The results regarding the amplitude of the oscillations given in the previous page hold only when the gravitational potential is constant. The reason why this hypothesis was done has to do with the presence of Dark Matter.

Jeans instability

If one considers a non relativistic fluid in a Minkowskian space (i.e., without expansion), the mass conservation and the Euler equation can be linerized as

where , , , and are the density, the density perturbation, the pressure perturbation, the velocity and the gravitational potential, respectively. Assuming that the density is constant and using the Poisson equation to express the gravitational potential in term of the density perturbation, one obtains

,

where the sound speed is defined as

Above the Jeans lengh , gravity dominates presure and the waves become unstable and grow exponantially. At fixed density, what determines the Jeans lenght is the soud speed, and hence the pressure of the fluid. The higher the pressure, the larger the Jeans length.

Dark matter perturbations in an expanding universe

Taking into account the expansion of the universe modifies the above equations in three ways:

 Expansion plays the role of a friction force. Although the physical velocity remains constant for a free particle, the expansion of the universe makes an object situated at a distance of the particle to recess at the rate , where is the Hubble parameter. In the same time, the particle can travel the distance . Formally (i.e., in comooving coordinates), this is equivalent to say that the object remains at fixed distance whereas the velocity has decreased by an amount

.

 Second, the Poisson equation is modified to take into account the fact that physical lengths depend with time. In comoving coordinates, it is written

.

 Third, the density is not independant of the expansion rate. For a given species i, one can introduce the density parameter

,

where is the critical density.

After all theses manipulations, the Euler equation becomes

.

In the radiation era, the right handside is negligible and Dark Matter perturbations only grow as , however this was expected as perturbations can grow when the density is high. The gravitational potential decays with time.

In the matter era, the Jeans instability still occurs, but the friction term decrease the growth rate to a power law, . It is then easy to check that the Poisson equation predicts that the gravitational potential is then constant (whereas it was growing in the Minkowskian case).

Influence on the CMB

Although not well known , the ratio between matter and radiation energy density lies between 10³ and 10⁴, so that the radiation-to matter transition arises before recombination. The fact that recombination occurs within the matter era explains why the gravitational potential play a significant role in determining the amplitude and the offset of the photon-baryon density perturbations. Should recombination happen within or close to the radiation era, then the formula would transform into

.

so that in this case no contrast between even and odd peaks.

In addition, there is a second important effect depending on the Dark Matter density. The redshift of recombination is fixed because it is almost independent from the baryon density and completely independent from other parameters. However, the age of the universe at this redshift does depend on the expansion rate of the universe. In particular, it is sensitive on how close it happens to the radiation era. Therefore, varying the Dark Matter density modifies the sound horizon at recombination. Since the wavevector which corresponds to the the first peak is given by

,

the position of the peaks depend on the Dark Matter density.

Influence on structure formation

Since Dark Matter is decoupled from other species, Dark Matter perturbations can grow as soon as the matter era starts. This is not the case for baryons the fluctuation of which can grow only after they have decoupled from photons. However, it is easy to show that immediately after decoupling between photons and baryons occurs, baryons can fall into the already existing potential wells created by Dark Matter, so that the fact that photons prevented baryons from forming structure does not have significant consequences on structure formation. However, what has an influence on structure formation is the edpth of Dark Matter potential wells at recombination. This is determined by the amount of time that elapsed between radiation-to-matter transition, which is determined by the amount of Dark Matter.

Summary

Dark matter plays a role both on CMB anisotropies and on structure formation. For the former, the main effects are:

 A decrease of the contrast between even and odd peaks for low Dark Matter density models,

 A modification of the observed size of the fluctuations as the corresponding physical size explicitely depend on the age of the Universe and hence its expansion history at recombination.

Regarding structure formation:

 The amount of structure that can be formed directly depend on the begining of the matter era from which structure can grow.

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