We can extend the excursion set methods that have been discussed and used previously to study voids, and associate with a void with a region that crosses a barrier of sufficient underdensity. One subtlety that we have to concern ourselves with, however, is that even if a given walk crosses our barrier, at some larger scale it may cross the barrier associated with spherical collapse, indicating that rather than identifying it as a void, it should be identified as a halo. This results in void formation being a two barrier problem [60], and we will discuss this in more detail in section 5.3.1. Before we can apply excursion set methods, we need to calculate the critical density for void formation, which we will again do by considering the evolution of underdense regions with an initially spherical tophat density profile.
5.2.1 The spherical tophat model
For voids, much of the discussion in section 2.1 and section 3.2.1 remains unchanged. In standard gravity, the solution is not a cycloid, but similar: the sines and cosines are replaced by their hyperbolic counterparts. This leads to the difficulty that the idea of collapse is not as well defined for voids as for haloes, so that while it is clear that a halo is collapsed sometime around when the radius reaches zero, for a void, one must either consider the condition of “shell-crossing” which occurs when initially interior shells cross initially exterior shells. In standard gravity, the objects for which is true are about 0.2 times that of the background. Because of the ambiguity associated with shell crossing in modified gravity models, we will use this value, an underdensity of −0.8, to define voids even in our modified gravity model. To calculate the relevant critical density, we consider the evolution of a number of concentric shells within which the initial density was uniform, but slightly below that of the background. We then use the methods of [37] to calculate the evolution of the shells, until the initial region has expanded sufficiently more than
the background that the underdensity within it has become −0.8. The quantity of
interest is δv, the analog toδc for collapsing structures, which is the critical initial
gravity,δv is independent of void size (as isδc), but for modified gravity models, we
expect it to depend on void size for the same reason that δc depends on the mass
of the collapsed object. If, in the course of its evolution, the size of a patch never exceeds rs, then the large-scale modification to gravity is inconsequential. Hence, for sufficiently small voids, which form from smaller patches in the initial field, we expect δv to be the same as in standard gravity. For large voids, the effect of the
modification should be stronger. For positive α voids should be easier to form, so
we expect |δv|to be smaller, whereas for negative α the opposite should be true. To find the scale dependence of δv(Ri), we study the evolution of under-
dense patches starting from a large grid of initial underdensities and sizes. Once
we have the scale dependence of δv(Ri), we can relate it to a mass scale by M =
(4π/3)R3iρi¯(1 +δi)≈(4π/3)R3i, the last relation holding because δi is small. Thus, we can calculateδv(M), which is shown in figure 5.1. This has the expected depen-
dence onM and α. Note that we have chosen to express the scale dependence ofδv
in terms of M, for ease of comparison with the scale dependence ofδc; in fact, our
Figure forδv is very similar to figure 4.3 forδc. Of course, we could have expressed it in terms of the initial size Ri, or in terms of the final size rv. This is because we have defined voids as being 0.2 times the density of the background, making
rv = 51/3Ri ≈1.7Ri: the comoving radius of a void is 1.7 times larger than it was
initially.
One might wonder why the curves in figure 5.1 are so similar, and yet not identical, in particular, the δv starts to deviate from standard at a slightly lower
mass than δc even with the same α and rs. This occurs because while both haloes
and voids expand initially, voids continue expanding and haloes recollapse. As a result, for a given mass (which corresponds to an initial size), a void would crossrs
earlier than a halo, which pushes the modification to slightly lower masses for voids than for haloes.
5.2.2 Density profiles
Before we use the scale dependence ofδv to estimate void abundances, it is
interesting to consider the evolution of the density profile. We started with an initial
Figure 5.1: Ratio of initial density required for void formation at the present time to that in the standard gravity model, when the background cosmology is ΛCDM,
rs = 5h−1 Mpc and rc = 70h−1 Mpc. From top to bottom, curves show models
in which α = −1,−0.5,0,0.5 and 1 (note that α = 0 is standard gravity). For reference, the dashed lines are the barriers for haloes
in the central region remains constant as a consequence of Birkhoff’s theorem, but a ridge forms at the edge of the void that is caused by the expansion of the inner shells pushing against the exterior shells ([60] and references therein).
In modified gravity Birkhoff’s theorem no longer holds, so these two fea-
tures of the evolution are expected to depend on α. Figure 5.2 shows that the
density in the interior region does not remain constant – this is clearly more evident
in the negative α case, but also evident in the positive α case. Ridge formation at
the edge still occurs, but the modification to gravity affects the size of the ridge; in the positive α case the ridge is smaller in physical size, whereas the opposite is true for negativeα. In particular, for the negativeα case the ridge extends further and trails off much more slowly. The cause of these ridges and the interior profile is the same as for haloes: the outer shells have a strong effect on the interior shells (though not as strong as for haloes, as the density is much smaller in a void). As a note, because δ <0, the effect of modified gravity on void profiles is opposite to the effect on the density profiles of haloes, that is the inner shells in negativeα are forced inward, rather than outward.