Lattice Holographic Cosmology Kostas Skenderis STAG R R E S E - - PowerPoint PPT Presentation
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Lattice Holographic Cosmology Kostas Skenderis STAG R R E S E S E E A A R R C H C H CENTER CENTER C E N T E R Numerical
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
R E S E A R C H CENTER R E S E A R C H CENTER C E N T E R
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
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Introduction
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Holographic cosmology
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Perturbative QFT
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Holographic Lattice Cosmology
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Conclusions
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ The scale of inflation could be as high as 1014 GeV and as such it is highest energy scale which is directly observable. ➢ This is much larger than any energy scale we could achieve with accelerators. ➠ The physics of the Early Universe is a unique probe of physics beyond the standard model.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ The leading theoretical paradigm for the very early universe is the theory of inflation. ➢ It describes remarkably well existing observational data. ➢ It is based on gravity coupled to scalar field(s), perturbatively quantized around an accelerating FRW background.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ Despite its successes the theory of inflation still has a number of shortcomings: fine tuning, trans-Planckian issues etc. ➢ This description breaks down at some point since the background has a curvature singularity: the theory has to be embedded in a "UV complete theory" (the "initial singularity problem"). ➢ However, it has been very difficult to embed inflation in fundamental theory (such as string theory).
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ Holography provides a new framework that can accommodate: ➠ conventional inflation: strongly coupled dual QFT ➠ qualitatively new models for the very early Universe : QFT at weak and intermediate coupling. ➢ The new framework gives new insight into conventional inflation. ➢ The new models are falsifiable with current data.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ A holographic framework for cosmology was put forward in works with Paul McFadden and Adam Bzowski (2009 - on-going). ➢ 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]
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
In this talk I will discuss new work on ➢ further developing the models based on perturbative QFT. This part is based on work in progress with
➢ Claudio Coriano and Luigi Delle Rose
➢ using Lattice methods to construct models valid for any value of the coupling constant. This part is based on work in progress with
➢ Evan Berkovitz, Philip Powel, Enrico Rinaldi, Pavlos Vranas (Lawrence Livermore National Laboratory) ➢ Masanori Hanada (YITP Kyoto and SITP Stanford) ➢ Andreas Jüttner, Antonin Portelli, Francesco Sanfilippo (SHEP , Southampton)
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
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Introduction
2
Holographic cosmology
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Perturbative QFT
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Holographic Lattice Cosmology
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Conclusions
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
1
Introduction
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Holographic cosmology
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Perturbative QFT
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Holographic Lattice Cosmology
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Conclusions
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
In holographic cosmology one relates: ➢ cosmological observable such as the power spectra and non-gaussianities ➢ to correlation functions of the the energy momentum tensor of the dual QFT, upon a specific analytic continuation.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ The dual QFT is 3d QFT that admits a large N limit and our results apply to two classes of theories: ➠ QFTs with a non-trivial UV fixed point. ➠ A class of super-renormalizable QFTs. ➢ The results hold perturbatively in 1/N2. It is not clear whether these dualities hold non-perturbatively in 1/N2.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
These results hold for spacetimes that at late times approach ➠ de Sitter spacetime, ds2 → ds2 = −dt2 + e2tdxidxi, as t → ∞ ➠ power-law scaling solutions, ds2 → ds2 = −dt2 + t2ndxidxi, (n > 1) as t → ∞
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ The scalar power spectrum is given by ∆2
R(q) = − q3
4π2 1 Im T(q)T(−q), where T = Ti
i is the trace of the energy momentum tensor Tij
and we Fourier transformed to momentum space. The imaginary part is taken after the analytic continuation, q → −iq, N → −iN ➢ The power spectrum of tensors is related with the 2-point of the traceless part of T and non-gausiaities are related with higher-point functions of Tij.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
The underlying framework is gravity coupled to a scalar field Φ with a potential V(Φ), S = 1 2κ2
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)
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
Domain-wall/Cosmology correspondence
FRW solutions of ↔ Domain-wall solutions of the 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 continuation from de Sitter to Anti de Sitter. This theorem shows that this relation is not accidental.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ A special case of the correspondence is that between inflationary backgrounds and holographic RG flow spacetimes. ➢ Inflationary spacetimes are mapped to
➢ asympotically Anti-de Sitter spacetime, ds2 → ds2 = dr2 + e2rdxidxi, as r → ∞ ➢ power-law scaling solutions, ds2 → ds2 = dr2 + r2ndxidxi, (n > 1) as r → ∞ For special values of n these backgrounds are related to non-conformal branes.
