holographic cosmology meets lattice gauge theory - menu · holographic universe â in holographic...

IntroductionHolographic cosmology

Perturbative QFTHolographic Lattice Cosmology

Conclusions

Lattice Holographic Cosmology

Kostas Skenderis

STAG RESEARCHCENTERSTAG RESEARCH

CENTERSTAG RESEARCHCENTER

Numerical approaches to the holographicprinciple, quantum gravity and cosmology

Kyoto, Japan, 22 July 2015

Kostas Skenderis Holographic cosmology meets lattice gauge theory



Conclusions

Outline

1 Introduction

2 Holographic cosmology

3 Perturbative QFT

4 Holographic Lattice Cosmology

5 Conclusions




Conclusions

Introduction

â The scale of inflation could be as high as 1014 GeV and as such itis highest energy scale which is directly observable.

â This is much larger than any energy scale we could achieve withaccelerators.

à The physics of the Early Universe is a unique probe of physicsbeyond the standard model.




Conclusions

Inflation

â The leading theoretical paradigm for the very early universe isthe theory of inflation.

â It describes remarkably well existing observational data.â It is based on gravity coupled to scalar field(s), perturbatively

quantized around an accelerating FRW background.




Conclusions

Inflation: problems

â Despite its successes the theory of inflation still has a number ofshortcomings: fine tuning, trans-Planckian issues etc.

â This description breaks down at some point since thebackground has a curvature singularity: the theory has to beembedded in a "UV complete theory" (the "initial singularityproblem").

â However, it has been very difficult to embed inflation infundamental theory (such as string theory).




Conclusions

Holographic cosmology

â Holography provides a new framework that can accommodate:à conventional inflation: strongly coupled dual QFTà qualitatively new models for the very early Universe : QFT at

weak and intermediate coupling.

â The new framework gives new insight into conventional inflation.â The new models are falsifiable with current data.




Conclusions

References

â A holographic framework for cosmology was put forward in workswith Paul McFadden and Adam Bzowski (2009 - on-going).

â Related work:[Hull (1998)] ... (E-branes)[Witten (2001)] [Strominger (2001)] ... (dS/CFT correspondence)[Maldacena (2002)] ... (wavefunction of the universe)[Hartle, Hawking, Hertog (2012)] ... (quantum cosmology)

[Trivedi et al][Garriga et al] [Coriano et al] .... [Arkani-Hamed,Maldacena]




Conclusions

References

In this talk I will discuss new work on

â further developing the models based on perturbative QFT. Thispart is based on work in progress with

â Claudio Coriano and Luigi Delle Rose

â using Lattice methods to construct models valid for any value ofthe coupling constant. This part is based on work in progresswith

â Evan Berkovitz, Philip Powel, Enrico Rinaldi, Pavlos Vranas(Lawrence Livermore National Laboratory)

â Masanori Hanada (YITP Kyoto and SITP Stanford)â Andreas Jüttner, Antonin Portelli, Francesco Sanfilippo (SHEP,

Southampton)




Conclusions

Outline

1 Introduction


3 Perturbative QFT


5 Conclusions




Conclusions

Holographic cosmology in a nutshell

In holographic cosmology one relates:

â cosmological observable such as the power spectra andnon-gaussianities

â to correlation functions of the the energy momentum tensor ofthe dual QFT, upon a specific analytic continuation.




Conclusions

Holographic cosmology: the dual QFT

â The dual QFT is 3d QFT that admits a large N limit and ourresults apply to two classes of theories:

à QFTs with a non-trivial UV fixed point.à A class of super-renormalizable QFTs.

â The results hold perturbatively in 1/N2. It is not clear whetherthese dualities hold non-perturbatively in 1/N2.




