Lattice Holographic Cosmology

Transcription

1 Introduction Lattice Holographic Cosmology STAG STAG STAG RESEARCH RESEARCH RESEARCH CENTER CENTER CENTER Numerical approaches to the holographic principle, quantum gravity and cosmology Kyoto, Japan, 22 July 2015 meets lattice gauge theory

2 Outline 1 Introduction meets lattice gauge theory

3 Introduction The scale of inflation could be as high as 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. meets lattice gauge theory

4 Inflation 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. meets lattice gauge theory

5 Inflation: problems 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). meets lattice gauge theory

6 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. meets lattice gauge theory

7 References A holographic framework for cosmology was put forward in works with Paul McFadden and Adam Bzowski ( 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] meets lattice gauge theory

8 References 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) meets lattice gauge theory

9 Outline 1 Introduction meets lattice gauge theory

10 Outline 1 Introduction meets lattice gauge theory

11 in a nutshell 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. meets lattice gauge theory

12 : the dual QFT 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/N 2. It is not clear whether these dualities hold non-perturbatively in 1/N 2. meets lattice gauge theory

13 : the bulk These results hold for spacetimes that at late times approach de Sitter spacetime, ds 2 ds 2 = dt 2 + e 2t dx i dx i, as t power-law scaling solutions, ds 2 ds 2 = dt 2 + t 2n dx i dx i, (n > 1) as t meets lattice gauge theory

14 Holographic formula for the scalar power spectrum The scalar power spectrum is given by 2 R(q) = q3 4π 2 1 Im T(q)T( q), where T = T i i is the trace of the energy momentum tensor T ij 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 T ij. meets lattice gauge theory

15 Sketch of derivation I The underlying framework is gravity coupled to a scalar field Φ with a potential V(Φ), S = 1 2κ 2 d 4 x g(r ( Φ) 2 2κ 2 V(Φ)) There is 1-1 correspondence [KS, Tonwsend (2006)] between: Domain-wall solutions FRW spacetimes ds 2 = dr 2 + e 2A(r) dx i dx i Φ = Φ(r) ds 2 = dt 2 + a 2 (t)dx i dx i Φ = Φ(t) meets lattice gauge theory

16 Domain-wall/Cosmology correspondence 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. meets lattice gauge theory

17 Inflation/holographic RG correspondence 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, ds 2 ds 2 = dr 2 + e 2r dx i dx i, as r power-law scaling solutions, ds 2 ds 2 = dr 2 + r 2n dx i dx i, (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. meets lattice gauge theory

18 Holographic formulae for cosmology [McFadden, KS] 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. meets lattice gauge theory

19 Remarks 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. meets lattice gauge theory

20 Sketch of derivation II: the wavefunction approach The partition function of the dual QFT (after analytic continuation) computes the wavefunction of the Universe [Maldacena (2002)]: ψ[φ] = Z QFT [Φ] Cosmological observables are computed as Φ(x 1 ) Φ(x n ) = DΦ ψ 2 Φ(x 1 ) Φ(x n ) The partition has an expansion in correlation functions: ( ) Z QFT [Φ] = exp O(x 1 ) O(x n ) Φ(x 1 ) Φ(x n ) n Applying this to 2-point function of T leads to the formula quoted earlier. meets lattice gauge theory

21 Gauge/gravity duality 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 spacetime. This is a non-geometric phase. meets lattice gauge theory

22 Holographic Universe 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. meets lattice gauge theory

23 Holographic Universe 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. meets lattice gauge theory

24 Holographic inflation 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 obtain support or falsify these models. meets lattice gauge theory

25 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 of holographic dualities. 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. meets lattice gauge theory

26 Take-home message If one can simulate 3d QFTs which in IR have either a fixed point or become super-renormalizable then one has interesting 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. meets lattice gauge theory

27 Outline 1 Introduction meets lattice gauge theory

28 Super-renormalizable theories A class of models that admit a holographic description: S = 1 [ 1 g 2 d 3 xtr YM 2 F ijf ij (DφJ ) 2 + ψ K /Dψ K + λ J1J 2J 3J 4 φ J1 φ J2 φ J3 φ J4 + µ αβ JL 1L 2 φ J ψ L1 α ψ L2 β All fields are massless and in the adjoint of SU(N), λ J1J 2J 3J 4, µ αβ JL 1L 2 are dimensionless couplings while g 2 YM has mass dimension 1. An example of such theory is the maximally supersymmetric SYM theory in d = 3. ]. meets lattice gauge theory

