Silke Weinfurtner- Emergent spacetimes

8/3/2019 Silke Weinfurtner- Emergent spacetimes

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Emergent spacetimes

By Silke Weinfurtner

and collaborators:Matt Visser, Stefano Liberati, Piyush Jain, Anglea White, Crispin Gardiner and Bill Unruh


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Emergent spacetimes

By Silke Weinfurtner

and collaborators:Matt Visser, Stefano Liberati, Piyush Jain, Anglea White, Crispin Gardiner and Bill Unruh


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Outlook

Spacetime geometry and general relativity

A world full of effective spacetimes

Concept of emergence

....

Emergent spacetimes bearing gifts:

From emergent spacetimes to emergent gravity...?


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Spacetime geometry and general relativity


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space ( d ) + time ( 1) spacetime ( d +1)

Geometry of spacetime:

independent components (without dynamical equations)

In general relativity free particles are freely falling particles - noexternal force but remain under the inuence of the spacetimegeometry. The kinematical equations of motion for free testparticles following geodesics:

Spacetime geometryEinstein: Gravity a consequence of spacetime geometry!


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General relativity also identies (in a coordinate covariantmanner) density and ux of energy and momentum in the n

dimensional spacetime as the source of the gravitational eld .

Stress-energy tensor:

Einstein tensor:

Einstein eld equations


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Spacetime and gravity

spacetime


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spacetime

Broad class ofsystems withcompletely different dynamics:

electromagneticwaveguide, uids,ultra-cold gas ofBosons andFermions;


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spacetimeEinsteindynamics:




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spacetimeEinsteindynamics:



xRestricted dynamics: Emergent gravity..?


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A world full of effective spacetimes


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A simple example: Sound waves in a uid ow

The kinematicequations for small -classical or quantum - perturbations

(i.e., sound waves) in barotropic,invicid and irrotational uid aregiven by

Exact analogy to amassless minimallycoupled scalar eld inan effective curved spacetim:

The rst analogue model: Schwarzschild bh


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Task:

Problems:

Results:

Relevance:

Can we use a uid to mimic a Kerr black hole?

+ Physical acoustics only for wavelength with:

+ Only 1 degrees of freedom for acoustic spacetime+ Conformal atness of spatial slice in acoustic metric

Penrose process; Superradiant

Scattering; Hawking radiationThe equatorial slice through arotating Kerr black hole is formallyequivalent to geometry felt by phononsentrained in a rotating uid vortex.

c k || v ||

Acoustic Kerr black hole


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Natural occurence..?


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v 1 / r

v r

Sound of superradiance?A tornado is a violently rotating column of air,and outside its core it is a perfect example of anirrotational vortex:

The highest wind speeds recorded do not exceed150 m/s (about 0.5 Mach), due to different physics in the core of the vortex:

Idea:

Problem:

Possibility of supersonic wind ows in tornados..?

Natural occurence of acoustic bh


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Propagation of ripples in a basin lled with liquid is governed by

velocity parallel to the basin

Advantage: Extreme ease with whichone can adjust the velocity of thesurface waves

Via the depth of the basin!

Gravity wave analogs of black holesAuthors: Ralf Schtzhold, William G. Unruh

Journal-ref: Phys.Rev. D66 (2002) 044019

Shallow water waves - theory


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Dumb holes: analogues for black holesAuthors: William G. Unruh

Journal-ref: Phil. Trans. R. Soc. A 366 (2008) 2905--2913

Figure 1. The left-to-right ow is supersonic (faster than the velocity of low-wavenumber gravitywaves) to the left of the stanchions and subsonic to the right. The jump in average level at thewhite hole horizon (where the decreasing velocity of the uid just equals the velocity of gravitywaves) can be regarded as a zero-wavenumber wave impinging on the white hole from the right.The undulating wave (which camps out due to the viscosity of the water) going off to the right haszero-phase velocity, but a non-zero group velocity, carrying energy away from the white holehorizon. Quantum mechanically, this would mean that the radiation from a white hole horizonwould not be thermal around zero wavenumber, but rather would be a non-thermal, high-frequencyradiation emitted by the white hole horizon.

ow direction

Shallow water waves - experiment


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Manipulating electromagnetic waves in a wave-guide so thatthey experience an effective curved spacetime in the form ofa (2+1) dimensional Painleve-Gullstrand-Lemaitre geometry.

