scholarly journals From locality and unitarity to cosmological correlators

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Sadra Jazayeri ◽  
Enrico Pajer ◽  
David Stefanyszyn
Keyword(s):  
Vacuum State ◽  
Tree Level ◽  
Local Theory ◽  
Optical Theorem ◽  
Translation Invariance ◽  
De Sitter ◽  
Integral Theorem ◽  
Vast Number ◽  
Time Translation ◽  
Cauchy’S Integral Theorem

Abstract In the standard approach to deriving inflationary predictions, we evolve a vacuum state in time according to the rules of a given model. Since the only observables are the future values of correlators and not their time evolution, this brings about a large degeneracy: a vast number of different models are mapped to the same minute number of observables. Furthermore, due to the lack of time-translation invariance, even tree-level calculations require an increasing number of nested integrals that quickly become intractable. Here we ask how much of the final observables can be “bootstrapped” directly from locality, unitarity and symmetries.To this end, we introduce two new “boostless” bootstrap tools to efficiently compute tree-level cosmological correlators/wavefunctions without any assumption about de Sitter boosts. The first is a Manifestly Local Test (MLT) that any n-point (wave)function of massless scalars or gravitons must satisfy if it is to arise from a manifestly local theory. When combined with a sub-set of the recently proposed Bootstrap Rules, this allows us to compute explicitly all bispectra to all orders in derivatives for a single scalar. Since we don’t invoke soft theorems, this can also be extended to multi-field inflation. The second is a partial energy recursion relation that allows us to compute exchange correlators. Combining a bespoke complex shift of the partial energies with Cauchy’s integral theorem and the Cosmological Optical Theorem, we fix exchange correlators up to a boundary term. The latter can be determined up to contact interactions using unitarity and manifest locality. As an illustration, we use these tools to bootstrap scalar inflationary trispectra due to graviton exchange and inflaton self-interactions.

scholarly journals Graviton two-point function in 3 + 1 static de Sitter spacetime

2016 ◽  
Vol 25 (09) ◽  
pp. 1641016 ◽  
Author(s):  
Rafael P. Bernar ◽  
Luís C. B. Crispino ◽  
Atsushi Higuchi
Keyword(s):  
Vacuum State ◽  
De Sitter Spacetime ◽  
Cosmological Horizon ◽  
De Sitter ◽  
Point Function ◽  
Gauge Invariant ◽  
Gravitational Perturbations ◽  
Time Translation ◽  
Sitter Spacetime ◽  
Arbitrary Dimensions

In [R. P. Bernar, L. C. B. Crispino and A. Higuchi, Phys. Rev. D 90 (2014) 024045.] we investigated gravitational perturbations in the background of de Sitter spacetime in arbitrary dimensions. More specifically, we used a gauge-invariant formalism to describe the perturbations inside the cosmological horizon, i.e. in the static patch of de Sitter spacetime. After a gauge-fixed quantization procedure, the two-point function in the Bunch–Davies-like vacuum state was shown to be infrared finite and invariant under time-translation. In this work, we give details of the calculations to obtain the graviton two-point function in 3 + 1 dimensions.

scholarly journals CFT unitarity and the AdS Cutkosky rules

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
David Meltzer ◽  
Allic Sivaramakrishnan
Keyword(s):  
Flat Space ◽  
Tree Level ◽  
Optical Theorem ◽  
Conformal Field Theories ◽  
Strongly Coupled ◽  
Conformal Field ◽  
Weakly Coupled ◽  
S Matrix ◽  
Space Limit ◽  
Diagrammatic Method

Abstract We derive the Cutkosky rules for conformal field theories (CFTs) at weak and strong coupling. These rules give a simple, diagrammatic method to compute the double-commutator that appears in the Lorentzian inversion formula. We first revisit weakly-coupled CFTs in flat space, where the cuts are performed on Feynman diagrams. We then generalize these rules to strongly-coupled holographic CFTs, where the cuts are performed on the Witten diagrams of the dual theory. In both cases, Cutkosky rules factorize loop diagrams into on-shell sub-diagrams and generalize the standard S-matrix cutting rules. These rules are naturally formulated and derived in Lorentzian momentum space, where the double-commutator is manifestly related to the CFT optical theorem. Finally, we study the AdS cutting rules in explicit examples at tree level and one loop. In these examples, we confirm that the rules are consistent with the OPE limit and that we recover the S-matrix optical theorem in the flat space limit. The AdS cutting rules and the CFT dispersion formula together form a holographic unitarity method to reconstruct Witten diagrams from their cuts.

scholarly journals An Efficient Method to Find Solutions for Transcendental Equations with Several Roots

