The main contribution to the observed CMB anisotropies comes from density waves. These are just density waves which propagate in a photon-baryon plasma. The effect of these waves is essencially threefold:

 Density wave generate density fluctuations, which observationally translate into temperature fluctuations. In addition these density fluctuations generate gravitational potentials through the Poisson equation. These gravitational potentials can also generate gravitational redshift or blueshift which also leave some imprint on the observed temperature fluctuations. The combination of the density perturbations and the gravitational redshift or blueshift is usually called the Sachs-Wolfe effect.

 The presence of density fluctuations also translate into pressure fluctuations. Through Euler equation, velocity fields can be generated by pressure gradients. Since the observed CMB photons have last-scattered on free electrons and since there exists velocity fields in the free electron fluid, one expects to see Doppler effects contributing to the CMB anisotropies.

 Although the main contribution to the CMB anisotropies comes from the Sachs-Wolfe and Doppler effects, there is a supplementary contribution coming from gravitational effects after the photon last scattering. The most intuitive one is gravitational lensing.This effect arises only at small angular scales, because the deflection angles are small. In addition to deflection effects, gravity can affect photons through energy exchanges: a photon which travels time-varying gravitational potential will eithe gain or lose energy, depending on whether the well becomes shallower or deeper. This effect is usually called the Integrated Sachs-Wolfe effect and mostly affects the large angular scales.

In addition, gravitational waves can also contribute to the CMB anisotropies. Gravitational waves are a prediction of general relativity and do not have any analog in Newtonian physics. Although no direct detection of gravitational waves has been achieved to date, a number of consequences of the existence of gravitational waves have been observed, the most spectacular of which being probably the ``gravitational damping’’ in a binary system, which make two celestial bodies slowly narrow their orbit and finally coalesce. The effect of a gravitational wave on the medium it travels is to shrink and elongate spacetime in directions orthogonal to the direction of propagation of the wave. For example, a circle orthogonal to the wave vector would remain in the same plane but would be transformed into an ellipse. When looking at the CMB we do not observe directly gravitational waves directly, but we can see the imprint of gravitational waves on the CMB photons. The effect of a gravitational wave is to shrink and extend space in direction perpendicular to the direction of propagation of the gravitational wave. When a graviatitional wave travels the path of the CMB photons, this shrinking and extending effect can translate into a gain or a loss of energy of the photons, exactly as a photon can gain or lose energy when it crosses a time varying graviational potential. Thus, one expects to have a contribution to the ISW effect because of gravitational waves.

Finally, it is possible to show that Thomson scattering in a locally anisotropic media (such as a media with density fluctuations or a media travelled by gravitational waves) generates polarization. Gravitational waves can generate any polarization pattern. On the contrary, some of the polarization patterns, the so-called B-modes cannot be generated by density fluctuations. In the (likely) hypothesis where the density waves represent the largest contribution to the CMB anisotropies, one expects to detect a large amount of E-modes (those that can be generated both by density fluctuations and gravitational waves) and a significantly smaller amount of B-modes.

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