Wave-guide consistsof a ladder circuit with:Time-dependent capacitanceCoil in each loop has constant inductanceCurrent in each circuit is given byEffective potential , such thatFor wave-length the discretness of the x-axis isnegligibly small, hence :

where

R. Schuetzhold and W. Unruh. Phys. Rev. Lett. , 95:14, 2005.

Electromagnetic wave guide - lattice structure


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Bose-Einstein condensate -a quantum model


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Concept of emergence


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The concept of emergence

Emergent spacetimes involveA microscopic system of fundamental objects(e.g. strings, atoms or molecules);

a dominant mean eld regime, where the microscopicdegrees of freedom give way to collective variables;

a geometrical object (e.g. a symmetric tensordominating the evolution of linearized classical and

quantum excitations around the mean eld;An emergent Lorentz symmetry for the long-distancebehavior of the geometrical object;

high temperature p

low temperature p

f i r

s t

- o

r d

e r

ph

a s e

tr

an

si

ti

on


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Example BEC [microscopic degrees of freedom]

A microscopic system of fundamental objects:

ultra-cold ! lute gas of weakly interaMicroscopic theory well understood:

Emergent spacetimes from Bose-gas

H = d x 2

2 m2 + V ext +

U

2

= exp( i )SO(2) symmetry


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Example BEC [macroscopic variables]

A dominant mean eld regime:

Bose-Ein " ein condensate Spontaneous symmetry breaking:


(t, x ) = (t, x ) = n 0 (t, x ) exp( i 0 (t, x )) = 0


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Example BEC [geometrical object]

Small perturbations - linear in density and phase - in the

macroscopic mean-eld emerging from an ultra-coldweakly interacting gas of bosons are inner observersexperiencing an effective spacetime geometry,

where


gab =c0

U/

2d 1

c20 v2 vx vy vz vx 1 0 0 vy 0 1 0 vz 0 0 1

;


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An emergent Lorentz symmetry:

Bogoliubov !# ersion relation for

Macroscopicvariables

Microscopicvariables

Example BEC [Emergent Lorentz symmetry]


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On inner and out observer and absolute

Inner observer :Small excitations in the systemexperience an effective spacetimegeometry represented by themacroscopic mean-eld variables!

Outer observer :

Live in the preferred frame - thelaboratory frame, such that thecondensate parameters are functions of lab-time (absolute time).

(t, x ), (t, x ) = i (x x )

n (t, x ), (t, x ) = i (x x )


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Semi-classical quantum geometry

Small perturbations around some background solution

In a generic Lagrangian , depdending only a singleScalar eld and its rst derivatives yields an effectiveSpacetime geometry

For the classical/ quantum uctuations. The equation of Motion for small perturbations around the backgroundAre then given by

C. Barcelo, S. Liberati, and M. Visser. Analog gravity from eld theory normal modes?Class. Quant. Grav. , 18:35953610, 2001.

Effective curved-spacetime quantum field theory description of the linearization process:

Kinematics versus dynamics!


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....

Emergent spacetimes bearing gifts:


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Signature of spacetime


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Signature of spacetime - why (-,+,+,+)?

Spacetime foliation into non-intersecting spacelikehypersurfacese (Lapse and Shift)

Signature of spacetime is a certain pattern ofEigenvalues of the metric tensor at each point of themanifold [Loretzian (-,+++) or Riemannian (+,+++)]

There is no driving mechanism within GR that driveschanges in the signature of the geometry...

