Photometric redshifts and intrinsic alignments: Degeneracies and biases in the $3\times 2\mathrm{pt}$ analysis

We present a systematic study of cosmological parameter bias in weak lensing and large-scale structure analyses for upcoming imaging surveys induced by the interplay of intrinsic alignments (IA) and photometric redshift (photo-z) model misspecification error. We first examine the degeneracies between the parameters of the tidal alignment–tidal torquing (TATT) model for IA and of a photo-z model including a mean shift ($\mathrm{\Delta }\overline{z}$) and variance (${\sigma }_z$) for each tomographic bin of lenses and sources, under a variety of underlying true IA behaviors. We identify strong degeneracies between: (1) the redshift scaling of the tidal alignment amplitude and the mean shift and variances of source bins, (2) the redshift scaling of the tidal torquing amplitude and the variance of the lowest-$z$ source bin, and (3) the IA source density weighting and the mean shift and variance of several source bins. We then use this information to guide our exploration of the level of cosmological parameter bias which can be induced given incorrect modeling of IA, photo-z, or both. We find that marginalizing over all the parameters of TATT is generally sufficient to preclude cosmological parameter bias in the scenarios we consider. However, this does not necessarily mean that IA and photo-z parameters are themselves unbiased, nor does it mean that the best-fit model is a good fit to the data. We also find scenarios where the inferred parameters produce ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2$ values indicative of a good fit but cosmological parameter bias is significant, particularly when the IA source density weighting parameter is not marginalized over.

Figure 1

Fiducial lens galaxy (left) and source galaxy (right) redshift distributions in our forecasting analysis setup.

Figure 2

Degeneracies between redshift-dependent NLA parameters $a_1$, ${\eta }_1$ and source photo-z parameters ${\overline{z}}_s$, ${\sigma }_{z,s}$ under IA parameter set 4 (where $a_2$, ${\eta }_2$, $b_{\mathrm{TA}}$ are all set to 0).

Figure 3

Degeneracy between TATT parameter ${\eta }_2$ and photo-z parameters ${\sigma }_{z,s}$ under IA parameter set 1 (which has $a_2$, ${\eta }_2$, and $b_{\mathrm{TA}}$ all nonzero and positive).

Figure 4

Degeneracy between TATT parameter $b_{\mathrm{TA}}$ and photo-z parameters ${\overline{z}}_s^1$, ${\sigma }_{z,s}^2$, and ${\sigma }_{z,s}^3$ under IA parameter set 10 (which has all five TATT parameters nonzero, $a_1$ negative, and the remainder positive).

Figure 5

Effect of misspecifying ${\sigma }_{z,s}^{1-3}$, with true and assumed IA scenario described by redshift-dependent NLA (IA parameter set 4). The assumed value is ${\sigma }_{z,s}^{1-3}=0.05$. The gray shaded region indicates ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2\le 1.1$. The blue shaded region indicates parameter bias of at least $0.3\sigma$. See Sec. 4a1 for more details.

Figure 6

Bias to a subset of noncosmological parameters due to misspecifying ${\sigma }_{z,s}^{1-3}$, with true and assumed IA scenario described by redshift-dependent NLA (IA parameter set 4). We see that although cosmological parameter bias levels are acceptable (see Fig. 5), IA and photo-z parameters are severely biased. Error bars represent posterior forecast uncertainties rather than Fisher instability uncertainties in this case. See Sec. 4a1 for more details.

Figure 7

Inferred cosmological parameters biases and goodness of fit of inferred parameters in the scenario where the true IA is given by a redshift-dependent NLA model but the analysis assumes a redshift-independent NLA model with ${\eta }_1=0$ (indicated on the color bar by a dashed black line). The gray shaded region indicates ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2\le 1.1$. The blue shaded region indicates parameter bias of at least $0.3\sigma$. See Sec. 4a2 for more details.

Figure 8

Effect of misspecification of ${\sigma }_{z,s}^1$, with true and assumed IA model given by TATT with all parameters positive nonzero (IA parameter set 1) and ${\sigma }_{z,s}^{1,\text{assumed}}=0.05$. The blue shaded region indicates parameter bias of at least $0.3\sigma$. The ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2$ is greater than our threshold of 1.1 for all points, hence no gray shading is included. See Sec. 4b1 for more details.

Figure 9

Effect of misspecification of ${\sigma }_{z,s}^1$, with true IA model given by TATT with all parameters positive nonzero (IA parameter set 1), assumed IA model given by NLA-z with the same $a_1$ and ${\eta }_1$ values as in the true TATT model (IA parameter set 4), and ${\sigma }_{z,s}^{1,\text{assumed}}=0.05$. The blue shading indicates parameter bias of at least $0.3\sigma$ (true for all points in this scenario). ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2$ is greater than our threshold of 1.1 for all points, hence no gray shading is included. See Sec. 4b1 for more details. Note that we maintain the same color bar scaling as Fig. 8 for easy comparison.

Figure 10

Effect of the misspecification of ${\eta }_2$ under a truth IA model of TATT (IA parameter set 1) and an assumed IA model of redshift-dependent NLA (with $a_2$ and $b_{\mathrm{TA}}$ assumed to be 0, along with ${\eta }_2$ as indicated by the dashed line on the color bar). The blue shading indicates parameter bias of at least $0.3\sigma$ (true for all points in this scenario). ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2$ is greater than our threshold of 1.1 for all points, hence no gray shading is included. For further details, see Sec. 4b2.

Figure 11

Effect of the misspecification of ${\sigma }_{z,s}^{2-3}$ under a truth IA model of TATT with all other parameters nonzero ($a_1$ negative and all others positive) and an assumed IA model of TATT. The assumed, fixed ${\sigma }_{z,s}^{2-3}$ is 0.05. The gray shaded region indicates ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2\le 1.1$. The bias in both $S_8-{\mathrm{\Omega }}_M$ and $w_0-w_a$ is below than our threshold of $0.3\sigma$ for all points, hence no blue shading is included. For further details, see Sec. 4c1.

Figure 12

Effect of the misspecification of ${\sigma }_{z,s}^{2,3}$ under a truth IA model of TATT with all other parameters nonzero ($a_1$ negative and all others positive) and an assumed IA model of TATT with $b_{\mathrm{TA}}$ fixed to its fiducial value of 1. The assumed ${\sigma }_{z,s}^{2,3}$ is 0.05. The gray shaded region indicates ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2\le 1.1$. The blue shaded region indicates parameter bias of at least $0.3\sigma$. For further details, see Sec. 4c1.

Figure 13

Effect of the misspecification of $b_{\mathrm{TA}}$ under a truth IA model of TATT with all other parameters nonzero ($a_1$ negative and all others positive, IA parameter set 10). The assumed value of $b_{\mathrm{TA}}$ is 1. The gray shaded region indicates ${\chi }_{\mathrm{d}.\mathrm{o}.\mathrm{f}.}^2\le 1.1$. The blue shaded region indicates parameter bias of at least $0.3\sigma$. For further details, see Sec. 4c2.