Conclusions

Holographic cosmology: the bulk

These results hold for spacetimes that at late times approach

à de Sitter spacetime,

ds2 → ds2 = −dt2 + e2tdxidxi, as t→∞

à power-law scaling solutions,

ds2 → ds2 = −dt2 + t2ndxidxi, (n > 1) as t→∞




Conclusions

Holographic formula for the scalar power spectrum

â The scalar power spectrum is given by

∆2R(q) = − q3

4π2

1Im 〈T(q)T(−q)〉

,

where T = T ii is the trace of the energy momentum tensor Tij

and we Fourier transformed to momentum space. The imaginarypart is taken after the analytic continuation,

q→ −iq, N → −iN

â The power spectrum of tensors is related with the 2-point of thetraceless part of T and non-gausiaities are related withhigher-point functions of Tij.




Conclusions

Sketch of derivation IThe underlying framework is gravity coupled to a scalar field Φ with apotential V(Φ),

S =1

2κ2

∫d4x√−g(R− (∂Φ)2 − 2κ2V(Φ))

There is 1-1 correspondence [KS, Tonwsend (2006)] between:Domain-wall solutions

ds2 = dr2 + e2A(r)dxidxi

Φ = Φ(r)

FRW spacetimes

ds2 = −dt2 + a2(t)dxidxi

Φ = Φ(t)




Conclusions

Domain-wall/Cosmology correspondence

Domain-wall/Cosmology correspondence

FRW solutions of ↔ Domain-wall solutions ofthe theory with potential V(Φ) the theory with potential −V(Φ).

This correspondence can be understood as analytic continuation.An example of this correspondence is the analytic continuationfrom de Sitter to Anti de Sitter. This theorem shows that thisrelation is not accidental.




Conclusions

Inflation/holographic RG correspondence

â A special case of the correspondence is that between inflationarybackgrounds and holographic RG flow spacetimes.

â Inflationary spacetimes are mapped toâ asympotically Anti-de Sitter spacetime,

ds2 → ds2 = dr2 + e2rdxidxi, as r →∞

â power-law scaling solutions,

ds2 → ds2 = dr2 + r2ndxidxi, (n > 1) as r →∞

For special values of n these backgrounds are related tonon-conformal branes.

â For these backgrounds there is an established holographicdictionary.




Conclusions

Holographic formulae for cosmology [McFadden, KS]

â Given an FRW, compute cosmological observables usingstandard cosmological perturbation theory.

â Corresponding to this FRW there is a domain-wall.â Use holography to compute energy-momentum tensor

correlators for the QFT dual to the domain-wall.

à Comparing the two results leads to the holographic formulae.




Conclusions

Remarks

â This derivation holds in the regime the gravity approximation isvalid.

à Conventional inflation is holographic.

â We will next present an alternative derivation which does notmake this assumption but postulates the form of the duality.




Conclusions

Sketch of derivation II: the wavefunction approach

â The partition function of the dual QFT (after analytic continuation)computes the wavefunction of the Universe [Maldacena (2002)]:

ψ[Φ] = ZQFT [Φ]

â Cosmological observables are computed as

〈Φ(x1) · · ·Φ(xn)〉 =

∫DΦ|ψ|2Φ(x1) · · ·Φ(xn)

â The partition has an expansion in correlation functions:

ZQFT [Φ] = exp

(∑n

〈O(x1) · · ·O(xn)〉Φ(x1) · · ·Φ(xn)

)â Applying this to 2-point function of T leads to the formula quoted

earlier.Kostas Skenderis Holographic cosmology meets lattice gauge theory



Conclusions

Gauge/gravity duality

An important feature of the holographic correspondence is that it is aweak/strong duality.

weakly coupled gravity ⇔ strongly coupled QFT

à QFT correlation functions at strong coupling are related toEinstein gravity.

Strongly coupled gravity ⇔ weakly coupled QFT

à Strongly coupled gravity here means that there is no notion ofspacetime. This is a non-geometric phase.




Conclusions

Holographic Universe

â In holographic cosmology:Cosmological evolution = inverse RG flow

â In our Universe we are currently living in an accelerating phase(driven by dark energy) and we believe that the Universeunderwent a period of inflation at early times.

