Interpretation of the Hubble diagram in a nonhomogeneous universe

In the standard cosmological framework, the Hubble diagram is interpreted by assuming that the light emitted by standard candles propagates in a spatially homogeneous and isotropic spacetime. However, the light from “point sources”—such as supernovae—probes the Universe on scales where the homogeneity principle is no longer valid. Inhomogeneities are expected to induce a bias and a dispersion of the Hubble diagram. This is investigated by considering a Swiss-cheese cosmological model, which (1) is an exact solution of the Einstein field equations, (2) is strongly inhomogeneous on small scales, but (3) has the same expansion history as a strictly homogeneous and isotropic universe. By simulating Hubble diagrams in such models, we quantify the influence of inhomogeneities on the measurement of the cosmological parameters. Though significant in general, the effects reduce drastically for a universe dominated by the cosmological constant.

Figure 1

The standard interpretation of SNe data assumes that light propagates in purely homogeneous and isotropic space (top). However, thin light beams are expected to probe the inhomogeneous nature of the actual Universe (bottom) down to a scale where the continuous limit is no longer valid.

Figure 2

Swiss-cheese models (bottom) allow us to model inhomogeneities beyond the continuous limit, while keeping the same dynamics and average properties as the FL model (top).

Figure 3

The junction hypersurface as seen from the FL point of view with equation $\chi ={\chi }_{\mathrm{h}}$ (left); and from the Kottler point of view with equation $r=r_{\mathrm{h}}(t)$ (right).

Figure 4

A light ray propagates alternatively in FL and Kottler regions. The main geometrical quantities defined and used in Sec. 4 are depicted in this simplified view of a single hole.

Figure 5

An arbitrary initial condition is rotated so that the geodesic lies in the equatorial plane $\theta =\pi /2$. Left: ${\mathcal{E}}_{\mathrm{out}}$ is brought (a) to $\phi =0$ by the rotation ${\mathit{R}}_z(-{\phi }_{\mathrm{out}})$, and (b) to $\theta =\pi /2$ by the rotation ${\mathit{R}}_y(\pi /2-{\theta }_{\mathrm{out}})$. The resulting event and wave vector are denoted ${\mathcal{E}}_{\mathrm{out},\mathrm{eq}}$ and $k_{\mathrm{out}}^{\prime \mu }$. Middle: $k_{\mathrm{out}}^{\prime \mu }$ is brought to the equatorial plane by the rotation ${\mathit{R}}_x(-\psi )$. Right: Final situation.

Figure 6

Null geodesic in the Kottler region. Depicted with the Kottler coordinate system, the hole grows so that the ray enters with $r_{\mathrm{in}}$ and exits with $r_{\mathrm{out}}>r_{\mathrm{in}}$.

Figure 7

Geometry and hierarchy of distances for a typical Swiss-cheese hole: $r_{\mathrm{S}}\ll r_{\mathrm{phys}}\ll r_{\mathrm{h}}\ll r_{\mathrm{Hubble}}$.

Figure 8

Setup for evaluating the effect of one hole on the redshift and luminosity distance.

Figure 9

Relative correction to the redshift $z$, due to the hole in the line of sight, as a function of the direction of observation $\beta$, for a source at $z_{\mathrm{source}}=0.05$. Top panel: The mass of the hole is $M={10}^{15}{M}_{\odot }$, and three positions between the source and the observer are tested, $z_{\mathrm{h}}^{(\mathrm{FL})}/z_{\mathrm{source}}=0.1$ (blue, dot-dashed), 0.5 (purple, dashed), and 0.9 (red, solid). Bottom panel: The hole is at $z_{\mathrm{h}}^{(\mathrm{FL})}=0.5z_{\mathrm{source}}$ and three values for the mass are tested, $M/{10}^{15}{M}_{\odot }=3$ (blue, dot-dashed), 2 (purple, dashed), and 1 (red, solid).

Figure 10

Same as Fig. 9, but plotted in terms of the centered and normalized observation direction $(\beta -{\beta }_{\min })/({\beta }_{\max }-{\beta }_{\min })$.

Figure 11

Relative correction to the luminosity distance $D_{\mathrm{L}}$, due to the hole in the line of sight, as a function of the direction of observation $\beta$, for a source at $z_{\mathrm{source}}=0.05$. Top panel: The mass of the hole is $M={10}^{15}{M}_{\odot }$, and three positions between the source and the observer are tested, $z_{\mathrm{h}}^{(\mathrm{FL})}/z_{\mathrm{source}}=0.1$ (blue, dot-dashed), 0.5 (purple, dashed), and 0.9 (red, solid). Bottom panel: The hole is at $z_{\mathrm{h}}^{(\mathrm{FL})}=0.5z_{\mathrm{source}}$ and three values for the mass are tested, $M/{10}^{15}{M}_{\odot }=3$ (blue, dot-dashed), 2 (purple, dashed), and 1 (red, solid).

Figure 12

Same as Fig. 11, but plotted in terms of the centered and normalized observation direction $(\beta -{\beta }_{\min })/({\beta }_{\max }-{\beta }_{\min })$.

Figure 13

Hexagonal lattice of identical holes. On the left, the arrangement is close-packed, so that the smoothness parameter is $f=f_{\min }\approx 0.26$. On the right, $f=0.7$.

Figure 14

Relative corrections to the redshift $z$ (top panel) and luminosity distance $D_{\mathrm{L}}$ (bottom panel) as functions of $z$, for an arbitrary light beam traveling through a Swiss-cheese universe. All holes are identical; their mass is $M={10}^{11}{M}_{\odot }$. Three different smoothness parameters are tested: $f=0.26$ (blue, dot-dashed), 0.6 (purple, dashed), and 0.9 (red, solid).

