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 Friedmann-Robertson-Walker 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 time-dependent 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 Slow-roll inflation: energy density of the universe dominated by potential energy of a scalar fieldφ; Scale factor: Slow-roll 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: Slow-roll 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 Post-inflationary behaviour Simplified model: Radiation + Matter with instantaneous transition General solutions: Radiation: Matter: Primordial Gravitational Waves J. G. Rosa Slide 10/19

11 Post-inflationary 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 Post-inflationary behaviour Smooth radiation-matter 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 Post-inflationary behaviour y = 0.09 y = 5.34 smooth instantaneous Transfer function coefficients: Primordial Gravitational Waves J. G. Rosa Slide 13/19

14 Post-inflationary 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 Post-inflationary 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 Post-inflationary behaviour Transfer function coefficients for phase transition: New time variable: General solution for second radiation-domination period: From continuity: Compute coefficients C k and D k after matter-radiation transition Primordial Gravitational Waves J. G. Rosa Slide 16/19

17 Post-inflationary behaviour Example: QCD Phase Transition (Laine, 2001) τ * = 1.4 x 10-8 τ eq (T = 170 MeV) g *i = (quark-gluon plasma) r = g *f = (hadrons) Relevant scales: k * = 7.1 x 10 7 k eq f ~ 10-8 Hz Primordial Gravitational Waves J. G. Rosa Slide 17/19

18 Post-inflationary 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 radiation-matter 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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