29 Generalized conformal structure What is special with this theory? Let us consider this theory in general d: S = 1 [ 1 g 2 d d xtr YM 2 F ijf ij (DφJ ) 2 + ψ K /Dψ K + λ J1J 2J 3J 4 φ J1 φ J2 φ J3 φ J4 + µ αβ JL 1L 2 φ J ψ L1 α ψ L2 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 g 2 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 of the theory but nevertheless controls many of its properties. ]. meets lattice gauge theory

30 2-point functions Introduction The form of 2-point functions is fixed by generalized conformal invariance [Kanitscheider, KS, Taylor (2008)]. In momentum space, Φ(q)Φ( q) = q 2 d c(g 2 ) where g = g 2 YM/q d 4 is the dimensionless coupling constant and in the perfurbative regime c(g) = c 0 g + c 1 g 2 + c 2 g 3 + c i (d) is the i-loop contribution. When Φ = {A, φ, ψ}, = {1, 1, 3/2}. 1 In even/odd dimensions c 1 (d)/c 2 (d) has a pole d (2k(+1)) and this induces g n log g in c(g) [Coriano, Delle Rose, KS (to appear)]. meets lattice gauge theory

31 2-point functions: renormalization 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: g 2 YM effectively acts as an IR regulator. meets lattice gauge theory

32 Energy-momentum tensor To simplify the presentation, I focus on the 2-point function of the trace of T. At large N, T(q)T( q) = q d N 2 f (g 2 eff), where g 2 eff = g2 YMN/q is the effective dimensionless t Hooft coupling and f (g 2 eff ) is a general function of g2 eff. When g 2 eff is small, the function f (g2 eff ) has the form f (g 2 eff) = f 0 (d) + f 1 (d)g 2 eff + meets lattice gauge theory

33 Energy-momentum tensor f 0 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. f 1 is determined at 2-loops. It is finite in even dimensions and has a UV divergence in odd dimensions. f 1 is IR finite, unless the scalars are non-minimally coupled. meets lattice gauge theory

34 Energy-momentum tensor: renormalization We now focus on d = 3. The UV infinities can be cancelled by adding the counteterm a CT d 3 xr where a CT is an appropriately chosen coefficient. After renormalization, where f (g 2 eff) = f 0 (1 f 1 g 2 eff ln g 2 eff + f 2 g 2 eff + O[g 4 eff]). f 2 = α 0 + α 1 log g2 YMN µ UV + α 2 log g2 YMN µ IR where α i are constants that depend on the field content. meets lattice gauge theory

35 Holographic power spectrum To compute the holographic scalar power spectrum we need to analytically continue T(q)T( q). The analytic continuation acts as and therefore g 2 eff g 2 eff, N 2 q 3 in 2 q 3 2 R(q) = q3 4π 2 1 Im T(q)T( q) = 1 4π 2 N 2 1 f (g 2 eff ) Thus, for this class of theories and in the perturbative regime: ( ) R(q) = 4π 2 N 2 f 0 1 f 1 g 2 eff ln g2 eff + f 2g 2 eff meets lattice gauge theory

36 Holographic power spectrum 2 q Blue curve: g > 0, Red curve: g < 0 meets lattice gauge theory

37 Confronting with data 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. meets lattice gauge theory

38 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 ruled in or out. meets lattice gauge theory

39 Fitting to data Assuming f 2 is negligible and redefining variables, f 1 g 2 YM N = 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 obtain the final formula: 2 R(q) = 2 1 R 1 + (gq /q) ln q/gq, 2 R = 1/(4π2 N 2 f 0 ). Smallness of the amplitude is related with the large N limit: matching with observations implies N When (gq /q) 1 one may rewrite the spectrum in the power-law form 2 R(q) = 2 Rq ns 1, n s (q) 1 = gq /q Thus the small deviation from scale invariance is related with the coupling constant of the dual QFT being small. meets lattice gauge theory

40 Holographic model vs slow-roll inflation The effective spectral index, 2 R (q) qns(q) 1, has the property: dn n s (n s 1) = gq q = dn s d ln q = = ( 1)n+1 d ln q n. 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. meets lattice gauge theory

41 Holographic model vs ΛCDM The power-law ΛCDM model depends on six parameters. Four describe the composition and expansion of the universe and the other two are the tilt n s and the amplitude 2 R of primordial curvature perturbations. The holographic ΛCDM model depends on the same set of parameters, except that the tilt n s 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. meets lattice gauge theory