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Rogelio Luck ◽  
Gregory J. Zdaniuk ◽  
Heejin Cho
Keyword(s):  
Null Space ◽  
Heat Conduction Problem ◽  
Hankel Matrix ◽  
Polynomial Equation ◽  
Transient Heat Conduction ◽  
Transcendental Equation ◽  
Integral Theorem ◽  
Transient Heat Conduction Problem ◽  
Cauchy’S Integral Theorem ◽  
Roots Of A Polynomial

This paper presents a method for obtaining a solution for all the roots of a transcendental equation within a bounded region by finding a polynomial equation with the same roots as the transcendental equation. The proposed method is developed using Cauchy’s integral theorem for complex variables and transforms the problem of finding the roots of a transcendental equation into an equivalent problem of finding roots of a polynomial equation with exactly the same roots. The interesting result is that the coefficients of the polynomial form a vector which lies in the null space of a Hankel matrix made up of the Fourier series coefficients of the inverse of the original transcendental equation. Then the explicit solution can be readily obtained using the complex fast Fourier transform. To conclude, the authors present an example by solving for the first three eigenvalues of the 1D transient heat conduction problem.

scholarly journals Phase Transition at the Metric Elastic Universe

1996 ◽  
Vol 168 ◽  
pp. 569-570
Author(s):  
Alexander Gusev
Keyword(s):  
Phase Transition ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Vacuum State ◽  
Space Time ◽  
Deformation Tensor ◽  
De Sitter ◽  
Initial State ◽  
Deformation State ◽  
The Universe

At the last time the concept of the curved space-time as the some medium with stress tensor σαβon the right part of Einstein equation is extensively studied in the frame of the Sakharov - Wheeler metric elasticity(Sakharov (1967), Wheeler (1970)). The physical cosmology pre- dicts a different phase transitions (Linde (1990), Guth (1991)). In the frame of Relativistic Theory of Finite Deformations (RTFD) (Gusev (1986)) the transition from the initial stateof the Universe (Minkowskian's vacuum, quasi-vacuum(Gliner (1965), Zel'dovich (1968)) to the final stateof the Universe(Friedmann space, de Sitter space) has the form of phase transition(Gusev (1989) which is connected with different space-time symmetry of the initial and final states of Universe(from the point of view of isometric groupGnof space). In the RTFD (Gusev (1983), Gusev (1989)) the space-time is described by deformation tensorof the three-dimensional surfaces, and the Einstein's equations are viewed as the constitutive relations between the deformations ∊αβand stresses σαβ. The vacuum state of Universe have the visible zero physical characteristics and one is unsteady relatively quantum and topological deformations (Gunzig & Nardone (1989), Guth (1991)). Deformations of vacuum state, identifying with empty Mikowskian's space are described the deformations tensor ∊αβ, wherethe metrical tensor of deformation state of 3-geometry on the hypersurface, which is ortogonaled to the four-velocityis the 3 -geometry of initial state,is a projection tensor.

ON THE SUPERSTRING HAMILTONIAN IN THE FRIEDMANN SPACE–TIME

2006 ◽  
Vol 15 (06) ◽  
pp. 845-868 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Cosmological Solution ◽  
Vacuum State ◽  
Einstein Condensate ◽  
De Sitter ◽  
Superstring Theory ◽  
Bose Einstein Condensate ◽  
Higher Derivative ◽  
Effective Lagrangian ◽  
State Formula ◽  
De Sitter Vacuum

The ten-dimensional effective Lagrangian [Formula: see text] for the gravitational sector of the heterotic superstring theory is known up to quartic higher-derivative order [Formula: see text]. In cosmology, the reduced, four-dimensional line element assumes the Friedmann form ds2 = dt2 - a(t)2dx2, where t is comoving time and a(t) ≡ a0eα(t) is the radius function of the three-space dx2, whose curvature is k = 0, ± 1. The four-Lagrangian can then be expressed as the power-series [Formula: see text], where ˙ ≡ d/dt, from which the field equation can be derived by the method of Ostrogradsky. Here, we determine the coefficients Λ0, An, Bn, Cn, and Kn, which are all non-vanishing in general. We recover the previously obtained, high-curvature, anti-de Sitter vacuum state [Formula: see text] with effective cosmological constant Λ = {18/[175ζ(3) - 1/2]}1/3A r κ-2, whose existence makes it possible to envisage a singularity-free and horizon-free cosmological solution, stable to linear perturbations. It is interesting that all the coefficients of quartic origin arise from the near-cancellation of sums of opposite sign but magnitude f ≈ (28.6–369) times larger than the answer. They thus exhibit a slight asymmetry with regard to positive and negative energies, the anti-de Sitter vacuum being characterized by positive Nordström energy, and therefore only accessible at high curvatures. This vacuum state is a Bose–Einstein condensate of non-interacting gravitons at zero temperature, which, referred to comoving time, can only be formulated after the Wick rotation t → ±iτ, resulting in an imaginary horizon.

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