In the picture on the right the different types of universes can be seen. The line with Ωm=1 describes a flat universe. it will expand but the universe doesn't accalerate. The line with Ωm=0.3 and Ωv=0 (which is ΩΛ) describes an open universe. The line with Ωm=0.3 and Ωv=0.3 describes the Λ dominated universe, it expands rapidly.
The time evolution of a halo with mass M and radius R is given by the Tolman-Bondi equation,
From this equation we can get the overdensities needed for different types of universes. Here the equation with ΩΛ=0 stands for an open universe, the equation with Ω0 + ΩΛ=1 stands for a Λ dominated universe with Ω0 the constand of matter. The equation with Ω0=1, ΩΛ=0 stands for a flat universe, also called an Einstein-de Sitter (EdS) universe.
These equations suggest that for a collapse in an open universe a higher overdensity is needed than for an EdS universe as Ωz (the constant of matter at a redshift z) is always smaller or equal to 1. But there is an important difference between the models based on an EdS universe an the models with Ω<1, concernig the time evolution of the denstity inhomogeneities. At hight redshifts ,when linear theory is valid, the growth of overdensities can be described by,
δ(x,z) is the local density at a redshift z, δ0(x) the linear extrapolated value at the present time and D(z) the linear growth factor which gives the growth rate of the overdensities in linear theory. The linear theory means that the universe evolves linear at hight redshifts and it becomes non-linear at lower redshifts. In an EdS universe D(z) can be desrcibed as,
In a universe with Ω<1, the slope of D(z) is shallower for low z. Since the growth factor is normalized to D(0)=1, D(z) is larger at higher redshifts and the non-linear regime is reached earlier than in the models based on a EdS universe. This effect neutralizes the effect of a higher overdensity needed for a collapse in an open than in an EdS universe. In an open universe the collapse will even take place at higher redshifts then in an EdS universe. The timepath differs in different universes, the formation process although is the same..
Looking at different kinds of matter distribution in the universe, there is a large difference between the formation process in an CDM, CHDM and HDM universe. This can be seen by looking at the power spectrum on the right, obtained from the same simulation as used in merging. The amount of power can be seen at the y-axis. At the x-axis the scale can be seen going from large scales at the left of the axis to small scales on the right of the axis. As in an HDM universe the matter is hot and moves around with almost the speed of light, matter will smooth out. Because of this smoothing there won't be overdensity regions. Sometimes overdensities, however are sufficiently high and exceed the δc(z), the critical overdensity needed for a collapse at a redshift z. Large-scale modes become non-linear first in an HDM universe, leading to the formation of clusters before galaxy-sized objects are present. Since there is no power on small scales in an HDM universe as can be seen in the power spectrum, the formation of a halo should be similar to the collapse of an overdensity region in a spherical top-hat model, merging doesn't take place. The formation process in an HDM universe is non-hierarchical.In a CHDM universe the formation process will be hierarchical, but it will take place at lower redshifts. The hot component in an CHDM universe, will smooth out and decrease the overdensities. It will take longer till the overdensity reaches the critical value needed for a collapse in an CHDM universe than in an CDM universe.
It can be said that the formation process in the CDM an CHDM universes are the same, but the time path isn't. The formation process in an HDM universe differs from the process in CDM and CHDM universes. In HDM the formation is non-hierarchical, in CDM and CHDM it is hierarchical.
All these differences can be seen in the following figure, it's from the same simulation as shown at merging. From the left to the right the collums represent an EdS CDM universe, Λ dominated CDM universe, open universe, CHDM universe and an HDM universe.
The figure shows an earlier collapse in an open universe than in an EdS universe, consistent with the predictions of the linear theory. At redshift z= 1.84 the collapse in the open universe is further evaluated than in the EdS universe. It also shows the later collapse in an CHDM universe, because of the hot component. At redshift z=1.84 the collapse has started in the three CDM universes, but is hasn't in the CHDM universe, in CHDM it starts at redshift z=0.53. Looking at the HDM model, it can be seen that there is no hierarchical clustering, only a collapse. In the other models small structure can be seen moving to the center of the cluster, because of gravitational interaction.
As said, the formation process in the CDM and CHDM universes is the same, only the time path isn't. The formation process in the HDM universe differs from the CDM and CHDM. Beside that the collapse will start earlier in an universe with Ωm<1.
The question remains which matter distribution is most realistic, the CDM, HDM or CHDM distribution. The figure on the right shows a power spectrum. In this spectrum the scale on the x-axis goes from small scale on the left to large scale on the right. The dots are COBE (Cosmic Background Explorer) measurements and the lines are theoretical values.Power can be seen on small scales which excludes an HDM universe. The light blue line describes an CDM universe. The dots and this line fit well with each other. This suggests the CDM universe to be the most realistic.