I was rummaging around looking for some things related to a paper I’m struggling to finish before term starts and I found some vintage diagrams. They brought back a lot of memories of working on the textbook I wrote with Francesco Lucchin way back in the 1990s. In particular I remember how long it took to make these figures, when nowadays it would take a few minutes. In fact I’m thinking of setting this as a Computational Physics project for next year. These are not full computations either, just a simple fluid-based approach.

The curves show the evolution of fluctuations in both matter δ_{m} and radiation δ_{r} on a particular scale (i.e. a Fourier mode of given wavelength) defined as δ_{m}=δρ_{m}/ρ_{m}, etc. The x-axis shows the cosmic scale factor, which represents the expansion of the Universe and in both cases the universe is flat, i.e. it has a critical density. The first graph shows a universe with only baryonic matter:

Notice the strongly coupled oscillations in matter and radiation until a scale factor of around 10^{-3}, corresponding to a redshift of a thousand or so, which is when matter and radiation decouple. The y-axis is logarithmic so the downward spikes represent zero points.

It is these oscillations which are responsible for the bumps and wiggles in the spectrum of the cosmic microwave background spectrum, as different Fourier modes arrive at the last scattering surface at a different phase of its oscillation. Of course going from the Figure above to the CMB fluctuation spectrum (see below) involves more calculations, and there is now a well-established machinery for doing these with full physical descriptions, but I think the above diagram makes the physical origin of these features clear.

The CMB power spectrum from Planck

The second diagram shows what happens if you add a third component called `X’ in the Figure below which we take to be cold non-baryonic matter. Because this stuff doesn’t interact directly with radiation (while baryons do) it doesn’t participate in the oscillations but the density perturbations just carry on growing:

Notice too that at late times (i.e. after the baryonic matter and radiation have decoupled) the baryonic component grows much more quickly than in the first Figure. This is because, when released from the effect of the photon background, baryons start to feel the gravitational pull of the dark matter perturbations.

There’s nothing new in this of course – these Figures are thirty years old and similar were produced even earlier than that – but I still think pictures like these are pedagogically useful,