Parampreet Singh Institute for Gravitational Physics and ...gravity.psu.edu/events/conferences/Quantum_GravityIII/Talks/Singh.p… · Dirac Observables: , Quantum constraint: with

Recent Advances in Loop Quantum Cosmology

Parampreet Singh

Institute for Gravitational Physics and Geometry, Penn State

Quantum Gravity in Americas III

QG in Americas – p.1

Outline

Introduction & Motivation

Loop Quantum Cosmology: Basics and Early Results (Bojowald, ...)

Recent Advances: Detailed analysis of simple models, Answers tosome key questions

Physical Applications and Extensions

Summary and Open Issues


Introduction & Motivation

General Relativity inadequate to provide correct physics at Planckscale. The backward evolution of our Universe leads to a Big Bangsingularity (result of powerful singularity theorems).

Occurrence of singularity � � limit of validity of the theoryNeed of new physics from Quantum Gravity

What shall quantum gravity tell about the early Universe ?Is there a big bang ?Does the model provides a non-singular evolution through theclassical singularity ? What is on the other side – a quantum foamor a classical spacetime ?What is the scale at which the spacetime ceases to beclassical ? Does the spacetime continuum exists at all scales,especially when primordial fluctuations were generated duringinflation ?What are the modifications to the Friedmann dynamics in the earlyUniverse and at what scales these become important ?Does the theory make testable predictions ?


Various Innovative Ideas

Pre-Big Bang Models: Based on the scale factor duality of the stringdilaton action. Envisions existence of a classical pre-big bang phasewhich undergoes inflation. No generic non-singular evolutionthrough the big bang.

?

Pre−Big Bang Post−Big Bang

Curvature

time

Similar limitations with Ekpyrotic/Cyclic models based on branedynamics in a bulk.

Large extra dimensions: Modifications expected to Friedmannequation. Example: Randall-Sundrum Model:

� � �� ;� � � � � �� . As � � �

,

�� . Leads to singular evolution asin standard cosmology.


Missing Elements:

Non-perturbative quantum gravitational effectsFeatures of Quantum Spacetime

Our Strategy: Construct a quantum cosmological model based onLoop Quantum Gravity � � Loop Quantum Cosmology

Caveats:Systematic derivation from LQG to be worked out

Objectives:

Glimpses of singularity resolution and physics of very earlyUniverseTesting ground for model building, phenomenologicalapplications, making testable predictionsValuable insights to complete the program in LQG and othernon-perturbative approaches


LQC: Homogeneous and Isotropic setting

Spatial homogeneity and isotropy: fix a fiducial triad

� � �� and co-triad� � �� . Symmetries ��

Basic variables: � and satisfying

� �� .

Relation to scale factor:� � � � �

(two possible orientations for the triad)� � � � � (on the space of physical solutions of GR).

Elementary variables – Holonomies:�� !#" �� %$ � , � & ��

Elements of form �' ( �) � � ��

– generate algebra of almost periodicfunctionsHilbert space:

*,+- . � / � �0 1� 2� 1

Orthonormal basis:3 � � � �' ( � ) � � ��

;

4 3 �� 3 �� 5 6 � 798;: 8 <

Hilbert Space different from the Wheeler-DeWitt theory


Hamiltonian Constraint

��

2 �� 3� �� Procedure: Express

�� in terms of elementary variables and theirPoisson brackets

Use Thiemann’s trick:

�� 8 �� 8 � � �� $ �

(

��

, ! can be a function of � )

Express field strength in terms of holonomies. " #%$ & ' ( Limit of the holonomy

around a loop divided by the area of the loop, as area shrinks to zero.

Limit well defined in full theory on Diff-Inv states.In LQC: Diff-Inv fixed. New strategy (Ashtekar, Bojowald, Lewandowski (2003))

Exploit the area gap in LQG :

) � � *� � � + �-,

Area associated with a square loop (with respect to physicalgeometry) = .� � � �

/�10 � � 6 � � �� + �2, �3 � � � .� � � � 6

Put eigenvalue =

) � � .� �4� *� �� 5 � 6 �

(Ashtekar, Pawlowski, PS (2006))


Action of

�� 8� 6 �

: Lie drag of the state by a unit affine parameterdistance along the vector field .� �� 8 . � is affine parameter proportionalto eigenvalues of the Volume operator:

� � � �6 � � �� 3 � 6 � � � � � �� + �, � �6 � � � � � �" �� 6 �� *� �� *�

Convenient to switch to

� � � :

�� 8� 6 � � � � � � � � � �

Quantization of the constraint:

� ��

� � � � � �

� ��

where ��

� 3

��3

� + ,

� � 6 ��

��

��

� � � ��

� � � � � � � ��

��

� �


Features of the quantum constraint:� �� is self-adjoint and negative definite.Evolution in constant steps of eigenvalues of the volume operator.Evolution non-singular across � � �

for all states.� ��

�� with natural factor ordering for

� � � � � .

