Szekeres Swiss-cheese model and supernova observations

We use different particular classes of axially symmetric Szekeres Swiss-cheese models for the study of the apparent dimming of the supernovae of type Ia. We compare the results with those obtained in the corresponding Lemaître-Tolman Swiss-cheese models. Although the quantitative picture is different the qualitative results are comparable, i.e., one cannot fully explain the dimming of the supernovae using small-scale ($\sim 50\mathrm{Mpc}$) inhomogeneities. To fit successfully the data we need structures of order of 500 Mpc size or larger. However, this result might be an artifact due to the use of axial light rays in axially symmetric models. Anyhow, this work is a first step in trying to use Szekeres Swiss-cheese models in cosmology and it will be followed by the study of more physical models with still less symmetry.

Figure 1

Density profiles along the axial geodesic for the models studied in this paper—model 1 (left), models 2 and 3 (center), models 4 and 5 (right).

Figure 2

The residual Hubble diagram for model 1 (solid line, red). For comparison, the results for the corresponding Lemaître-Tolman model (i.e. with $M$ and $k$ as in model 1 but with $E^{\prime }=0$) are also presented (dashed line, green). The Union supernova data set [45] is depicted in the inset, for clarity only measurements for $z>0.2$ are given.

Figure 3

The residual Hubble diagram for model 2 (dotted line, blue), model 3 (solid line, red), and for the corresponding Lemaître-Tolman model (i.e. with $M$ and $k$ as in models 2–3 but with $E^{\prime }=0$) (dashed line, green). Supernovae measurements are extracted from the Union data set [45].

Figure 4

The residual Hubble diagram for model 4 (blue, SC—Swiss-cheese, MV—minimal void), model 5 (red, SC—Swiss-cheese, MV—minimal void) and for the corresponding Lemaître-Tolman model (i.e. with $M$ and $k$ as in models 4 and 5 but with $E_{,r}=0$) are also presented (green). For clarity, the Union supernova data set [45] presented in the inset is for $z>0.2$ only.

Figure 5

The residual Hubble diagram for models 6 (grey line, green), 7 (solid line, red), and $\Lambda \mathrm{CDM}$ (dashed line). For clarity the supernova data are presented for $z>0.1$.