➢ For these backgrounds there is an established holographic dictionary.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ Given an FRW, compute cosmological observables using standard cosmological perturbation theory. ➢ Corresponding to this FRW there is a domain-wall. ➢ Use holography to compute energy-momentum tensor correlators for the QFT dual to the domain-wall. ➠ Comparing the two results leads to the holographic formulae.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ This derivation holds in the regime the gravity approximation is valid. ➠ Conventional inflation is holographic. ➢ We will next present an alternative derivation which does not make this assumption but postulates the form of the duality.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ The partition function of the dual QFT (after analytic continuation) computes the wavefunction of the Universe [Maldacena (2002)]: ψ[Φ] = ZQFT[Φ] ➢ Cosmological observables are computed as Φ(x1) · · · Φ(xn) =
➢ The partition has an expansion in correlation functions: ZQFT[Φ] = exp
O(x1) · · · O(xn)Φ(x1) · · · Φ(xn)
earlier.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
An important feature of the holographic correspondence is that it is a weak/strong duality. weakly coupled gravity
strongly coupled QFT ➠ QFT correlation functions at strong coupling are related to Einstein gravity. Strongly coupled gravity
weakly coupled QFT ➠ Strongly coupled gravity here means that there is no notion of
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ In holographic cosmology: Cosmological evolution = inverse RG flow ➢ In our Universe we are currently living in an accelerating phase (driven by dark energy) and we believe that the Universe underwent a period of inflation at early times. ➢ This translates into specific properties of the dual QFT.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➠ The dual QFT should have a strongly coupled UV fixed point corresponding to the current dark energy era. ➠ In the IR the theory should either flow to:
➢ an IR fixed point (corresponding to de Sitter inflation), or ➢ a phase governed by a super-renormalizable theory (corresponding to power-law inflation).
➠ To correctly model the history of our Universe we would have to correctly account for the rest of the cosmological periods (radiation domination, matter domination).
➢ The holographic description of these eras is not currently known.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
We focus from now on the IR of the theory (inflationary era). ➢ If the QFT is strongly coupled in the IR then this would correspond to perturbative gravity in the early Universe.
➠ Such theories correspond to conventional inflationary models. ➠ Checking the QFT predictions against standard cosmological perturbation theory would provide a test of holography.
➢ If the QFT is not strongly coupled in the IR then this would correspond to a non-geometric phase in the early Universe.
➠ These are qualitative new modes for the very early Universe. ➠ Checking the QFT predictions against observations one can either
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ Use Lattice methods to compute the QFT observables. ➠ If the QFT is weakly coupled, one can check the Lattice results against perturbative QFT computations. ➠ If the QFT is strongly coupled, the Lattice provides a first principles derivation of the correlators. This would provide a test
➠ If the coupling is of intermediate strength we get new models for the Very Early Universe. The predictions of these models can be tested against CMB data.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ If one can simulate 3d QFTs which in IR have either a fixed point
holographic models for the very early universe. ➢ Depending on the nature of the IR theory (strongly or weakly coupled) one can either test holography or obtain predictions that can be checked against observations.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
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Introduction
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Holographic cosmology
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Perturbative QFT
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Holographic Lattice Cosmology
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Conclusions
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ A class of models that admit a holographic description: S = 1 g2
YM
1 2FijFij + 1 2(DφJ)2 + ¯ ψK / DψK + λJ1J2J3J4φJ1φJ2φJ3φJ4 + µαβ
JL1L2φJψL1 α ψL2 β
All fields are massless and in the adjoint of SU(N), λJ1J2J3J4, µαβ
JL1L2
are dimensionless couplings while g2
YM has mass dimension 1.
➢ An example of such theory is the maximally supersymmetric SYM theory in d = 3.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
What is special with this theory? Let us consider this theory in general d: S = 1 g2
YM
1 2FijFij + 1 2(DφJ)2 + ¯ ψK / DψK + λJ1J2J3J4φJ1φJ2φJ3φJ4 + µαβ
JL1L2φJψL1 α ψL2 β
➢ In this action all terms scale the same way if one assigns "4d dimensions" to the fields: [φ] = [A] = 1, [ψ] = 3/2. ➢ If we promote g2
YM to a field which transforms under conformal
transformations then the theory would be conformally invariant
[Yevicki, Kazama, Yoneya (1998)].
➢ This generalised conformal structure is not a bona fide symmetry
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ The form of 2-point functions is fixed by generalized conformal invariance [Kanitscheider, KS, Taylor (2008)]. In momentum space, Φ(q)Φ(−q) = q2∆−dc(g2) where g = g2
YM/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}. ➢ In even/odd dimensions c1(d)/c2(d) has a pole
1 d−(2k(+1)) and this
induces gn log g in c(g) [Coriano, Delle Rose, KS (to appear)].
