constant.Since their equations of motion are identical, they are all shifted bythe same amount ∆Π by the LQC bounce:

Πi = Πi + ∆Π.

Transformation Rule for Kasner Exponents

Since the Πi and Πi each correspond to a classical Kasner solutionaway from the bounce (respectively far before and far after), bothmust satisfy the general relativity Hamiltonian constraint CH = 0.This constrains ∆Π:

(2∑i

Πi + 3∆Π

There are two solutions: ∆Π = 0 corresponding to the pre-bouncesolution, and the post-bounce solution

∆Π = −2

(Π1 + Π2 + Π3

In terms of the Kasner exponents, this gives

ki → ki =2

3− ki ,

which agrees with the results of numerical simulations [Gupt, Singh].

Transformation Rule for Kasner Exponents

Since the Πi and Πi each correspond to a classical Kasner solutionaway from the bounce (respectively far before and far after), bothmust satisfy the general relativity Hamiltonian constraint CH = 0.This constrains ∆Π:

(2∑i

Πi + 3∆Π

There are two solutions: ∆Π = 0 corresponding to the pre-bouncesolution, and the post-bounce solution

∆Π = −2

(Π1 + Π2 + Π3

In terms of the Kasner exponents, this gives

ki → ki =2

3− ki ,

which agrees with the results of numerical simulations [Gupt, Singh].E. Wilson-Ewing (UNB) The LQC bounce as a Kasner transition May 23, 2018 15 / 28

So the loop quantum cosmology bounce acts, in effect,as a rapid transition between two Kasner solutions withthe simple transition rule

ki → ki =2

3− ki

relating the values of the Kasner exponents before andafter the LQC bounce.

Bianchi IX Solution

The Hamiltonian constraint for the Bianchi IX space-time hasmultiple exponential potentials:

CH ∼ C(B.I )H + e4α1 + e4α2 + e4α3 ,

keeping only the dominant terms in the potential for large curvatures,note these are three copies of the Bianchi II potential.

The Bianchi IX space-time can be viewed as a sequence of Kasnersolutions (or ‘epochs’). In the vacuum case there are an infinitenumber of Kasner epochs during the approach to the singularity.

During each Kasner epoch, it is convenient to order the Kasnerexponents as

−13< kmin < 0 < kmid <

23< kmax < 1,

and then to denote their respective logarithmic scale factors byαmin, αmid, αmax.

Bianchi IX Solution

The Hamiltonian constraint for the Bianchi IX space-time hasmultiple exponential potentials:

CH ∼ C(B.I )H + e4α1 + e4α2 + e4α3 ,

keeping only the dominant terms in the potential for large curvatures,note these are three copies of the Bianchi II potential.

The Bianchi IX space-time can be viewed as a sequence of Kasnersolutions (or ‘epochs’). In the vacuum case there are an infinitenumber of Kasner epochs during the approach to the singularity.

During each Kasner epoch, it is convenient to order the Kasnerexponents as

−13< kmin < 0 < kmid <

23< kmax < 1,

and then to denote their respective logarithmic scale factors byαmin, αmid, αmax.

Kasner Parameters

Each Kasner epoch can be parametrized by

u =kmax

, pΩ ∼∑i

v =kmid

· kminαmax − kmaxαmin

kminαmid − kmidαmin

+ 1, κ = kmax

(αmin

− αmid

which are constant for a Kasner solution.

At each transition between two Kasner epochs, following from theBianchi II transformation rule for the Kasner exponents given earlier:

u → u =

u − 1 if u − 1 > 1,

(u − 1)−1 otherwise,

pΩ → pΩ = pΩu2 − u + 1

u2 + u + 1.

Transition Rules

Simple (although approximate) transition rules for v and κ can bederived under the assumption that the transition between subsequentKasner epochs occurs instantaneously when the largest αi = 0.

v → v =

v + 1 if u − 1 > 1,

1 + 1/v otherwise,

κ→ κ =

κ if u − 1 > 1,

v κ/(u − 1) otherwise.

Can analogous transition rules for (u, v , pΩ, κ) be derived for theLQC bounce?

Transition Rules

Simple (although approximate) transition rules for v and κ can bederived under the assumption that the transition between subsequentKasner epochs occurs instantaneously when the largest αi = 0.

v → v =

v + 1 if u − 1 > 1,

1 + 1/v otherwise,

κ→ κ =

κ if u − 1 > 1,

v κ/(u − 1) otherwise.

Can analogous transition rules for (u, v , pΩ, κ) be derived for theLQC bounce?

Spatial Curvature in LQC

In the presence of spatial curvature, the holonomy of the fieldstrength around a loop of minimal area is not almost-periodic in theconnection, so a different approach is required.