â This translates into specific properties of the dual QFT.




Conclusions

Holographic Universe

à The dual QFT should have a strongly coupled UV fixed pointcorresponding to the current dark energy era.

à In the IR the theory should either flow to:â an IR fixed point (corresponding to de Sitter inflation), orâ a phase governed by a super-renormalizable theory

(corresponding to power-law inflation).

à To correctly model the history of our Universe we would have tocorrectly account for the rest of the cosmological periods(radiation domination, matter domination).

â The holographic description of these eras is not currently known.




Conclusions

Holographic inflation

We focus from now on the IR of the theory (inflationary era).

â If the QFT is strongly coupled in the IR then this wouldcorrespond to perturbative gravity in the early Universe.

à Such theories correspond to conventional inflationary models.à Checking the QFT predictions against standard cosmological

perturbation theory would provide a test of holography.

â If the QFT is not strongly coupled in the IR then this wouldcorrespond to a non-geometric phase in the early Universe.

à These are qualitative new modes for the very early Universe.à Checking the QFT predictions against observations one can either

obtain support or falsify these models.




Conclusions

Holographic Lattice Cosmology

â Use Lattice methods to compute the QFT observables.

à If the QFT is weakly coupled, one can check the Lattice resultsagainst perturbative QFT computations.

à If the QFT is strongly coupled, the Lattice provides a firstprinciples derivation of the correlators. This would provide a testof holographic dualities.

à If the coupling is of intermediate strength we get new models forthe Very Early Universe. The predictions of these models can betested against CMB data.




Conclusions

Take-home message

â If one can simulate 3d QFTs which in IR have either a fixed pointor become super-renormalizable then one has interestingholographic models for the very early universe.

â Depending on the nature of the IR theory (strongly or weaklycoupled) one can either test holography or obtain predictions thatcan be checked against observations.




Conclusions

Outline

1 Introduction


3 Perturbative QFT


5 Conclusions




Conclusions

Super-renormalizable theories

â A class of models that admit a holographic description:

S =1

g2YM

∫d3xtr

[12

FijFij +12

(DφJ)2 + ψK /DψK

+ λJ1J2J3J4φJ1φJ2φJ3φJ4 + µαβJL1L2

φJψL1α ψ

L2β

].

All fields are massless and in the adjoint of SU(N), λJ1J2J3J4 , µαβJL1L2

are dimensionless couplings while g2YM has mass dimension 1.

â An example of such theory is the maximally supersymmetricSYM theory in d = 3.




Conclusions

Generalized conformal structure

What is special with this theory? Let us consider this theory ingeneral d:

S =1

g2YM

∫ddxtr

[12

FijFij +12

(DφJ)2 + ψK /DψK

+ λJ1J2J3J4φJ1φJ2φJ3φJ4 + µαβJL1L2

φJψL1α ψ

L2β

].

â In this action all terms scale the same way if one assigns "4ddimensions" to the fields: [φ] = [A] = 1, [ψ] = 3/2.

â If we promote g2YM to a field which transforms under conformal

transformations then the theory would be conformally invariant[Yevicki, Kazama, Yoneya (1998)].

â This generalised conformal structure is not a bona fide symmetryof the theory but nevertheless controls many of its properties.




Conclusions

2-point functions

â The form of 2-point functions is fixed by generalized conformalinvariance [Kanitscheider, KS, Taylor (2008)]. In momentum space,

〈Φ(q)Φ(−q)〉 = q2∆−dc(g2)

where g = g2YM/qd−4 is the dimensionless coupling constant and

in the perfurbative regime

c(g) = c0g + c1g2 + c2g3 + · · ·

ci(d) is the i-loop contribution.When Φ = {A, φ, ψ}, ∆ = {1, 1, 3/2}.

â In even/odd dimensions c1(d)/c2(d) has a pole 1d−(2k(+1)) and this

induces gn log g in c(g) [Coriano, Delle Rose, KS (to appear)].




