Mukhanov–Sasaki equation

Mukhanov–Sasaki equation
Name	Mukhanov–Sasaki equation
Field	Cosmology, General Relativity, Quantum Field Theory
Introduced	1985
Introduced by	Viatcheslav Mukhanov; Misao Sasaki
Related	Cosmological perturbation theory; Inflaton; Friedmann equations

Mukhanov–Sasaki equation

The Mukhanov–Sasaki equation is a linear second-order differential equation governing scalar perturbations of the metric and inflaton field in early-universe cosmology. It provides the dynamical evolution for the gauge-invariant curvature perturbation related to primordial fluctuations produced during inflation and connects ideas in Albert Einstein's General relativity with quantum field theory in curved spacetime studied by figures like Stephen Hawking and Leonard Susskind. The equation underpins predictions tested with observations by missions such as COBE, WMAP, and Planck.

Introduction

The Mukhanov–Sasaki equation arose from efforts by Viatcheslav Mukhanov and Misao Sasaki to combine perturbative treatments present in the work of James Bardeen and canonical quantization methods inspired by Paul Dirac and Richard Feynman. It is central to models of Alan Guth's and Andrei Linde's inflationary scenarios and to the calculation of the primordial power spectrum compared against analyses by collaborations like BICEP2 and experiments at Large Hadron Collider. The equation is formulated for a single scalar degree of freedom built from combinations of metric perturbations and inflaton fluctuations originally explored in studies by Yakov Zel'dovich and Rafael Sorkin.

Derivation

Starting from the action of a scalar field minimally coupled to gravity in a Friedmann–Lemaître–Robertson–Walker background used by Alexander Friedmann and Georges Lemaître, gauge-invariant variables were introduced following methods of James Bardeen and canonical procedures echoing ADM decomposition developed by Richard Arnowitt, Stanley Deser, and Charles Misner. The quadratic action for scalar perturbations yields a canonical variable (commonly denoted v or u) whose Euler–Lagrange equation is the Mukhanov–Sasaki equation, mirroring procedures from perturbation theory applied by Subrahmanyan Chandrasekhar and variational techniques associated with Emmy Noether. The derivation uses background equations such as the Friedmann equations and slow-roll approximations originally formulated by Andrei Linde and refined in the literature influenced by Guth and Alan Guth's collaborators.

Mathematical Properties

Mathematically the Mukhanov–Sasaki equation is a mode equation of the form v + (k^2 - z/z) v = 0, where primes denote conformal time derivatives and z encodes background quantities related to the scale factor used by Friedmann and the scalar field dynamics. It shares structural similarities with the Schrödinger equation as in work by Erwin Schrödinger and Sturm–Liouville theory applied in analyses by David Hilbert. The effective potential term z''/z reflects background evolution and slow-roll parameters introduced in research by Steinhardt and Paul Steinhardt and determines stability, WKB applicability associated with techniques from Hermann Weyl and adiabatic vacua constructions used by Leonard Parker.

Solutions and Quantization

Mode solutions are typically chosen to match the Bunch–Davies vacuum motivated by studies by T. S. Bunch and Paul Davies and are normalized following canonical commutation relations rooted in the quantization rules of Paul Dirac and Werner Heisenberg. For de Sitter space considered in work by William de Sitter, analytic solutions use Hankel functions explored in classical analysis by Ernest William Barnes and yield scale-invariant spectra first anticipated by Harrison and Zel'dovich and later quantified by Mukhanov and Sasaki. Quantization leads to computation of two-point correlators compared with observational likelihood analyses carried out by teams behind Planck and WMAP.

Applications in Cosmology

The equation is applied to compute the primordial curvature perturbation that seeds cosmic structure observed in galaxy surveys like Sloan Digital Sky Survey and the anisotropy measured by COBE, WMAP, and Planck. It informs models of single-field inflation proposed by Andrei Linde, multi-field extensions considered by David Wands and Martin Bucher, and alternatives such as ekpyrotic scenarios associated with Paul Steinhardt and Neil Turok. Predictions derived using the equation feed into parameter estimation frameworks developed by the Planck Collaboration and model-selection studies by researchers including Max Tegmark and Nicholas Kaiser.

Numerical Methods and Examples

Numerical integration of the Mukhanov–Sasaki equation employs methods and codes influenced by numerical relativity work by Miguel Alcubierre and Boltzmann solvers like CAMB and CLASS used by collaborations of Anthony Lewis and Julien Lesgourgues. Standard techniques include Runge–Kutta integration, adaptive step-size control discussed by John Butcher, and matching across horizon crossing employing WKB approximations used in semiclassical analyses by Masao Takahashi. Examples involve computing scalar power spectra for chaotic inflation models by Andrei Linde or plateau models discussed in the context of Katherine Freese and comparing outputs with likelihood pipelines utilized by Planck Collaboration.

Generalizations and Extensions

Extensions generalize the Mukhanov–Sasaki framework to multiple fields studied by D. Langlois and David Wands, non-minimal couplings explored by Yasunori Fujii and Keith Olive, and modified gravity contexts such as f(R) theories investigated by T. P. Sotiriou and Valerio Faraoni. Other extensions incorporate non-Bunch–Davies initial states analyzed in work by Daniel Green and features from effective field theory of inflation developed by Clifford Cheung and Leonard Senatore. The equation's structure also informs studies of reheating processes pursued by Lev Kofman, Andrei Linde, and Alexei Starobinsky.

Category:Cosmology