So far, there are two suggestions: ‘A’ and ‘K’ loop quantizationsbased on the parallel transport of the Ashtekar-Barbero connectionand the extrinsic curvature respectively [Vandersloot; Ashtekar, WE; Singh, WE].

In the closed Friedmann universe, the ‘K’ loop quantization providesa better approximation to the ‘field strength’ loop quantization [Corichi,

Karami; Singh, WE], so here I will consider the ‘K’ loop quantization.

The Bianchi IX effective constraint for the ‘K’ loop quantization is

CH ∼ C(LQC B.I )H + e4α1 + e4α2 + e4α3 .

LQC affects only the Bianchi I part, the potential is unchanged.

Spatial Curvature in LQC

In the presence of spatial curvature, the holonomy of the fieldstrength around a loop of minimal area is not almost-periodic in theconnection, so a different approach is required.

So far, there are two suggestions: ‘A’ and ‘K’ loop quantizationsbased on the parallel transport of the Ashtekar-Barbero connectionand the extrinsic curvature respectively [Vandersloot; Ashtekar, WE; Singh, WE].

In the closed Friedmann universe, the ‘K’ loop quantization providesa better approximation to the ‘field strength’ loop quantization [Corichi,

Karami; Singh, WE], so here I will consider the ‘K’ loop quantization.

The Bianchi IX effective constraint for the ‘K’ loop quantization is

CH ∼ C(LQC B.I )H + e4α1 + e4α2 + e4α3 .

LQC affects only the Bianchi I part, the potential is unchanged.

Approximations

Recall from the discussion above that the Kasner transitionsgenerated by the spatial curvature occur rapidly and it is reasonableto approximate these transitions as occuring instantaneously; awayfrom the transitions the spatial curvature is negligible.

The LQC bounce happens even more rapidly.

Therefore, unless initial conditions are carefully chosen so that thespatial curvature is important during the LQC bounce, for typicalsolutions:

The spatial curvature is negligible during the LQC bounce, and LQCeffects are negligible during the Mixmaster-type Kasner transitions.

For these solutions, the transition rule at the LQC bounce will be thesame as for Bianchi I, and the Mixmaster transition rules will be thesame as in general relativity.

Approximations

Recall from the discussion above that the Kasner transitionsgenerated by the spatial curvature occur rapidly and it is reasonableto approximate these transitions as occuring instantaneously; awayfrom the transitions the spatial curvature is negligible.

The LQC bounce happens even more rapidly.

Therefore, unless initial conditions are carefully chosen so that thespatial curvature is important during the LQC bounce, for typicalsolutions:

The spatial curvature is negligible during the LQC bounce, and LQCeffects are negligible during the Mixmaster-type Kasner transitions.

For these solutions, the transition rule at the LQC bounce will be thesame as for Bianchi I, and the Mixmaster transition rules will be thesame as in general relativity.

LQC Transition Rules for the Kasner Parameters

From the transition map ki → ki = 23− ki , it immediately follows that

u → u =u + 2

u − 1, pΩ → pΩ = −pΩ.

A little work is required to derive the transition maps for v and κ.Under the assumptions that (i) the last Mixmaster Kasner transitionbefore the bounce occured when one αi = 0, and (ii) the LQCbounce occurs instantaneously when θ = θcrit ∼ `−1

v → v =2(2u + 1) ln

(2 γ√

∆ |pΩ|)

2(u + 1) ln(

2 γ√

∆ |pΩ|)

+ vκ,

κ→ κ = − 2(u + 1)

u − 1ln(

2 γ√

∆ |pΩ|)− κv

u − 1.

u → u =u + 2

u − 1, pΩ → pΩ = −pΩ.

v → v =2(2u + 1) ln

(2 γ√

∆ |pΩ|)

2(u + 1) ln(

2 γ√

∆ |pΩ|)

+ vκ,

κ→ κ = − 2(u + 1)

u − 1ln(

2 γ√

∆ |pΩ|)− κv

u − 1.

u → u =u + 2

u − 1, pΩ → pΩ = −pΩ.

v → v =2(2u + 1) ln

(2 γ√

∆ |pΩ|)

2(u + 1) ln(

2 γ√

∆ |pΩ|)

+ vκ,

κ→ κ = − 2(u + 1)

u − 1ln(

2 γ√

∆ |pΩ|)− κv

u − 1.

Some Comments

The derivation of the transition rules for u and pΩ required lessassumptions and so are more robust.

The Planck scale only appears in the transition rules for v and κ;the transition rules for u and pΩ are independent of `Pl. Theonly input needed for deriving the transition rules for u and pΩ isthat the equations of motion for the Πi are all identical.