42 Angular power spectrum: ΛCDM vs holographic model C 2Π ΜK Red: ΛCDM, Green: holographic model meets lattice gauge theory

43 Parameter estimation 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 = meets lattice gauge theory

44 Is the perturbative treatment justified? The value of g leads to a small effective coupling, except potentially for the very low wavelength modes. Since g 2 eff = (1/f 1)(gq /q) one needs to know the value of the 2-loop factor f 1 when (gq /q) itself is not very small. A related issue is whether the parameter f 2 is important. If it is the power spectrum is modified as: 2 R(q) = 2 1 R 1 + (gq /q) ln q/βgq f 2 cannot be computed perturbatively when there are IR divergences. If g 2 eff is not small for all relevant momenta, one must include higher order terms in the computation of the 2-point function. meets lattice gauge theory

45 Model 1 SU(N) gauge theory coupled to N φ conformally coupled massless scalars (without self-interaction). The perturbative answer at 2-loops is [Coriano, Delle Rose, KS (to appear)], f 0 = 1 64, f 1 = 2 3π 2 (N φ 4), f 2 = 1 24π 2 (16 + 3π2 ) 8 3π 2 (N φ 1) log g2 YMN µ UV + 1 2π 2 N φ log g2 YMN µ IR We would need N φ 300 in order to satisfy the constrain that gravitational waves have not been observed so far (r 0.1 ). meets lattice gauge theory

46 Model 2 A non-minimally coupled massless scalar field in the adjoint of SU(N) with φ 4 self-interaction S = 1 ( 1 d 3 xtr λ 2 ( µφ) ) 4! φ4, and energy momentum tensor T ij = 1 λ Tr ( i φ j φ δ ij ( 1 2 ( φ) ! φ4 ) + ξ(δ ij i j )φ 2 ) The perturbative answer to 2-loops is [Coriano, Delle Rose, KS (to appear)] (1 8ξ)2 f 0 =, f 1 = 0, f 2 = We need ξ 1/ to satisfy r 0.1 [Kawai, Nakayama (2014)]. The fit of the perturbative model to data is in progress. meets lattice gauge theory

47 Outline 1 Introduction meets lattice gauge theory

48 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. meets lattice gauge theory

49 Prof of concept: put model 2 on Lattice One can straightforwardly discretize the action, S lattice = ã ( 3 Tr 1 φ n+ˆµ φ ) 2 n ã 4! φ n 4, n where ã = 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): δm 2 = δm 2 divergent meets lattice gauge theory

50 Finding the massless point: Binder Cumulant We also need to add and fine tune a finite mass δm 2 finite so that the renormalized mass vanishes in the continuum limit. If the mass in the continuum limit is positive then M n = 0 for any n, where M = n φ n. If the mass in the continuum limit is negative we are in the spontaneously broken phase, M n 0. To find the massless point one may compute U = M 2 2 / M 4 for different lattice sizes and find the intersection point. meets lattice gauge theory

51 Binder Cumulant for a single scalar field Vertical axis: U, Horizontal axis: δm 2 finite/λ 2, Lattice size: 24 3, 32 3, 48 3, 64 3 meets lattice gauge theory

52 Energy momentum tensor The discretized energy momentum tensor reads T µν, n = Tr ( µ φ n ν φ n ( 1 2 ( φ n ) δm2 φ 2 n + 1 4! φ4 n) δµν +ξ ( δ µν 2 µ ν ) φ 2 n ), where we use symmetric derivative on the lattice µ φ n φ n+ˆµ φ n ˆµ, 2a The δm 2 = δm 2 divergent + δm2 finite is the contribution of the mass counterterm. meets lattice gauge theory

53 Energy momentum tensor 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. µ T µν (x)φ(x 1 ) φ(x k ) = k δ(x x i ) i=1 x ν i φ(x 1 ) φ(x k ), It turns out that in our case this holds automatically once we take into account the contribution from δm 2 divergent. meets lattice gauge theory

54 To do list Compute TT for different values of N to extract the large N behavior. Consider λ eff = λn/q 1 and check with perturbative results. Consider λ eff 1 and compare with gravity dual ds 2 = dr 2 + r 2n dx i dx i, This should allow us to extract n. Consider λ eff 1 and compare with Planck data. meets lattice gauge theory

55 Outline 1 Introduction meets lattice gauge theory

56 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. meets lattice gauge theory

57 If one can simulate 3d QFTs which in IR have either a fixed point or become super-renormalizable then one has interesting 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. meets lattice gauge theory