Primordial Gravitational Waves


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1 Primordial Gravitational Waves João G. Rosa Rudolph Peierls Centre for Theoretical Physics University of Oxford First Annual School of the EU Network Universe Net The Origin of the Universe Mytilene, September 2007
2 Motivation Gravitational waves can be a powerful probe of the early universe: Produced during inflation Weak interactions with matter and radiation May enconde information about the history of the universe Primordial Gravitational Waves J. G. Rosa Slide 2/19
3 Classical Tensor Perturbations Flat FriedmannRobertsonWalker background: Metric perturbations (conformal time coordinate): Tensor perturbations are transverse and traceless; Linearised Einstein equations (synchronous gauge):, Evolution similar to scalar field case Primordial Gravitational Waves J. G. Rosa Slide 3/19
4 Classical Tensor Perturbations Fourier expansion: where the symmetric polarisation tensor is transverse and traceless and is normalised as Equation for the mode k: Power law expansion: General solution expressed in terms of Bessel functions: Primordial Gravitational Waves J. G. Rosa Slide 4/19
5 Quantisation and Power Spectrum In linear theory, one can use the analogy with the scalar field case to construct the quantum theory associated with the free tensor modes in a curved spacetime: physical timedependent operator Wronskian Normalisation Condition: Primordial Gravitational Waves J. G. Rosa Slide 5/19
6 Quantisation and Power Spectrum Power Spectrum: Energy density: Primordial Gravitational Waves J. G. Rosa Slide 6/19
7 Inflationary perturbations Slowroll inflation: energy density of the universe dominated by potential energy of a scalar fieldφ; Scale factor: Slowroll parameters: Equation for tensor modes: Solution: Primordial Gravitational Waves J. G. Rosa Slide 7/19
8 Inflationary perturbations Evolution of k=1 mode The solution exhibits two distinct behaviours: Subhorizon  redshifted plane wave: Horizon crossing: Superhorizon frozen amplitude: Primordial Gravitational Waves J. G. Rosa Slide 8/19
9 Inflationary perturbations Assume that at the end of inflation (τ=0) all modes of interest are well outside the horizon: Power spectrum: Energy density: Spectral index: Slowroll inflation produces a cosmic background of gravitational waves from quantum fluctuations with an almost scale invariant power spectrum Primordial Gravitational Waves J. G. Rosa Slide 9/19
10 Postinflationary behaviour Simplified model: Radiation + Matter with instantaneous transition General solutions: Radiation: Matter: Primordial Gravitational Waves J. G. Rosa Slide 10/19
11 Postinflationary behaviour Transfer function coefficients: Modes reenter the Hubble horizon during the radiation era or the matter era Primordial Gravitational Waves J. G. Rosa Slide 11/19
12 Postinflationary behaviour Smooth radiationmatter transition: Rescaled variables: Scale factor: Tensor modes equation: Solve numerically with initial conditions (at the end of inflation): Primordial Gravitational Waves J. G. Rosa Slide 12/19
13 Postinflationary behaviour y = 0.09 y = 5.34 smooth instantaneous Transfer function coefficients: Primordial Gravitational Waves J. G. Rosa Slide 13/19
14 Postinflationary behaviour If we neglect phase shift induced by radiation era: Fit to numerical data: Transfer function Energy density (averaged over several periods) Primordial Gravitational Waves J. G. Rosa Slide 14/19
15 Postinflationary behaviour Effect of phase transition atτ= τ * (radiation era): Number of relativistic d.o.f. changes from g *i to g *f < g *i ; Energy density of radiation fluid: Conservation of entropy Scale factor evolution (instantaneous transitions): Primordial Gravitational Waves J. G. Rosa Slide 15/19
16 Postinflationary behaviour Transfer function coefficients for phase transition: New time variable: General solution for second radiationdomination period: From continuity: Compute coefficients C k and D k after matterradiation transition Primordial Gravitational Waves J. G. Rosa Slide 16/19
17 Postinflationary behaviour Example: QCD Phase Transition (Laine, 2001) τ * = 1.4 x 108 τ eq (T = 170 MeV) g *i = (quarkgluon plasma) r = g *f = (hadrons) Relevant scales: k * = 7.1 x 10 7 k eq f ~ 108 Hz Primordial Gravitational Waves J. G. Rosa Slide 17/19
18 Postinflationary behaviour Results: Oscillations due to QCD phase transition H 0 = 1 Oscillations due to matterradiation transition Primordial Gravitational Waves J. G. Rosa Slide 18/19
19 Conclusions Tensor perturbations are powerful tools for understanding the evolution of our universe; Studying the cosmic background of gravitational waves may provide important information about the mechanism behind inflation; The tensor transfer function may encode information about the radiationmatter transition and other possible phase transitions where relativistic d.o.f. are lost; Primordial Gravitational Waves J. G. Rosa Slide 19/19
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