Kinematics of signature change (Lapse is a non-dynamical variable)


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Signature of spacetime - what is it really?Spacetime foliation into non-intersecting spacelike

hypersurfacese (Lapse and Shift)ds 2 = gab dx a dx b = ( n a n a ) N

2 dt 2 + h ij (dy i + N i dt ) (dy j + N j dt )


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BEC: Interactions spacetime signature

c 20 c (t )2

U U (t ) gab = c(t)U (t)/

2d 1

c(t)2 0 0 0

0 1 0 00 0 1 00 0 0 1

v

0

c20 =n 0 (t, x ) U (t )

m

U > 0 repulsive ;U < 0 attractive .


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BEC: Interactions spacetime signature

c 20 c (t )2

U U (t ) gab =c(t)U (t)/

2d 1

c(t)2 0 0 0

0 1 0 00 0 1 00 0 0 1

v

0

c20 =n 0 (t, x ) U (t )

m


gab + c(t )

2

0 0 00 +1 0 00 0 +1 00 0 0 +1

g ab

c(t )2

0 0 00 +1 0 00 0 +1 00 0 0 +1

Lorentzian signature Riemannian signature


l h


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Daily signature change events...

The bosenova experiment:

l h


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The bosenova experiment:

Lorentzian signatureRiemannian signature

ld h f ld


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Quantum eld theory on Riemannian manifolds

Q ld h Ri i if ld


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Quantum eld theory on Riemannian manifolds

D il i h


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Can we understand the bosenova experiment via the emergentspacetime programme?

Need to understand particleproduction process via

sudden variations in atomic-interactions...

Q ld h Ri i if ld


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Quantum eld theory on Riemannian manifoldsPhysical grasp on quantum field on Riemannian manifolds super-Hubble horizon modes in cosmology:

Mechanism responsible for enormous particle production worksanalogous to cosmological particle production during ination:

vk (t ) + e ff v k (t ) = 0 ,

2at = k

A+ m 2

Q ld h Ri i if ld


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vk (t ) + e ff v k (t ) = 0 ,

2at = k

A+ m 2

k (t ) + k2

e 2 Ht+ m 2 d

2

H 2

4 k (t ) = 0

m < dH

2

k < k HubbleHorizon

Q t ld th Ri i if ld


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vk (t ) + e ff v k (t ) = 0 ,

2at = k

A+ m 2

k (t ) + k2

e 2 Ht+ m 2 d

2

H 2

4 k (t ) = 0

m < dH

2

k < k HubbleHorizon

frozen modes all modes are *frozen*

Signicant particle production on ALL scales!???

T Pl ki b t i t


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Trans-Planckian beats signatureHydrodynamic approximation: Variations in the kinetic energyof the condensate are considered to be negligible, comparedto the internal potential energy of the Bosons.

2

2 m

2 n 0 + nn 0 + n U

U = U 2

4mn 0( n 0 )2 ( 2 n 0 )n 0

n 20

n 0

n 20+ 2

Keeping quantum pressure term leads to effective interaction seen by inner observer:

condensate in box[uniform number density]

harmonic trap[position dependent sound speed]

H = dx 2 m 2 + V ext + U 2

T Pl ki b t ig t


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condensate in box[uniform number density]

Trans-Planckian beats signature U = U

2

4mn 0( n 0 )2 ( 2 n 0 )n 0

n 20

n 0

n 20+ 2 U = U

2

4 mn 0

2 .

healing length:

k 2 m k

2 k c (t ) k

k

2k = c (t ) k 2 + 2m

2

k 4

Lorentz

symmetry

2 (t) = qpc(t)

2

= / 2mc(t)

2

c 2 (U ) = n 0 U m

c 2k ( U ) = c2 (U ) +

2m

2

k 2

T Pl ki b t ig t


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Trans-Planckian beats signature

hydrodynamic

approximationmodied hydrodynamics[including quantum pressure effects]

Number of quasiparticlesinnite!?