The transition rules are the same no matter the initial values of(u, v , pΩ, κ).

Chaos?

The classical Bianchi IX dynamics are known to be chaotic [Barrow, Chernoff;

Cornish, Levin]. What about the LQC effective dynamics?

Classically, the chaos arises due to the repeated Kasner era changeswhen u → u = 1/(u − 1), with u − 1 < 1; there are an infinitenumber of these for Bianchi IX in general relativity. An importantdifference in LQC is that there will be a finite number of such erachanges in any bounce/recollapse cycle.

Note that the usual BKL transition map is unmodified by LQCeffects; rather LQC adds the bounce, a new type of Kasner transition.

And there are an infinite number of bounce/recollapse cycles, so itappears likely that the Bianchi IX space-time will be chaotic in LQCalso. However, this is a statement about the full sequence ofbounce/recollapse cycles, not about one such cycle alone.

Chaos?

The classical Bianchi IX dynamics are known to be chaotic [Barrow, Chernoff;

Cornish, Levin]. What about the LQC effective dynamics?

Classically, the chaos arises due to the repeated Kasner era changeswhen u → u = 1/(u − 1), with u − 1 < 1; there are an infinitenumber of these for Bianchi IX in general relativity. An importantdifference in LQC is that there will be a finite number of such erachanges in any bounce/recollapse cycle.

Note that the usual BKL transition map is unmodified by LQCeffects; rather LQC adds the bounce, a new type of Kasner transition.

And there are an infinite number of bounce/recollapse cycles, so itappears likely that the Bianchi IX space-time will be chaotic in LQCalso. However, this is a statement about the full sequence ofbounce/recollapse cycles, not about one such cycle alone.

The BKL Conjecture

The BKL conjecture is that near a space-like singularity time-likederivatives dominate over space-like derivatives, and the dynamics ata generic space-time point will be those of a Bianchi IX model.

A number of caveats are important:

Not all singularities are space-like, in particular weak nullsingularities appear to be generic in general relativity,

The BKL behaviour arises asymptotically, potentially past thePlanck scale in which case it would be irrelevant for LQC,

‘Spike surfaces’ are known to arise where spatial derivativesremain important,

The effective equations may break down when considering thedynamics at a point as quantum fluctuations may play animportant role.

Potential Implications for Singularities

With these caveats in mind, what insight can we gain?

In regions where the BKL behaviour arises and quantum gravityeffects are captured by the LQC effective dynamics:

Singularities will be resolved point by point, with the expansionand shear bounded by the Planck scale,

The dynamics will be sensitive to initial conditions at eachKasner era change.

Will this generate uncontrollably large inhomogeneities?

This will depend on the number of Kasner era changes. Previousstudies indicate a small number between the onset of BKL behaviourand the Planck scale [Doroshkevich, Novikov; Weaver, Isenberg, Berger; Garfinkle].

This suggests that inhomogeneities will grow, but not uncontrollably.

Discussion

The bounce in the effective LQC dynamics can be viewed as aKasner transition with the simple rule ki → ki = 2

3− ki relating

the Kasner exponents of the pre- and post-bounce branches.

Transition rules for the LQC bounce can be derived for(u, v , pΩ, κ) that parameterize each Kasner epoch in theMixmaster dynamics; this provides a quantum gravity extensionto the Mixmaster map.

The Bianchi IX space-time is likely to be chaotic for thesequence of all bounce/recollapse cycles.

The BKL dynamics, where applicable, will likely not generateuncontrollably large inhomogeneities due to the limited numberof Kasner transitions near the LQC bounce.

Outlook

There remain many open problems here, two that I think areparticularly important are:

Study numerical solutions of the LQC effective dynamics for theBianchi II and Bianchi IX space-time to check the assumptionsused in the derivations.

Understand how to do a loop quantization of non-perturbativeinhomogeneities (which is tractable for calculations/numerics),perhaps building off the BKL conjecture. The reformulation ofthe BKL conjecture in LQG-like variables [Ashtekar, Henderson, Sloan] couldbe a good place to start.

Thank you for your attention!

Outlook

There remain many open problems here, two that I think areparticularly important are:

Study numerical solutions of the LQC effective dynamics for theBianchi II and Bianchi IX space-time to check the assumptionsused in the derivations.

Understand how to do a loop quantization of non-perturbativeinhomogeneities (which is tractable for calculations/numerics),perhaps building off the BKL conjecture. The reformulation ofthe BKL conjecture in LQG-like variables [Ashtekar, Henderson, Sloan] couldbe a good place to start.

Thank you for your attention!

the loop quantum cosmology bounce as a kasner transition · 2018. 7. 30. · there are two key...

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