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
We now focus on d = 3. ➢ One-loop is automatically finite when using dimensional regularization. ➢ At 2-loops, (only) the scalars have a UV divergence which can be removed by adding a bare mass term. ➢ At loops, all 2-point functions have an IR singularity. It was argued in [Jackiw,Templeton (1981)][Appelquist, Pisarski(1981)] that these type of theories are non-perturbative IR finite: g2
YM effectively acts as an IR regulator.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ To simplify the presentation, I focus on the 2-point function of the trace of T. At large N, T(q)T(−q) = qdN2f(g2
eff),
where g2
eff = g2 YMN/q is the effective dimensionless ’t Hooft
coupling and f(g2
eff) is a general function of g2 eff.
➢ When g2
eff is small, the function f(g2 eff) has the form
f(g2
eff) = f0(d) + f1(d)g2 eff + · · ·
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ f0 is determined at 1-loop. It is finite in odd dimensions and it diverges in even dimensions. It has been computed in [McFadden,
KS (2009)] for d = 3 and in [Coriano, Delle Rose, KS (to appear) in
general d. ➢ f1 is determined at 2-loops. It is finite in even dimensions and has a UV divergence in odd dimensions. ➢ f1 is IR finite, unless the scalars are non-minimally coupled.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
We now focus on d = 3. ➢ The UV infinities can be cancelled by adding the counteterm aCT
where aCT is an appropriately chosen coefficient. ➢ After renormalization, f(g2
eff) = f0(1 − f1g2 eff ln g2 eff + f2g2 eff + O[g4 eff]).
where f2 = α0 + α1 log g2
YMN
µUV + α2 log g2
YMN
µIR where αi are constants that depend on the field content.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ To compute the holographic scalar power spectrum we need to analytically continue T(q)T(−q). ➢ The analytic continuation acts as g2
eff → g2 eff,
N2q3 → −iN2q3 and therefore ∆2
R(q) = − q3
4π2 1 ImT(q)T(−q) = 1 4π2N2 1 f(g2
eff)
➢ Thus, for this class of theories and in the perturbative regime: ∆2
R(q) =
4π2N2f0
1 − f1g2
eff ln g2 eff + f2g2 eff
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
2 4 6 8 ln qgq 0.6 0.8 1.0 1.2 1.4 1.6 2
q2
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ In cosmology there are very few observables so the way we check the theory against data is different than in high energy physics. ➢ The main question one addresses is: Given a set of models, which one is preferred by the data? ➢ One way to answer this is to check how well the model fits the data: what is the probability for obtaining the data given the model. ➢ A better way is to compute the so-called Bayesian Evidence: what is the probability for the model given the data.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
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 ruled
in or out.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ Assuming f2 is negligible and redefining variables, f1g2
YMN = 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), we
∆2
R(q) = ∆2 R
1 1 + (gq∗/q) ln |q/gq∗|, → ∆2
R = 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 ∆2
R(q) = ∆2 Rqns−1,
ns(q) − 1 = gq∗/q Thus the small deviation from scale invariance is related with the coupling constant of the dual QFT being small.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ The effective spectral index, ∆2
R(q) ∼ qns(q)−1, has the property:
−(ns − 1) = gq∗ q = dns d ln q = · · · = (−1)n+1 dnns d ln qn . ➢ This is very different from slow-roll models where higher order running is suppressed by slow-roll parameters. ➢ Another difference is that one can easily accommodate any amount of tensors: the ratio of scalars-to-tensor r depends on the field content. ➢ Given the significant differences, we undertook a dedicated data analysis [Easther, Flauger, McFadden, KS (2011) [Dias (2011)]) to custom-fit this model to WMAP and other astrophysical data.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
The power-law ΛCDM model depends on six parameters. Four describe the composition and expansion of the universe and the
R of primordial
curvature perturbations. The holographic ΛCDM model depends on the same set of parameters, except that the tilt ns is replaced by the parameter g. We determined the best-fit values for all parameters for both models and used Bayesian evidence in order to make a model comparison.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
5 10 50 100 500 1000 4000 2000 2000 4000 6000 8000
Red: ΛCDM, Green: holographic model
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
The estimated values for the five common parameters of the two models are roughly within one standard deviation of each other. The data favor negative values of g (red spectrum) with central value g = −1.27 × 10−3.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ The value of g leads to a small effective coupling, except potentially for the very low wavelength modes. Since g2
eff = (1/f1)(gq∗/q) one needs to know the value of the 2-loop
factor f1 when (gq∗/q) itself is not very small. ➢ A related issue is whether the parameter f2 is important. If it is the power spectrum is modified as: ∆2
R(q) = ∆2 R
1 1 + (gq∗/q) ln |q/βgq∗| → f2 cannot be computed perturbatively when there are IR divergences. → If g2
eff is not small for all relevant momenta, one must include
higher order terms in the computation of the 2-point function.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ SU(N) gauge theory coupled to Nφ conformally coupled massless scalars (without self-interaction). ➠ The perturbative answer at 2-loops is [Coriano, Delle Rose, KS (to
appear)],
f0 = 1 64, f1 = − 2 3π2 (Nφ − 4), f2 = − 1 24π2 (16 + 3π2) − 8 3π2 (Nφ − 1) log g2
YMN
µUV + 1 2π2 Nφ log g2
YMN
µIR ➠ We would need Nφ ∼ 300 in order to satisfy the constrain that gravitational waves have not been observed so far (r ≤ 0.1 ).