Conclusions

2-point functions: renormalization

We now focus on d = 3.

â One-loop is automatically finite when using dimensionalregularization.

â At 2-loops, (only) the scalars have a UV divergence which can beremoved by adding a bare mass term.

â At loops, all 2-point functions have an IR singularity. It wasargued in [Jackiw,Templeton (1981)][Appelquist, Pisarski(1981)] thatthese type of theories are non-perturbative IR finite:g2

YM effectively acts as an IR regulator.




Conclusions

Energy-momentum tensor

â To simplify the presentation, I focus on the 2-point function of thetrace of T. At large N,

〈T(q)T(−q)〉 = qdN2f (g2eff),

where g2eff = g2

YMN/q is the effective dimensionless ’t Hooftcoupling and f (g2

eff) is a general function of g2eff.

â When g2eff is small, the function f (g2

eff) has the form

f (g2eff) = f0(d) + f1(d)g2

eff + · · ·




Conclusions

Energy-momentum tensor

â f0 is determined at 1-loop. It is finite in odd dimensions and itdiverges in even dimensions. It has been computed in [McFadden,KS (2009)] for d = 3 and in [Coriano, Delle Rose, KS (to appear) ingeneral d.

â f1 is determined at 2-loops. It is finite in even dimensions and hasa UV divergence in odd dimensions.

â f1 is IR finite, unless the scalars are non-minimally coupled.




Conclusions

Energy-momentum tensor: renormalization

We now focus on d = 3.

â The UV infinities can be cancelled by adding the counteterm

aCT

∫d3xR

where aCT is an appropriately chosen coefficient.â After renormalization,

f (g2eff) = f0(1− f1g2

eff ln g2eff + f2g2

eff + O[g4eff]).

where

f2 = α0 + α1 logg2

YMNµUV

+ α2 logg2

YMNµIR

where αi are constants that depend on the field content.




Conclusions

Holographic power spectrum

â To compute the holographic scalar power spectrum we need toanalytically continue 〈T(q)T(−q)〉.

â The analytic continuation acts as

g2eff → g2

eff, N2q3 → −iN2q3

and therefore

∆2R(q) = − q3

4π2

1Im〈T(q)T(−q)〉

=1

4π2N2

1f (g2

eff)

â Thus, for this class of theories and in the perturbative regime:

∆2R(q) =

(1

4π2N2f0

)1

1− f1g2eff ln g2

eff + f2g2eff




Conclusions

Holographic power spectrum

0 2 4 6 8ln Èq�gq*È0.6

0.8

1.0

1.2

1.4

1.6

D2RHqL�D2

R

Blue curve: g > 0, Red curve: g < 0Kostas Skenderis Holographic cosmology meets lattice gauge theory



Conclusions

Confronting with data

â In cosmology there are very few observables so the way wecheck the theory against data is different than in high energyphysics.

â The main question one addresses is:

Given a set of models, which one is preferred by the data?

â One way to answer this is to check how well the model fits thedata: what is the probability for obtaining the data given themodel.

â A better way is to compute the so-called Bayesian Evidence:what is the probability for the model given the data.




Conclusions

Protocol

1 Choose a model with desired IR behaviour.

2 Compute 2-point function of the energy momentum tensor.

3 Insert in holographic formula to obtain the holographic prediction.

4 Compute Bayesian Evidence to check whether the model is ruledin or out.




Conclusions

Fitting to data

â Assuming f2 is negligible and redefining variables, f1g2YMN = gq∗,

where q∗ is a reference scale that is taken to be q∗ = 0.05 Mpc−1

(the WMAP momentum range is 10−4 . q . 10−1 Mpc−1), weobtain the final formula:

∆2R(q) = ∆2

R1

1 + (gq∗/q) ln |q/gq∗|,

→ ∆2R = 1/(4π2N2f0). Smallness of the amplitude is related with the

large N limit: matching with observations implies N ∼ 104.→ When (gq∗/q)� 1 one may rewrite the spectrum in the

power-law form

∆2R(q) = ∆2

Rqns−1, ns(q)− 1 = gq∗/q

Thus the small deviation from scale invariance is related with thecoupling constant of the dual QFT being small.




