Stochastic Tools for Model Building

Short talk given at the DESY fellows meeting to introduce myself.

Transcript

1. Stochastic Tools for Model Building Jonny Frazer
2. Me Previous work mostly on inflation model independent e.g. model building/exploration e.g. DESY-16-171 Prepared for submission to JCAP Numerical evaluation of the bispectrum in multiple field inflation – the transport approach with code Mafalda Dias1 , 2 , Jonathan Frazer1 , 3 , 4 , David J. Mulryne5 and David Seery2 1 Theory Group, Deutsches Elektronen-Synchrotron, DESY, D-22603, Hamburg, Germany 2 Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK 3 Department of Theoretical Physics, University of the Basque Country, UPV/EHU 48040 Bilbao, Spain 4 ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain 5 School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, UK E-mail: mafalda.dias@desy.de, jonathan.frazer@desy.de, d.mulryne@qmul.ac.uk, D.Seery@sussex.ac.uk Abstract. We present a complete framework for numerical calculation of the power spectrum and bispectrum in canonical inflation with an arbitrary number of light or heavy fields. Our method includes all relevant e ects at tree-level in the loop expansion, including (i) interference between growing and decaying modes near horizon exit; (ii) correlation and coupling between species near horizon exit and on superhorizon scales; (iii) contributions from mass terms; and (iv) all contributions from coupling to gravity. We track the evolution of each correlation function from the vacuum state through horizon exit and the superhorizon regime, with no need to match quantum and classical parts of the calculation; when integrated, our approach corresponds exactly with the tree-level Schwinger or ‘in–in’ formulation of quantum field theory. In this paper we give the equations necessary to evolve all two- and three-point correlation functions together with suitable initial conditions. The final formalism is suitable Prepared for submission to JCAP MultiModeCode: An e cient numerical solver for multifield inflation Layne C. Price,1 Jonathan Frazer,2,3 Jiajun Xu,4 Hiranya V. Peiris,5 and Richard Easther1 1Department of Physics, University of Auckland Private Bag 92019, Auckland, New Zealand 2Department of Theoretical Physics, University of the Basque Country, UPV/EHU 48040 Bilbao, Spain 3IKERBASQUE, Basque Foundation for Science 48011 Bilbao, Spain 4Department of Physics, University of Wisconsin-Madison CO] 2 Oct 2014 A New Angle on Chaotic Inflation Thomas C. Bachlechner,1 Mafalda Dias,2 Jonathan Frazer,3 and Liam M 1Department of Physics, Cornell University, Ithaca, New York, 14853, U 2Astronomy Centre, University of Sussex, Falmer, Brighton, BN1 9QH, 3Department of Theoretical Physics, University of the Basque Country UPV/EHU, 480 N-flation is a radiatively stable scenario for chaotic inflation in which the displacem axions with decay constants f1  . . .  fN < MP lead to a super-Planckian e↵ective equal to the Pythagorean sum fPy of the fi . We show that mixing in the axion generically leads to the phenomenon of kinetic alignment, allowing for e↵ective dis large as p NfN fPy , even if f1, . . . , fN 1 are arbitrarily small. At the level of k necessary alignment occurs with very high probability, because of eigenvector deloc present conditions under which inflation can take place along an aligned direction. Ou sharply reduces the challenge of realizing N-flation in string theory. INTRODUCTION Inflationary scenarios producing detectable primor- f2 f1 r 2014 Inflationary perturbation theory is geometrical optics in phase space David Seery,1, ∗ David J. Mulryne,2, † Jonathan Frazer,1, ‡ and Raquel H. Ribeiro3, § 1Astronomy Centre, University of Sussex, Brighton BN1 9QH, United Kingdom 2Astronomy Unit, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom 3Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom A pressing problem in comparing inflationary models with observation is the accurate calculation of correlation functions. One approach is to evolve them using ordinary differential equations (“transport equations”), analogous to the Schwinger–Dyson hierarchy of in–out quantum field theory. We extend this approach to the complete set of momentum space correlation functions. A formal solution can be obtained using raytracing techniques adapted from geometrical optics. We reformulate inflationary perturbation the- eory is geometrical optics in phase space 2, † Jonathan Frazer,1, ‡ and Raquel H. Ribeiro3, § of Sussex, Brighton BN1 9QH, United Kingdom of Mathematical Sciences, Queen Mary, End Road, London E1 4NS, United Kingdom d Mathematics and Theoretical Physics, cal Sciences, University of Cambridge, ambridge CB3 0WA, United Kingdom nary models with observation