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ A non-minimally coupled massless scalar field in the adjoint of SU(N) with φ4 self-interaction S = 1 λ
1 2(∂µφ)2 + 1 4!φ4
and energy momentum tensor Tij = 1 λTr
2(∂φ)2 + 1 4!φ4) + ξ(δij − ∂i∂j)φ2
appear)]
f0 = (1 − 8ξ)2 256 , f1 = 0, f2 = − 1 24 ➠ We need |ξ − 1/8| ∼ 10−2 to satisfy r ≤ 0.1 [Kawai, Nakayama
(2014)] .
➢ The fit of the perturbative model to data is in progress.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
1
Introduction
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Holographic cosmology
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Perturbative QFT
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Holographic Lattice Cosmology
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Conclusions
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ Perturbative QFT yields interesting new models for the very Early Universe. ➢ Comparing with data suggests that we may need to go beyond leading order/need non-perturbative information. ➢ QFT at intermediate coupling may provide yet more interesting models. ➠ Use Lattice to compute the relevant QFT observables.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ One can straightforwardly discretize the action, Slattice = ˜ a3
Tr 1 2 ˜ φ
n+ˆ µ − ˜
φ
n
˜ a 2 + 1 4! ˜ φ4
, where ˜ a = aλ is a dimensionless lattice spacing (a is the lattice spacing). ➢ We need to add a mass counterterm to remove UV infinities (like in the continuum): δm2 = δm2
divergent
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ We also need to add and fine tune a finite mass δm2
finite so that
the renormalized mass vanishes in the continuum limit. ➢ If the mass in the continuum limit is positive then Mn = 0 for any n, where M =
n.
➢ If the mass in the continuum limit is negative we are in the spontaneously broken phase, Mn = 0. ➢ To find the massless point one may compute U = M22/M4 for different lattice sizes and find the intersection point.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
0.2 0.4 0.6
Vertical axis: U, Horizontal axis: δm2
finite/λ2, Lattice size: 243, 323, 483, 643
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ The discretized energy momentum tensor reads Tµν,
n =
Tr ¯ ∆µφ
n · ¯
∆νφ
n −
1
2( ¯
∆φ
n)2 + 1 2δm2φ2
4!φ4
+ξ
∆2 − ¯ ∆µ ¯ ∆ν
where we use symmetric derivative on the lattice ¯ ∆µφ
n ≡ φ n+ˆ µ − φ n−ˆ µ
2a , ➢ The δm2 = δm2
divergent + δm2 finite is the contribution of the mass
counterterm.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ Since the lattice breaks Poincaré invariance, the energy momentum tensor is not automatically conserved and may mix with other operators. ➢ We need to ensure that the discretized energy momentum tensor is conserved in the continuum limit. ∂µTµν(x)φ(x1) · · · φ(xk) =
k
δ(x − xi) · ∂ ∂xν
i
φ(x1) · · · φ(xk) , ➢ It turns out that in our case this holds automatically once we take into account the contribution from δm2
divergent.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ Compute TT for different values of N to extract the large N behavior. ➢ Consider λeff = λN/q ≪ 1 and check with perturbative results. ➢ Consider λeff ≫ 1 and compare with gravity dual ds2 = dr2 + r2ndxidxi, This should allow us to extract n. ➢ Consider λeff ∼ 1 and compare with Planck data.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
1
Introduction
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Holographic cosmology
3
Perturbative QFT
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Holographic Lattice Cosmology
5
Conclusions
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ Holography offers a unified framework for discussing the very Early Universe: ➠ Strongly couple QFT: conventional inflation. ➠ perturbative QFT: new non-geometric models. ➠ Intermediate coupling: Lattice Holographic Cosmology.
Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions
➢ If one can simulate 3d QFTs which in IR have either a fixed point
holographic models for the very early universe. ➢ Depending on the nature of the IR theory (strong or weakly coupled) one can either test holography or obtain predictions that can be checked against observations. ➢ We have initiated this program by studying a simple toy model.
Kostas Skenderis Holographic cosmology meets lattice gauge theory