6 General Relativistic Perturbation Theory6.1 Concept Questions1. Why do general relativistic perturbation theory using the tetrad f ormalism as opposedto t he coordinate approach?2. Why is the tetrad metric γmnassumed fixed in the presence of perturbations?3. Are the tetrad axes γmfixed under a perturbation?4. Is it true that the tetrad components ϕmnof a perturbation are (anti-)symmetric inm ↔ n if and only if its coordinate components ϕµνare (anti-)symmetric in µ ↔ ν?5. Does an unperturbed quantity, such as the unperturbed metric0gµν, change under aninfinitesimal coordinate gauge transformation?6. How can the vierbein perturbation ϕmnbe considered a tetrad tensor field if it changesunder a n infinitesimal coordinate gauge transformations?7. What properties of the unperturb ed spacetime allow decompo sition of perturbationsinto independently evolving Fourier modes?8. What properties of the unperturb ed spacetime allow decompo sition of perturbationsinto independently evolving scalar, vector, and tensor modes?9. In what sense do scalar, vector, and tensor modes have spin 0, 1, and 2 respectively?10. Tensor modes represent gravitational waves that propagate at the speed of light. Ifscalar and vector modes also propagate at the speed of light, do these not also constitutegravitational waves?11. Equation (81) defines the mass M of a body a s what a distant observer would measurefrom its gravitational potential. Similarly equation (89) defines the angular momentumL of a body as what a distant observer would measure from the dragging of inertialframes. In what sense are these definitions legitimate?12. Can an observer far fr om a body detect the difference between t he scalar potentia ls Ψand Φ produced by the body?13. If a gravitational wave is a wave of spacetime itself, distorting the very rulers andclocks that measure spacetime, how is it possible to measure gravitational waves atall?14. Have gravitational waves been detected?15. If gravitat io nal waves carry energy-momentum, then can gravitational waves be presentin a region of spacetime with va nishing energy-momentum tensor, Tmn= 0?116. Why do the wavelengths of perturbations in cosmology expand with the Universe,whereas perturbations in Minkowski spa ce do not expand?17. What does power spectrum mean?18. Why is the power spectrum a good way to char acterize the amplitude of fluctuations?19. Why is the power spectrum of fluctuations of the Cosmic Microwave Background(CMB) plotted as a function of harmonic number?20. What causes the acoustic peaks in the power spectrum of fluctuations of the CMB?21. Are there acoustic peaks in the power spectrum of matter (galaxies) today?22. What sets the scale of the first peak in the power spectrum of the CMB? [What setsthe physical scale? Then what sets the angular scale?]23. The odd peaks (including the first peak) in the CMB power spectrum are compressionpeaks, while the even peaks are rarefaction peaks. Why does a rarefaction produce apeak, not a trough?24. Why is the first peak the most prominent? Why do higher peaks generally get pro-gressively weaker?25. The third peak is about as strong as the second peak? Why?26. The matter power spectrum reaches a maximum at a scale that is slightly larger thanthe scale of the first baryonic acoustic peak. Why?27. The physical density of species x at the t ime of recombination is proportio na l to Ωxh2where Ωxis the ratio of the actual to critical density of species x at the present time,and h ≡ H0/100 km s−1Mpc−1is the present day Hubble constant. Explain.28. How does changing the baryon density Ωbh2affect the CMB power spectrum?29. How does changing the non- bar yonic cold dark matter density Ωch2, without changingthe baryon density Ωbh2, affect the CMB power spectrum?30. What effects do neutrinos have on perturbations?31. How does changing the curvature Ωkaffect the CMB power spectrum?32. How does changing the dark energy ΩΛaffect the CMB power spectrum?26.2 What’s important?This section of the notes describ es the elements of perturbation theo ry in GR, using thetetrad formalism. It covers gravitational waves, and cosmological perturbation theory.1. Getting your brain around coordinate and tetrad gauge transformations.2. A central aim of general relativistic perturbation theory is to identify the coordinateand tetrad gauge-invariant perturbations, since only these have physical meaning.3. A second central a im is to classify perturbations into independently evolving modes,to t he extent that this is possible.4. In background spacetimes with spatial translation and rotation symmetry, which in-cludes Minkowski space and the Fr iedmann-Roberston-Walker metric of cosmology,modes decompose into independently evolving scalar (spin-0), vector (spin-1), andtensor (spin-2) modes. In background spacetimes witho ut spatial translation and ro-tation symmetry, such as black holes, sca la r , vector, and tensor modes scatter off thecurvature of space, and therefore mix with each other.5. In backgro und spacetimes with spatial translation and rotation symmetry, there are6 algebraic combinations of metric coefficients that are coordinate and tetrad gauge-invariant, and therefore represent physical perturbations. There are 2 scalar modes,2 vecto r modes, and 2 tensor modes. A spin-m mode varies as eimχwhere χ is therotational angle a bout the spatial wavevector k of the mode.6. In background spacetimes without spatial tr anslation and rotation symmetry, the coor-dinate and tetrad gauge-invariant perturbations are not algebraic combinations of themetric coefficients, but r ather combinations that involve first and second derivatives ofthe metric coefficients. Gravitational waves are described by the Weyl tensor, whichcan be decomposed into 5 complex components, with spin 0, ±1, and ±2. The spin-±2components describe propagating gravitational waves, while the spin-0 and spin-±1components describe the non-propagating gravitational field near a source.7. The preeminent a pplication of general relativistic perturbation theory is to cosmol-ogy. Coupled with physics that is either well understood (such as photon-electronscattering) or straightforward to model even without a deep understanding (such asthe dynamical behavior of non-baryonic dark matter and dark energy), the theory hasyielded predictions that are in spectacular agreement with observations of fluctuationsin the CMB and in the large scale distribution of