Figure 15

Same as Fig. 14 but with $f=0.26$ and three different values for the masses: $M/{10}^{11}{M}_{\odot }=1$ (blue, dot-dashed), 2 (purple, dashed), and 3 (red, solid).

Figure 16

Probability distribution of the relative correction to the luminosity distance for random directions of observation, at $z=0.1$ (top panel) and $z=1$ (bottom panel). The smoothness parameter is $f=f_{\min }$ and two different values for the masses of the holes are tested: $M={10}^{11}{M}_{\odot }$ (blue, solid) and $M={10}^{15}{M}_{\odot }$ (purple, dashed).

Figure 17

Evolution, with redshift $z$, of the relative correction to the luminosity distance averaged over $N_{\mathrm{obs}}=200$ random directions of observation. Error bars indicate the dispersion ${\sigma }_{\delta D_{\mathrm{L}}}$ around the mean correction $⟨\delta D_{\mathrm{L}}⟩$. As in Fig. 16, we compare Swiss-cheese models with $M={10}^{11}{M}_{\odot }$ (blue, filled markers) and $M={10}^{15}{M}_{\odot }$ (purple, empty markers).

Figure 18

Hubble diagram of a Swiss-cheese universe (dots) with $f=f_{\min }$, $M={10}^{11}{M}_{\odot }$ and EdS background cosmology. For comparison, we also display the distance-redshift relations of purely FL universes, with EdS parameters (blue, solid) and $({\Omega }_{\mathrm{m}},{\Omega }_K,{\Omega }_{\Lambda })=(0.3,0,0.7)$ (black, dashed).

Figure 19

Comparison between background parameters (crosses) and apparent parameters (disks) for two Swiss-cheese models with $f=f_{\min }$ and $M={10}^{15}{M}_{\odot }$. In blue, $({\Omega }_{\mathrm{m}},{\Omega }_{\Lambda })=(1,0)$ leads to $({\overline{\Omega }}_{\mathrm{m}},{\overline{\Omega }}_{\Lambda })=(0.5,-0.3)$. In black, $({\Omega }_{\mathrm{m}},{\Omega }_{\Lambda })=(0.3,0.7)$ leads to $({\overline{\Omega }}_{\mathrm{m}},{\overline{\Omega }}_{\Lambda })=(0.2,0.6)$. The $1\sigma$ and $2\sigma$ contours are respectively the solid and dashed ellipses. The solid straight line indicates the configurations with zero spatial curvature.

Figure 20

Apparent cosmic parameters ${\overline{\Omega }}_{\mathrm{m}}$ (gray disks), ${\overline{\Omega }}_K$ (red squares) and ${\overline{\Omega }}_{\mathrm{m}}$ (black diamonds) versus smoothness parameter $f$, for a Swiss-cheese universe with EdS background $({\Omega }_{\mathrm{m}},{\Omega }_K,{\Omega }_{\Lambda })=(1,0,0)$. Solid lines and filled markers correspond to $M={10}^{11}{M}_{\odot }$, dashed lines and empty markers to $M={10}^{15}{M}_{\odot }$.

Figure 21

Apparent deceleration parameter $\overline{q}$ as a function of smoothness parameter $f$, for Swiss-cheese models with EdS background ($q=1/2$). Solid lines and filled markers correspond to $M={10}^{11}{M}_{\odot }$, dashed lines and empty markers to $M={10}^{15}{M}_{\odot }$.

Figure 22

Difference between apparent and background cosmological parameters ${\overline{\Omega }}_{\mathrm{m}}-{\Omega }_{\mathrm{m}}$ (gray disks), ${\overline{\Omega }}_K-{\Omega }_K$ (red squares) and ${\overline{\Omega }}_{\Lambda }-{\Omega }_{\Lambda }$ (black diamonds) versus background ${\Omega }_{\Lambda }$, for Swiss-cheese models with $f=f_{\min }$. Solid lines and filled markers correspond to $M={10}^{11}{M}_{\odot }$, dashed lines and empty markers to $M={10}^{15}{M}_{\odot }$.

Figure 23

Difference between apparent and background deceleration parameters $\overline{q}-q$ as a function of $q$, for Swiss-cheese models with $f=f_{\min }$. Solid lines and filled markers correspond to $M={10}^{11}{M}_{\odot }$, dashed lines and empty markers to $M={10}^{15}{M}_{\odot }$.

Figure 24

Comparison between the approximate luminosity-redshift relation $D_{\mathrm{L}}^{\mathrm{SC}}(z)=(1+z{)}^2{D}_{\mathrm{A}}^{\mathrm{SC}}(z)$ in a Swiss-cheese universe (solid lines), simulated observations (dots), and the Dyer-Roeder model $D_{\mathrm{L}}^{\mathrm{DR}}(z)$ with $\alpha (z)=f$ (dashed lines). Three different values of the smoothness parameter are tested, from top to bottom: $f=f_{\min }\approx 0.26$, $f=0.7$, $f=0.9$.

Figure 25

Fit of the Hubble diagram constructed from the SNLS 3 data set [77], by using the luminosity-redshift relation $D_{\mathrm{L}}^{\mathrm{SC}}(z|{\Omega }_{\mathrm{m}},f)$ of a spatially Euclidean Swiss-cheese model. The colored areas indicate (from the darkest to the lightest) the $1\sigma$, $2\sigma$ and $3\sigma$ confidence levels.