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## APPENDIX A: CONVERSION OF THE MAGNITUDE LIMIT FOR DESI INTO FLUXES

In order to compare the fluxes of our sources to the DESI BGS limit, we convert this magnitude limit into a flux limit. The flux cut $F_{\text{lim}}$ can be written in terms of a flux density $f_{\nu, \text{lim}}$ (in units of Jansky) as follows:

$$F_{\text{lim}}[L_{\odot}/\text{Mpc}^2] = f_{\nu, \text{lim}}[\text{Jy}] c \left( \frac{1}{\lambda_{\text{min}}} - \frac{1}{\lambda_{\text{max}}} \right) \frac{A_{\text{SI}} B_{\odot}}{C_{\lambda}} \quad \text{(A1)}$$

where c is the speed of light (in units of m s⁻¹), $\lambda_{\text{min}}$ and $\lambda_{\text{max}}$ are the minimum and maximum wavelengths in the r-band (in nanometres), and $A_{\text{SI}}$, $B_{\odot}$, and $C_{\lambda}$ are conversion factors between units: $A_{\text{SI}} = 10^{-26} \text{ W m}^{-2} \text{ Hz}^{-1} \text{ Jy}^{-1}$ converts a flux density in Jansky into the International System of Units (SI), $B_{\odot} \approx 2.474 \times 10^{18} L_{\odot} \text{ Mpc}^{-2} \text{ W}^{-1} \text{ m}^2$ converts a flux from the SI system into $L_{\odot} \text{ Mpc}^{-2}$, and $C_{\lambda} = 10^{-9} \text{ m nm}^{-1}$ converts wavelengths from nanometres into metres. The flux density $f_{\nu, \text{lim}}[\text{Jy}]$ is related to the magnitude in the r-band as follows:

$$r_{\text{min}} = \frac{5}{2} \log_{10} \left( \frac{f_{\nu, \text{lim}}[\text{Jy}]}{3631[\text{Jy}]} \right) \quad \text{(A2)}$$

Therefore, the flux limit in units of $L_{\odot} \text{ Mpc}^{-2}$ can be computed as a function of $r_{\text{min}}$ as follows:

$$F_{\text{lim}}[L_{\odot}/\text{Mpc}^2] = 3631 \times 10^{-\frac{2}{3}r_{\text{min}}} c \left( \frac{1}{\lambda_{\text{min}}} - \frac{1}{\lambda_{\text{max}}} \right) \frac{A_{\text{SI}} B_{\odot}}{C_{\lambda}} \quad \text{(A3)}$$

$\lambda_{\text{min}} = \lambda_{\text{eff}} - \Delta\lambda/2$ and $\lambda_{\text{max}} = \lambda_{\text{eff}} + \Delta\lambda/2$ are computed assuming an effective wavelength midpoint $\lambda_{\text{eff}} = 616.6 \text{ nm}$ and bandwidth $\Delta\lambda = 120 \text{ nm}$.

## APPENDIX B: DETAILS ON THE MEASUREMENT OF GALAXY BIAS AND MAGNIFICATION BIAS

In this section, we report more details on the procedure we have adopted to estimate the specifics of the faint and bright catalogues, needed to evaluate the theoretical prediction and covariance for the dipole measured in our N-body simulations.

The effective redshift is estimated as the average redshift over the number of objects in the full-sky catalogue, that is

$$\bar{z}_{\text{eff}} = \frac{1}{N_{\text{sources}}} \sum_{i} z_{i} \quad \text{(B1)}$$

The galaxy bias is estimated from the angular power spectrum of the number counts of the bright/faint objects. We construct the pixelised map of the number counts and estimate the angular power spectrum of the map using the HEALPix routine anafast. Note that our catalogues cover the full sky, thus we do not need to correct for the effect of a mask at this stage. The angular power spectra are then fitted to a theory model that includes the contributions of density and linear RSD. The linear galaxy bias is the only free parameter in this model. The linear RSD model is expected to be sufficiently accurate for our bias estimate, as we are considering redshift bins with width $\sigma_z = 0.1$ (see Lepori et al. 2023). The accuracy degrades for thinner redshift bins, see for example Jalilvand et al. (2020) and Matthewson & Durrer (2022). We use the curve_fit routine from
the scipy package to obtain the best-fitting linear bias and its error. The error is estimated assuming the uncertainty on the simulated data has cosmic variance and shot-noise contribution. In our fit, we include scales between $\ell_{min} = 20$ and $\ell_{max} = 80$. We have tested the dependence of our best-fitting estimation on the range of scales used in the fit. We find that a fit performed on scales $\ell_{min} = 100$ and $\ell_{max} = 200$ leads to changes in the bias estimates of a few percent, which would not significantly affect the theory prediction for the dipole and its covariance.