Conclusions

Holographic model vs slow-roll inflation

â The effective spectral index, ∆2R(q) ∼ qns(q)−1, has the property:

−(ns − 1) =gq∗q

=dns

d ln q= · · · = (−1)n+1 dnns

d ln qn .

â This is very different from slow-roll models where higher orderrunning is suppressed by slow-roll parameters.

â Another difference is that one can easily accommodate anyamount of tensors: the ratio of scalars-to-tensor r depends onthe field content.

â Given the significant differences, we undertook a dedicated dataanalysis [Easther, Flauger, McFadden, KS (2011) [Dias (2011)]) tocustom-fit this model to WMAP and other astrophysical data.




Conclusions

Holographic model vs ΛCDM

The power-law ΛCDM model depends on six parameters. Fourdescribe the composition and expansion of the universe and theother two are the tilt ns and the amplitude ∆2

R of primordialcurvature perturbations.The holographic ΛCDM model depends on the same set ofparameters, except that the tilt ns is replaced by the parameter g.We determined the best-fit values for all parameters for bothmodels and used Bayesian evidence in order to make a modelcomparison.




Conclusions

Angular power spectrum: ΛCDM vs holographic model

5 10 50 100 500 1000-4000

-2000

0

2000

4000

6000

8000

{

{H{

+1L

C{�2

Π@Μ

K2

D

Red: ΛCDM, Green: holographic modelKostas Skenderis Holographic cosmology meets lattice gauge theory



Conclusions

Parameter estimation

The estimated values for the five common parameters of the twomodels are roughly within one standard deviation of each other.The data favor negative values of g (red spectrum) with centralvalue g = −1.27× 10−3.




Conclusions

Is the perturbative treatment justified?

â The value of g leads to a small effective coupling, exceptpotentially for the very low wavelength modes. Sinceg2

eff = (1/f1)(gq∗/q) one needs to know the value of the 2-loopfactor f1 when (gq∗/q) itself is not very small.

â A related issue is whether the parameter f2 is important. If it isthe power spectrum is modified as:

∆2R(q) = ∆2

R1

1 + (gq∗/q) ln |q/βgq∗|

→ f2 cannot be computed perturbatively when there are IRdivergences.

→ If g2eff is not small for all relevant momenta, one must include

higher order terms in the computation of the 2-point function.




Conclusions

Model 1

â SU(N) gauge theory coupled to Nφ conformally coupledmassless scalars (without self-interaction).

à The perturbative answer at 2-loops is [Coriano, Delle Rose, KS (toappear)],

f0 =164, f1 = − 2

3π2 (Nφ − 4),

f2 = − 124π2 (16 + 3π2)− 8

3π2 (Nφ − 1) logg2

YMNµUV

+1

2π2 Nφ logg2

YMNµIR

à We would need Nφ ∼ 300 in order to satisfy the constrain thatgravitational waves have not been observed so far (r ≤ 0.1 ).




Conclusions

Model 2

â A non-minimally coupled massless scalar field in the adjoint ofSU(N) with φ4 self-interaction

S =1λ

∫d3xTr

(12

(∂µφ)2 +14!φ4),

and energy momentum tensor

Tij =1λ

Tr(∂iφ∂jφ− δij(

12

(∂φ)2 +14!φ4) + ξ(δij�− ∂i∂j)φ

2)

â The perturbative answer to 2-loops is [Coriano, Delle Rose, KS (toappear)]

f0 =(1− 8ξ)2

256, f1 = 0, f2 = − 1

24à We need |ξ − 1/8| ∼ 10−2 to satisfy r ≤ 0.1 [Kawai, Nakayama

(2014)] .â The fit of the perturbative model to data is in progress.




