Studying the precision of ray tracing techniques with Szekeres models

The simplest standard ray tracing scheme employing the Born and Limber approximations and neglecting lens-lens coupling is used for computing the convergence along individual rays in mock N-body data based on Szekeres swiss cheese and onion models. The results are compared with the exact convergence computed using the exact Szekeres metric combined with the Sachs formalism. A comparison is also made with an extension of the simple ray tracing scheme which includes the Doppler convergence. The exact convergence is reproduced very precisely as the sum of the gravitational and Doppler convergences along rays in Lemaitre-Tolman-Bondi swiss cheese and single void models. This is not the case when the swiss cheese models are based on nonsymmetric Szekeres models. For such models, there is a significant deviation between the exact and ray traced paths and hence also the corresponding convergences. There is also a clear deviation between the exact and ray tracing results obtained when studying both nonsymmetric and spherically symmetric Szekeres onion models.

Figure 1

Present time 1D density profiles of LTB and nonsymmetric Szekeres models with $r_b=60\mathrm{Mpc}$ and varied $m$. For the nonsymmetric models, the profiles are shown along the direction of highest degree of anisotropy. The comoving coordinates are normalized at present time in units of 0.1 Mpc.

Figure 2

Present time 1D density profiles of the onion models. The comoving coordinates are normalized at present time in units of 0.1 Mpc. The nonsymmetric models’ density profiles are shown along the axis of largest anisotropy.

Figure 3

Density and convergence along a light ray in a swiss cheese model based on compensated LTB single void models with $m=4$ and $r_b=60\mathrm{Mpc}$. The ray path has been computed using both the Born approximation and the geodesic equations of the exact LTB metric. The two resulting density distributions are indistinguishable in the figure. Four convergences are shown. The line labeled “${\kappa }_v+{\kappa }_{\delta }$” shows the sum of the Doppler convergence and the gravitational convergence obtained using the simple ray tracing scheme. The two components are also shown individually. The line labeled “${\kappa }_{\text{exact}}$” is the exact convergence as obtained from the exact set of ODEs. The figure includes a close-up of the region where the gravitational convergence begins to be non-negligible.

Figure 4

Convergence along a radial ray in the swiss cheese model based on the LTB single void model specified by $m=4$ and $r_b=60\mathrm{Mpc}$. The line labeled “${\kappa }_v+{\kappa }_{\delta }$” shows the sum of the Doppler convergence and the gravitational convergence obtained using the simple ray tracing scheme. The line labeled “${\kappa }_{\text{exact}}$” is the exact convergence as obtained from the exact set of ODEs. The figure includes close-ups at different redshift intervals.

Figure 5

Luminosity distance plotted as a function of the observed redshift and fitted to low-redshift FLRW approximations of $D_L(z)$. The results from four luminosity distances are shown corresponding to the exact convergence, the sum of the gravitational and Doppler convergences, the gravitational convergence and the background solution. The results are shown for LTB swiss cheese models specified by $m=4$ and both $r_b=30\mathrm{Mpc}$ and $r_b=100\mathrm{Mpc}$.

Figure 6

Density and convergence along nonradial light rays in an LTB single void model specified by $m=4$ and $r_b=60\mathrm{Mpc}$. The observer has been placed at $(t,r,\theta ,\varphi )=(t_0,70\mathrm{Mpc},8/5\pi ,3/5\pi )$, looking in various directions determined by $(k^t,k^r,k^p,k^q)=(-\frac{1}{c},k_0^r,k_0^p,0)$, where $k_0^p$ is varied as $k_0^p=-\frac{2}{c},-\frac{3}{c},-\frac{4}{c}$ and $k_0^r$ is determined from the null condition. Note that $k_0^p$ does not have the same units as $\frac{1}{c}$ so it is only the numerical value of $c$ that is implied in the values of $k_0^p$. Since the LTB model reduces exactly to the EdS model outside the structure, these initial conditions are for both the exact Szekeres coordinate system and the ray tracing coordinate system. The exact and ray traced rays are labeled by “exact” and “Born,” respectively.

Figure 7

Density along geodesics in a swiss cheese model based on compensated nonsymmetric Szekeres single void models. The geodesics have been computed using both the Born approximation and the exact geodesic equations according to the exact Szekeres metric.

Figure 8

Convergences along rays in swiss cheese models based on nonsymmetric Szekeres single void models. Four convergences are shown in each figure. The line labeled “${\kappa }_v+{\kappa }_{\delta }$” shows the sum of the Doppler convergence and the gravitational convergence. The two components are also shown individually. The line labeled “${\kappa }_{\text{exact}}$” is the exact convergence as obtained from the exact set of ODEs.

Figure 9

Luminosity distance plotted as a function of the observed redshift and fitted to low-redshift FLRW approximations of $D_L(z)$. The results from four luminosity distances are shown corresponding to the luminosity distance obtained from the exact equations, the sum of the gravitational and Doppler convergences, the exact convergence and the background solution.

Figure 10

Density and convergence along exact and ray traced radial rays in the onion models $O{1}_{\mathrm{ltb}}$, $O{2}_{\mathrm{ltb}}$ and $O{3}_{\mathrm{ltb}}$. The exact convergence is shown together with the sum of the Doppler and gravitational convergences. The gravitational convergence is also shown alone but its deviation from zero is too small to be visible in the figure. For model $O{3}_{\mathrm{ltb}}$, a close-up of the convergence has been included to highlight that the exact and approximate convergences are not exactly identical.

Figure 11

Density and convergence along exact and ray traced rays in onion model $O{3}_{\mathrm{sz}}$. Four convergences are shown: Doppler, gravitational, the sum of these two, and the exact. The gravitational convergence is negligible in the low-redshift interval studied here, so the lines showing the Doppler convergence and the sum of the Doppler convergence and the gravitational convergence are indistinguishable.

Figure 12

Luminosity distance plotted as a function of the observed redshift and fitted to the low-redshift FLRW approximation for $D_L(z)$. The results from four luminosity distances are shown corresponding to the exact convergence, the sum of the gravitational and Doppler convergences, the gravitational convergence and the background solution. The results are shown along three different lines of sight for a central observer in onion model $O{3}_{\mathrm{sz}}$.

Figure 13

Density and convergence along exact and ray traced rays in onion model $O{2}_{\mathrm{sz}}$. Note that ${\kappa }_{\delta }\approx 0$ so that the lines representing ${\kappa }_v$ and the sum ${\kappa }_v+{\kappa }_{\delta }$ are indistinguishable in the figure.

Figure 14

Luminosity distance plotted as a function of the observed redshift and fitted to low-redshift FLRW approximations for $D_L(z)$. The results from four luminosity distances are shown corresponding to the exact convergence, the sum of the gravitational and Doppler convergences, the gravitational convergence and the background solution.

Figure 15

Density and convergence along a radial geodesic in an LTB swiss cheese model. The density profiles along the exact and ray traced rays are indistinguishable. The same is true for the exact convergence and the sum of the gravitational and Doppler convergence.

Figure 16

Convergence along a radial ray in the LTB swiss cheese model based on the LTB single void model specified by $m=4$ and $r_b=60\mathrm{Mpc}$. The exact and ray traced convergences are computed using various FLRW backgrounds specified by their values of ${\mathrm{\Omega }}_{m,0}$. The proper background is the EdS model.