In earlier works, constraint based on different association of the areaof the square loop: .� � � � � � *� ��

. Quantum evolution in steps of�� . Evolution non-singular across � � �for all states (Bojowald (2001).

Similar non-singular constraint obtained for closed and anisotropicmodels (Bojowald, Date, Hossain, Vandersloot (2003-2004)). Many physical applicationsbased on effective theory primarily dealing with modifications tomatter (Nunes’s talk).Problems with gravitational sector – Unphysical predictions!

Pending questions:What is the physics of singularity resolution ?

How to extract trustable physical predictions in LQC ?QG in Americas – p.9

Algorithm to extract Physics in LQC

Seek a dynamical variable that can play the role of internal time(physical interpretations more transparent).

Introduce Inner Product, find Physical Hilbert Space.

Construct Dirac Observables, raise them to operators.

Construct physical semi-classical states representing a large classicalUniverse at late times.Evolve these states backward towards Big Bang using quantumconstraint. Analyze the behavior of expectation values andfluctuations of Dirac observables.Compare with classical dynamics, obtain predictions.


Massless Scalar Field ModelAshtekar, Pawlowski, PS (2006)

Phase space:

� ��

,

� ��

� �� 3 � �

� ��

� � � 6 � � �

� � �� " � �� " ��

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 1*104 2*104 3*104 4*104 5*104

v

φ

All solutions are singular


�

is a monotonic function, can play the role of internal timeEvolution refers to relational dynamics – the way geometry changeswith ‘time’ (or as

�

evolves).Dirac Observables: � ,

� � � ��

Quantum constraint:

� ��

� � � ��

��

� � � � ��

��

with

� � � � eigenvalues of inverse triad operator.Constraint similar to the massless Klein-Gordon equation in staticspacetime.

�

plays the role of time,

�

of Laplacian-type operator.

�

is self-adjoint and positive definite.

Inner Product:Demand that action of operators corresponding to Diracobservables is self-adjointGroup averaging


Result:Physical considerations require symmetric states:� � ��

� ��

.Inner Product:

� � ��

are positive frequency: �) ��

* � �

4 � � � � �6 �

�

� � � . � � � ��

Action of Dirac observables:

� � � � �) � ��

Numerics:Initial data at

� � � � consists of� � �� and

�� .Choose semi-classical states peaked at a large value of � � �� atlate times. Fix a point

� � �� on the classical trajectory for � � � �

(large classical Universe).Using quantum constraint follow its evolution backward andcompare with classical trajectory.

(More details ( Pawlowski’s talk)


Quantum Bounce

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-1.2

-1

-0.8

-0.6

-0.4

-0.2

5*103

1.0*104

1.5*104

2.0*104

2.5*104

3.0*104

3.5*104

4.0*104

0

0.5

1

1.5

|Ψ(v,φ)|

v

φ

|Ψ(v,φ)|


Comparison of Evolution

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 1*104 2*104 3*104 4*104 5*104

v

φ

LQCclassical


Results of Loop Quantum Evolution

States remain sharply peaked through out the evolution.

Expectation values of

� � � � and � are in good agreement with classicaltrajectories until energy density becomes of the order of a criticaldensity�

�� - � ( � 0.82� � � )

The state bounces at critical density from expanding branch to thecontracting branch with same value of

4 � � 6 . This phenomena isgeneric. Big bang replaced by a big bounce at Planck scale.

Norm and expectation value of

� � remain constant.

Fluctuations of observables remain small. Some differences arise nearthe bounce point depending on the method of specification of theinitial state.The old constraint shares similar features but� �� - � is not a constant(� �� - � � � � � ).

Non-singular evolution achieved naturally


Effective Theory

An effective Hamiltonian description can be obtained by using thegeometric methods of quantum mechanics (Ashtekar, Bojowald, Willis (2004); Bojowald,

Skirzewski (2005))

Useful tool to understand the underlying quantum dynamics.Bounce unaffected for more general states (Bojowald (2006))

Results in an effective Friedmann equation:

� � ��

� � ��

��

�� - � � �� - � �

*�

� 3 � � � � � � �

Features:Similar to the Friedmann equation in Randall-Sundrum braneworlds,except for the (-) sign.Small difference in Friedmann equation � Profound implications forPhysics.

For� � �� - � , classical Friedmann dynamics is recovered.

Bounce occurs at� � � �� - � . Quantum geometric modifications tomatter not important for the bounce.