Conclusions

Outline

1 Introduction


3 Perturbative QFT


5 Conclusions




Conclusions

Holographic Lattice Cosmology

â Perturbative QFT yields interesting new models for the very EarlyUniverse.

â Comparing with data suggests that we may need to go beyondleading order/need non-perturbative information.

â QFT at intermediate coupling may provide yet more interestingmodels.

à Use Lattice to compute the relevant QFT observables.




Conclusions

Prof of concept: put model 2 on Lattice

â One can straightforwardly discretize the action,

Slattice = a3∑~n

Tr

12

(φ~n+µ − φ~n

a

)2

+14!φ4~n

,

where a = aλ is a dimensionless lattice spacing (a is the latticespacing).

â We need to add a mass counterterm to remove UV infinities (likein the continuum):

δm2 = δm2divergent




Conclusions

Finding the massless point: Binder Cumulant

â We also need to add and fine tune a finite mass δm2finite so that

the renormalized mass vanishes in the continuum limit.

â If the mass in the continuum limit is positive then 〈Mn〉 = 0 forany n, where M =

∑~n φ~n.

â If the mass in the continuum limit is negative we are in thespontaneously broken phase, 〈Mn〉 6= 0.

â To find the massless point one may compute U = 〈M2〉2/〈M4〉 fordifferent lattice sizes and find the intersection point.




Conclusions

Binder Cumulant for a single scalar field

-0.03 -0.028 -0.026 -0.024 -0.022

0

0.2

0.4

0.6

Vertical axis: U, Horizontal axis: δm2finite/λ

2, Lattice size: 243, 323, 483, 643




Conclusions

Energy momentum tensor

â The discretized energy momentum tensor reads

Tµν,~n = Tr(∆µφ~n · ∆νφ~n −

( 12 (∆φ~n)2 + 1

2δm2φ2~n + 1

4!φ4~n

)δµν

+ξ(δµν∆2 − ∆µ∆ν

)φ2~n

),

where we use symmetric derivative on the lattice

∆µφ~n ≡φ~n+µ − φ~n−µ

2a,

â The δm2 = δm2divergent + δm2

finite is the contribution of the masscounterterm.




Conclusions

Energy momentum tensor

â Since the lattice breaks Poincaré invariance, the energymomentum tensor is not automatically conserved and may mixwith other operators.

â We need to ensure that the discretized energy momentum tensoris conserved in the continuum limit.

〈∂µTµν(x)φ(x1) · · ·φ(xk)〉 =

k∑i=1

δ(x− xi) ·∂

∂xνi〈φ(x1) · · ·φ(xk)〉 ,

â It turns out that in our case this holds automatically once we takeinto account the contribution from δm2

divergent.




Conclusions

To do list

â Compute 〈TT〉 for different values of N to extract the large Nbehavior.

â Consider λeff = λN/q� 1 and check with perturbative results.â Consider λeff � 1 and compare with gravity dual

ds2 = dr2 + r2ndxidxi,

This should allow us to extract n.â Consider λeff ∼ 1 and compare with Planck data.




Conclusions

Outline

1 Introduction


3 Perturbative QFT


5 Conclusions




Conclusions

Conclusions

â Holography offers a unified framework for discussing the veryEarly Universe:

à Strongly couple QFT: conventional inflation.à perturbative QFT: new non-geometric models.à Intermediate coupling: Lattice Holographic Cosmology.




Conclusions

Conclusions

â If one can simulate 3d QFTs which in IR have either a fixed pointor become super-renormalizable then one has interestingholographic models for the very early universe.

â Depending on the nature of the IR theory (strong or weaklycoupled) one can either test holography or obtain predictions thatcan be checked against observations.

â We have initiated this program by studying a simple toy model.


holographic cosmology meets lattice gauge theory - menu · holographic universe â in holographic...

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