T Pl ki b t ig t


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hydrodynamic


Trans Planckian beats signature


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hydrodynamic


U| ik U k = U +2

4 mn 0k 2



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Trans Planckian beats signature but why?


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Let s do quantum gravity phenomenology, in the sense of anultra-high energy breakdown of Lorentz symmetry

d +1 F ( d ) = m where

Trans-Planckian beats signature - but why?



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d +1 is the spacetime DAlembertian d +1 =

1

g d +1 a g d +1 g

ab b




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d is a purely spatial DAlembertian d =

1

g d i g d g ij j


1

g d +1 a g d +1 g

ab b




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d is a purely spatial DAlembertian d =

1

g d i g d g ij j


1

g d +1 a g d +1 g

ab b

2e ff ective = A

d

m2

+ F (k2

/A ) + k2

/A

i sinh E m 2 + k2 /A + F (k2 /A ) Ad/ 2 dt




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i sinh E m 2 + k2 /A + F (k2 /A ) Ad/2 dt

Particle production in *real* world with naive LIV terms:



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i sinh E m 2 + k2 /A + F (k2 /A ) Ad/2 dt

Particle production in *real* world with naive LIV terms:

i sin E B m 2 + 2qp k4 + B k 2 /A A d/ 2 dt

Particle production in analogue *world* - a BEC - withquantum pressure correction to the mean-eld:

Conclusions for signature change events


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* quantum modes on a Riemannian manifold have like super-Hubble horizon modes during ination=> Explains particle production

* Signature change events in the *real* universe show seriousproblems: driving the production of an innite number ofparticle, with innite energy, which are not removed bydimension, rest mass, or even reasonable sub-class of LIV



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* quantum modes on a Riemannian manifold have like super-Hubble horizon modes during ination=> Explains particle production

* Signature change events in the *real* universe show seriousproblems: driving the production of an innite number ofparticle, with innite energy, which are not removed bydimension, rest mass, or even reasonable sub-class of LIV

If there is a way to drive sig. change events within therealm QG, there should be a mechanism to regularize the

innities..! (Analogue to the situation in the BEC)


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Quantum gravity phenomenology

Quantum gravity phenomenology [LIV]


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Quantum gravity phenomenology [LIV]QGP: Summarizes all possible phenomenological

consequences from quantum gravity . While differentquantum gravity candidates may have completely distinct

physical motivation, they can yield similar observableconsequences, e.g.

.

1) Presense of a preferred frame;2) All frames equal, but transformation

laws between frames are modied.

D. Mattingly. Modern tests of Lorentz invariance. Living Reviews in Relativity , 8(5), 2005.

Analogue Lorentz symmetry breaking


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Analogue Lorentz symmetry breaking

Bose-Einstein condensate:

Electromagnetic waveguide:

Symmetry breaking mechanism in different analogue modelslead to model-specic modications:

Despite all fundamental differences similar modications:

Any emergent spacetimes based on analogue models per denhave a preferred frame: The external observer.

Generating a mass and Goldstones theorm


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[*] Back to the equation of motion -Can we extend the class of elds...

[*] Mass generating mechnism - Explicit symmetry breaking...

Generating a mass and Goldstone s theorm

[*] Goldstone

s theorem - Spontaneous symmetry breaking...

Goldstones theorem


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Goldstone s theorem...

... whenever a continuos symmetry is spontaneouslybroken, massless elds, known as Nambu-Goldstonebosons, emerge. [Quantum Field Theory in a Nutshell, A. Zee]

Per denition Bose-Einstein condensation always (spontaneously) breaks SO(2) symmetry of many-body

Hamiltonian!!!