is the accurate calculation of volve them using ordinary differential equations (“transport son hierarchy of in–out quantum field theory. We extend this pace correlation functions. A formal solution can be obtained Simple emergent power spectra from com Mafalda Dias,1, ⇤ Jonathan Frazer,1, † and M 1Deutsches Elektronen-Synchrotron, DESY, Notkestraße 2Department of Applied Mathematics and Theor University of Cambridge, Cambridge, CB3 0W We construct ensembles of random scalar potentials for N equilibrium random matrix theory, and use these to study small-field inflation. For Nf = O(few), these heavily featured spectra that are highly non-linear, at odds with observatio evolution of the perturbations is generically substantial, yet th and become more predictive, with most realisations being w spectrum. This provides proof of principle that complex inflat emergent power spectra. We explain how these results can universality of random matrix theory. DESY-16-073 Simple emergent power spectra from complex inflationary physics Mafalda Dias,1, ⇤ Jonathan Frazer,1, † and M.C. David Marsh2, ‡ 1Deutsches Elektronen-Synchrotron, DESY, Notkestraße 85, 22607 Hamburg, Germany 2Department of Applied Mathematics and Theoretical Physics, DAMTP, University of Cambridge, Cambridge, CB3 0WA, United Kingdom We construct ensembles of random scalar potentials for Nf interacting scalar fields using non- equilibrium random matrix theory, and use these to study the generation of observables during small-field inflation. For Nf = O(few), these heavily featured scalar potentials give rise to power spectra that are highly non-linear, at odds with observations. For Nf 1, the superhorizon evolution of the perturbations is generically substantial, yet the power spectra simplify considerably and become more predictive, with most realisations being well approximated by a linear power spectrum. This provides proof of principle that complex inflationary physics can give rise to simple emergent power spectra. We explain how these results can be understood in terms of large Nf Prepared for submission to JCAP Multifield consequences for D-brane inflation Mafalda Dias, Jonathan Frazer and Andrew R. Liddle Astronomy Centre, University of Sussex, Brighton BN1 9QH, United Kingdom E-mail: m.dias@sussex.ac.uk, j.frazer@sussex.ac.uk, a.liddle@sussex.ac.uk Abstract. We analyse the multifield behaviour in D-brane inflation when contributions from the bulk are taken into account. For this purpose, we study a large number of realisations of the potential; we find the nature of the inflationary trajectory to be very consistent despite the complex construction. Inflation is always canonical and occurs in the vicinity of an inflection -ph.CO] 18 Mar 2013 Prepared for submission to JCAP Multifield consequences for D-brane inflation Mafalda Dias, Jonathan Frazer and Andrew R. Liddle Astronomy Centre, University of Sussex, Brighton BN1 9QH, United Kingdom E-mail: m.dias@sussex.ac.uk, j.frazer@sussex.ac.uk, a.liddle@sussex.ac.uk Abstract. We analyse the multifield behaviour in D-brane inflation when contributions from CO] 18 Mar 2013 Prepared for submission to JCAP Multifield consequences for D inflation Mafalda Dias, Jonathan Frazer and Andrew R. Lidd Astronomy Centre, University of Sussex, Brighton BN1 9QH, United E-mail: m.dias@sussex.ac.uk, j.frazer@sussex.ac.uk, a.liddle@sussex.a Abstract. We analyse the multifield behaviour in D-brane inflation whe the bulk are taken into account. For this purpose, we study a large num the potential; we find the nature of the inflationary trajectory to be very -ph.CO] 18 Mar 2013 see Mafalda’s talk see https://transportmethod.com http://modecode.org
3. Talk Overview Motivation Kac polynomials and ab-polynomials Gaussian Random Fields Random Matrix Theory Hopes and dreams
4. Why use a statistical approach to model building? Complementary to “minimal models” Concrete framework for asking about of fine tuning Emergent simplicity is ubiquitous in complex systems Can treat theoretical and experimental uncertainty with the same language If we are going to take the “string landscape” seriously then new tools are needed
5. Kac polynomials https://terrytao.wordpress.com/tag/kac-polynomials/ deterministic: ci = 1 p i! ci = s✓ n i ◆ ci = 1 Flat (a.k.a Weyl) Elliptic Kac independent and identically distributed (iid) f(z) = n X i=0 ci⇠izi N(0, 1)C
6. Kac polynomials f(z) = n X i=0 ci⇠izi < =
7. Kac polynomials: ab-polynomials 2 4 6 8 10 2 4 6 8 10 a b f(z) = ⇠0 + ⇠aza + ⇠bzb
8. Gaussian Random Fields m(t) = E{f(t)} C(s, t) = E{(f(s) m(s))(f(t) m(t))} A Gaussian (random) field is determined purely by it’s mean and covariance function f(t) t 2 Rd C(s, t) = C(|s t|) = e (s t).(s t) 2⇠2 m(t) = 0 Example: t1 t2 f