Applications:Attractors of dynamics in effective theory have been found. Usefulto construct more general cosmological models, e.g. quintessenceand relate distinct models. Mathematical dualities found withbraneworld models. (PS (2006))

Effective theory + Potentials from Cyclic/Bi-Cyclic models:Non-singular bounce occurs in generic conditions for Bi-Cyclicmodel potentials. Fine tuning problems can be alleviated. (PS,

Vandersloot, Vereshchagin (2006)), Vandersloot’s talk

Effective dynamics can successfully get rid of a future big ripsingularity, if the universe is dominated by an exotic field at latetimes. (M. Sami, PS, S. Tsujikawa (2006))

Extensions (Robustness of results):

� � � model: Knowledge of inner product leads to overcominglimitations noticed by Green and Unruh. Universe bounces at thecritical density and re-collapses at the classical scales as predictedby GR.(Ashtekar, Lewandowski, Szulc, Vandersloot (2006); Ashtekar, Pawlowski, PS (2006))

Quantum theory analyzed for the Bianchi I model with a masslessscalar field. Singularity resolution obtained. (Chiou (2006))


Summary and Open Issues

Loop quantum cosmology provides a new picture of the Universenear and at the big bang and beyond.

Big bang not the beginning, big crunch not the end.Two classical regions of spacetime joined by a

quantum geometric bridge.

Quantum gravity makes curvature non-local at Planck scale. Thisplays an important role at to yield non-singular evolution across theclassical singularity.

Gravity becomes repulsive at Planck scale. Bounce occurs in genericconditions. No need to introduce exotic matter or ad-hoc assumptions.

Important insights gained in quantization of simple models:Ambiguities can be narrowed down by physical considerations.Example: In � � evolution, critical density not constant

� � � � �

,bounce could occur at small curvatures !


How generic are the results ?Positive indications from the works on anisotropic models.A massive scalar field model will serve as an important step toconsider more realistic cosmological scenariosKey question: Does the picture of bounce survive on incorporatinginhomogenities ? Can perturbations be propagated across thebounce ? Work on these issues started (Kagan’s talk)

Fundamental Issues:What is the connection with full theory ? Preliminaryinvestigations (Engle’s talk)

Link to other ideas in quantum cosmology: The pre “big bang”phase may be envisioned as approximating the Euclidean patch inHartle-Hawking scenario.Quantum to Classical transition ?

Testable predictions ? Confirmation with observations ?


Thanks.


Problem: What is the action of

�� 8� 6 �

?

Idea: Since �� 8� � 6 � � � � � ��

action is to drag the state a unit affine parameter distance along the vectorfield � � �� 8

Set

�� 8� 6 � � � ��

as Lie drag of the state by a unit affine parameter distancealong the vector field .� �� 8 .Affine parameter of this vector field:

� � � � �" �� 6 �� *� � � � *�

� are proportional to the eigenvalues of the Volume operator. � ��

isinvertible,

� �

function of � .

� � � �6 �

�� 3

� 6 � � � �� + �, � �6

Orthonormal basis in* +- . : 4 � � � � �6 � 7

��: ��

Consider

� � � ��

: �� 8� 6 � � � � � � � � � �


For

� � � � � ,

� � � � B

� � � � � � � � � � .Wheeler-DeWitt limit for the constraint:

� ��

� � ��

In geometrodynamics classical constraint:

� ��

In quantum theory natural choice of factor ordering � Laplacian for� ��

.LQC Hamiltonian constraint automatically yields the natural factorordering for Wheeler-DeWitt.

Action of parity operator:

�� large gauge transformation on the space of solutions. No

observable which can detect the change in orientation of the triad.Physical considerations require symmetric states.


Numerics

Evolution in

�

:

Initial value problem in

�

. Solve

� ��

� � � countable number ofcoupled ODE’s.

Initial data at

� � � � consists of

� � �� and�� . Initial data

specified in three different ways using classical equations.

For semi-classical states choose a large value of � � �� . Fix a point� � �� on the classical trajectory for � � � � (large classical Universe).Using quantum constraint follow its evolution backward.Example of initial state:

� � ��

��

2 � � 5 � � ��


Does the theory provides non-singular evolution through the big bang? Yes, for all solutions.What is on the other side of the big bang ? Quantum foam or aclassical spacetime ? Universe escapes big bang and bounces at Planckscale to a pre-big bang classical contracting branch.

What is the scale at which the spacetime ceases to be classical ? Doesthe spacetime continuum exists at all scales ? When the energy densitybecomes of the order of a critical density ( � 0.82� , � � . �+ ), deviationsfrom classical dynamics are significant and spacetime ceases to beclassical. Picture of a continuum spacetime breaks down in the deepPlanck regime.

What about the modifications to Friedmann equation ? � �

modifications at high energy scales with a negative sign.


Documents

Parampreet Singh Institute for Gravitational Physics and ...gravity.psu.edu/events/conferences/Quantum_GravityIII/Talks/Singh.p… · Dirac Observables: , Quantum constraint: with