= d x 2

2 m2 + V ext +

U

2

= exp( i )SO(2) symmetry

Mass generating mechnism - all about symmetries


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Mass generating mechnism all about symmetries

Explicit symmetry breaking through

transitions in a 2-component system

M. Visser and S. W. Phys. Rev. , D72:044020, 2005.

U AB

U AA

U BB



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U AB

U AA

U BB



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U AB

U AA

U BB

Mono-metricity


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Mono metricityMore complicated hyperbolic wave equation:

Emergent massive elds


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Emergent massive elds

the acoustic metrics are given by

In-phase perturbtion (= mass zero particle):

Anti-phase perturbtion (= mass non-zero particle):

Collective excitations + microscopic ngerprints


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Collective excitations microscopic ngerprintsBeyond the hydrodynamic approximation...

[1] S. Liberati, M. Visser, and S. W.. Class. Quant. Grav. , 23:31293154, 2006.[2] S. Liberati, M. Visser, and S. W.. Phys. Rev. Lett. , 96:151301, 2006.

Bogoliubov dispersion realtion for 2-comp. sys.


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ogo ubov d spe s o ea t o o co p. sys.

Naturalness problem in emergent spacetime


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p g p

Dispersion relation obtained from our CMS has the form:

* CPT invariant (LIV in the boost subgroup)

* has the form as suggested in many (non-renormalizable) effective eld theory approaches* natural suppression of low-order modications in our model!

* analogue LIV scale is given by the microscopic variables:* not a tree-level result. results directly computed from fundamental Hamiltonian* -coefcients are different for


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Cosmolgoy

On robustness...


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Robustness (against microscopic modications) of quantum eld

theory in emergent spacetimes?

Emergent FRW universe emerging from Bose gas:Microscopic substructure induces Lorentz symmetry breaking

as suggested by many Effective eld theories, i.e. , preferredframe induces non-linear dispersion in the boost-subgroup.

Particle production in general is not robust and signicantmodication may appear.

S. Liberati, M. Visser, and S. W.. Naturalness in emergent spacetime. Phys. Rev. Lett. , 96:151301, 2006.

P. Jain, S. W., M. Visser, and C. Gardiner. Analogue model of an expanding FRWuniverse in BoseEinstein condensates: Application of the classical eld method.arXiv:0705.2077, 2006. Accepted for publication in Phys. Rev. A.

Application for real cosmology?


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Are there any general lessons to be learnt from emergent FRW

spacetimes - incorporating model-specic modications - for cosmology?

Planck-modications

Rainbow geometry k-dependent commutator relation for matter fields

Group and phasevelocity

Dispersionrelation

Cosmologicalhorizons

Modications in:

pp gy

Rainbow geometries


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Emergent spacetime exhibits explicit momentum-dependence:

Macroscopic variables determine metric components:

Speed of sound for perturbations in the condensate:

Background velocity as gradient of the phase of the condensate:

Where represents the UV correction!

g

Rainbow FRW geometries


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In (2+1) dimensions - for time-dependent atomic interactions - we get:

Where the time- and momentum dependence enter as follows:

The closest to a de Sitter like expansion (exact de Sitter inon infrared scales) is given by:

g

Rainbow FRW geometries


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ak (t) = exp(2 H t) + (k / K) 2

In (2+1) dimensions - for time-dependent atomic interactions - we get:

Where the time- and momentum dependence enter as follows:

The closest to a de Sitter like expansion (exact de Sitter inon infrared scales) is given by:

g

Emergent scale factor during ination


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t|k |

a k

01

23

10 40 50100

150200

10 2

10 0

10 2

(a) Scale factor without quantum pressure effects ;t s = 1 10 5 .

t

a k

0 1 2 3 10 4

10 2

10 2

10 0

10 0

10 2

10 2

(b) Scale factor without quantum pressure effects;t s = 1 10 5 .

t|k |

a k

01

23

10 40 50100

150200

10 2

10 1

(c) Scale factor quantum pressure effects ; t s =1 10 5 .