9. Gaussian Random Fields: Euler Characteristic ' = C H + . . .
10. Gaussian Random Fields: Euler Characteristic ' = C H + . . .
11. Gaussian Random Fields: Euler Characteristic ' = C H + . . .
12. 12 1 Introduction Fig. 1.4.3. The observed Euler characteristics of the set of high density regions of the CfA Galaxy survey plotted against the density threshold. High thresholds pro- duce a meatball topology (magnification, and Figure 1.4.2(a)). Medium thresholds produce a sponge topology (Figure 1.4.2(b)), and low thresholds produce a bubble topology (Figure 1.4.2(c)). Also shown are the expected values of these Euler char- acteristics for galaxies generated as the excursion sets of Gaussian random fields. 1.5 Brain Imaging One of the earliest experiments in brain imaging was conducted in 1990 at Montreal Neurological Institute (cf. [52].) In this experiment, subjects were injected with a radio isotope emitting positrons, which annihilate with nearby electrons to release gamma rays that are detected by Positron Emission To- mography (PET). By careful reconstruction, it was possible to build up an 1.6 Beyond the Euler Characteristic 17 Fig. 1.5.5. Plot of the observed Euler characteristic of the set of high Z regions for the PET data and the expected Euler characteristic from (1.4.9) if there is no activation due to the linguistic task, plotted against the threshold u. The most interesting part is when u > 3, showing a higher Euler characteristic than expected, confirming that some components of the excursion set are due to the linguistic task and not the random noise. In particular, at u = 3.3 we expect an Euler characteristic of 1, but we observe 4 (visible in Figure 1.5.4(g)). At the 5% critical value of u = 4.22, we expect 0.05 but we observe 2 components (visible in Figure 1.5.4(h)). A remaining question is whether or not the empirical and expected curves in Figure 1.5.5 are su ciently close to allow us to feel comfortable about the assumptions of Gaussianity of the field, and its stationarity and isotropy. The curves are certainly closer8 in this case than they were for the CfA galaxy data in Figure 1.4.3 but we do not yet have su cient experience to know if, in this case, “close enough is good enough”. In fact, there are good physiological Gaussian Random Fields: Example applications R. Adler, J. Taylor, K. Worsley: Applications of Random Fields and Geometry
13. Random Matrix Theory: Gaussian Orthogonal Ensemble p(M)dM = p(M0)dM0 M0 = OT MO d ⇥ d entries are (again) independent and identically distributed (iid)
14. eigenvalue repulsion Eigenvalues behave like a gas of electrons in , confined to a line and subject to a quadratic potential Random Matrix Theory: Gaussian Orthogonal Ensemble p( 1, . . . , n) = Ce 1 2 W W = 1 2 n X i=1 2 i X i6=j ln | i j | R2
15. 0.0 0.5 1.0 1.5 2.0 -2 -1 0 1 2 DfêLh l 0.0 0.5 1.0 1.5 2.0 -2 -1 0 1 2 DfêLh l Random Matrix Theory: Dyson Brownian Motion Mij = Aij Mij t ⇠ t/⇠ t/⇠ uniquely determined stochastic contribution
16. Final Remarks Stochastic methods can be powerful tools for model building, enabling a high degree of computability Formally results are usually obtained in some large n limit, assuming iid random variates but • Results often continue to hold at low n • Universality theorems allow assumptions about the statistics of the model to be relaxed Emergent simplicity is ubiquitous in complex systems