t

a k

0 1 2 3 10 4

10 2

10 1

(d) Scale factor quantum pressure effects; t s =1 10 5 .

g g

Emergent scale factor during ination


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t|k |

a k

01

23

10 40 50100

150200

10 2

10 0

10 2

(a) Scale factor without quantum pressure effects ;t s = 1 10 5 .

t

a k

0 1 2 3 10 4

10 2

10 2

10 0

10 0

10 2

10 2

(b) Scale factor without quantum pressure effects;t s = 1 10 5 .

t|k |

a k

01

23

10 40 50100

150200

10 2

10 1

(c) Scale factor quantum pressure effects ; t s =1 10 5 .

t

a k

0 1 2 3 10 4

10 2

10 1

(d) Scale factor quantum pressure effects; t s =1 10 5 .

g g

Cosmological horizons


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Phase velocity:

Group velocity:

Beyond the hydrodynamic limit: phase velocity group velocity

spaceMaximum distance a signal sent at t 0 can travel:

within the hydrodynamic limit

No cosmological horizon..

How far can a signal travel in areal BEC..?

No cosmological horizon..

g

Hubble parameter?


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Modied Hubble parameter:

Early times

Late times

Hubble frequency?

time

Frequency ratio


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Characteristic value of particle production in thehydrodynamic limit:

Equation of quantum modes in the hydrodynamic limit:

Sign determines nature of quantum modes!

Modied frequency ratio


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Modied frequency ratio:

Turning point:

Freezing and melting of quantum modes


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Hubble horizon

Crossing andre-entering ofHubble horizonduring inationaryepoch! Inationcomes naturally toan end.

Can one alwaysunderstand the

particle productionin terms of thefrequency ratios?

Simulations versus quantitative predictions


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t|k|

N k

01

23

10 40

50 100150 200

0

5

10

15

20

25

30

(a) N k (t ).

t|k|

R k

01

23

10 40

50100

150

100

105

1010

(b) R k (t ).

t

N k

0 1 2 3

10 4

0

5

10

15

20

25

30

(c) N k (t ) projected onto the t - N k plane. (d) R k (t ) projected onto the t - R k plane.

Planck-length during ination


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In some sense we are dealing with a relative scale-shiftbetween the two frames, a time-dependent Planck-scale,

that enters the scale factorof the universe and the effective Hubble parameter

in an unexpected way. (Thenon-perturbative corrections have to be included at thelevel of the hydrodynamic uid equations.

Lorentz symmetry breaking due to a preferred frame (in ourspecic spacetime) has non-negligible effects on the particle

production process as the preferred frame is static!

a FRW (t ) = a FRW ,0bn(" )

ba (" )

12(d 1)

Planck (t ) =# qp

c(t )=

# qp

c0 bn(t )ba (t )

= Planck ,0

bn(t )ba (t )

However it is possible to mimic an expanding universe and keep theeffective Planck length constant:


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From emergent spacetimes to emergent gravity...?

Emergent gravity


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spacetime

Einstein

dynamics: x



(1+4) dimensionalquantum Halleffect, quantumrotor model;

spin-2particle?

Restricted dynamics: Emergent gravity..?

Emergent gravity


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spacetime

Einstein

dynamics: x



(1+4) dimensionalquantum Halleffect, quantumrotor model;

spin-2particle?

Restricted dynamics: Emergent gravity..?


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Id like to thank Matt and everyone at VIC for the warm welcome!

books on the Emergent Spacetime / Analogue Models for Gravity:Articial black holes (Novello, Visser, Volovik)The universe in a helium droplet (Volovik)Quantum Analogues: From phase transitions to black holes and cosmology (Unruh, Schuetzhold)


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books on the Emergent Spacetime / Analogue Models for Gravity:Articial black holes (Novello, Visser, Volovik)The universe in a helium droplet (Volovik)

Silke Weinfurtner- Emergent spacetimes

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