The magnification bias for the bright populations is estimated from the cumulative luminosity function of the bright population. Special care has to be taken, as the flux threshold that separates bright and faint populations is given by a redshift-dependent function. Therefore, we estimate the values of magnification bias in the 20 sub-bins, where the redshift-dependent flux threshold is computed for a given ratio of bright and faint source. In each sub-bin, the magnification bias is given by the slope of the cumulative flux distribution at the flux threshold. From these 20 values $s_B^i$, we compute an effective value for the magnification bias as follows:

$$s_B = \frac{\sum N_B^i s_B^i}{\sum N_B^i}, \qquad (B2)$$

where $N_B^i$ is the number of bright objects in the sub-bin $i$, and the sum runs over the 20 sub-bins.

The error of this estimate is computed as the square root of the variance of a weighted mean

$$(\Delta s_B)^2 = \frac{\sum N_B^i}{(\sum N_B^i)^2 - (\sum N_B^i)^2} \sum N_B^i (s_B^i - s_B)^2. \qquad (B3)$$

The magnification bias of the faint population is computed from the values of magnification bias of the full catalogue and the bright catalogue, as in equation (14). Its error is estimated from the error on $s_B$ using the linear propagation of the uncertainty, that is

$$\Delta s_F = \frac{\bar{N}_B(z)}{\bar{N}_F(z)} \Delta s_B. \qquad (B4)$$

## APPENDIX C: IMPACT OF EVOLUTION BIAS ON THE FLUX-LIMITED CATALOGUE

In our analysis, we do not attempt to model the redshift distribution within the redshift bin used for dipole measurements. As a consequence, the evolution bias of our catalogues cannot provide a realistic estimate of the evolution bias for the DESI

<figure><img src="image_9.png" /><figcaption>**Figure C1.** Redshift distribution in the three redshift bins, for the bright populations and the equal split. Coloured histograms show the redshift distribution for our baseline case, black dashed histograms show the redshift distribution for the flux-limited catalogues.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure C2.** Impact of evolution bias on the theory prediction for the flux-limited catalogues at $\bar{z} = 0.25$. Top and bottom panels refer to the 10–90 per cent and 50–50 per cent splits between faint and bright populations, respectively. Dashed lines show the theory prediction used in our analysis, which sets the evolution bias to zero. Continuous lines includes the evolution bias, modelled as described in the text. The measurements of the dipole are highlighted in red. The shaded region represents the uncertainty of the theoretical prediction due to errors in the bias measurements (we assume here the evolution bias to be perfectly known). The grey vertical dotted line marks the scale cut used in our analysis, corresponding to the bin $d = 27 \text{ Mpc h}^{-1}$.</figcaption></figure>

BGS sample. For this reason, we neglect this effect in our fiducial analysis. However, we can still attempt to model this effect for our catalogue.

In the case without a flux limit, estimating the evolution bias is straightforward. Our selection of bright/faint populations corresponds roughly to a sharp luminosity threshold. Therefore, we can evaluate the evolution bias from the slope of the redshift distribution of bright/faint objects, which is by construction the same for the two populations. As shown in Fig. C1, the slope is roughly a constant within the redshift bin. We correct for a geometrical factor, as shown in equation (15), and find that the evolution bias of the bright and faint populations is identical, with values ranging between –1 and 1 in the redshift bins we consider. The impact of the evolution bias on the baseline catalogues is completely negligible as a result.

Modelling the evolution bias for the flux-limited catalogue is less straightforward. By applying a fixed flux limit in each redshift bin, we are removing the faintest objects in the bin, preferentially at high-redshift, leading to a redshift distribution that is not necessarily smooth (see dashed histograms in Fig. C1). Furthermore, since we are not applying a sharp luminosity threshold, the evolution bias of the faint sample also depends on the slope of the luminosity function, i.e. the magnification bias, as in equation (17). In order to take into account these two effects, we estimate the evolution bias of the bright/faint populations $f_{B/F}^{\text{evo}}(z)$ as a function of redshift inside the bin, modelling the change of slope at a transition redshift $z_t$ whose value is computed through a fit. The selection of the bright population is not directly affected by the flux limit. Therefore, we estimate the redshift-dependent evolution bias from equation (17), that is

$$f_{B}^{\text{evo}}(z) = - \left. \frac{\partial \ln \bar{N}_{B}}{\partial \ln(1+z)} \right|_{\text{sel}} + \frac{\dot{H}}{H^2} + \frac{2}{rH} - 1, \qquad (C1)$$

where we denote with $\left. \frac{\partial \ln \bar{N}_B}{\partial \ln(1+z)} \right|_{\text{sel}}$ the logarithmic slope of the redshift distribution for the bright population. For the faint population, we
the scipy package to obtain the best-fitting linear bias and its error. The error is estimated assuming the uncertainty on the simulated data has cosmic variance and shot-noise contribution. In our fit, we include scales between $\ell_{min} = 20$ and $\ell_{max} = 80$. We have tested the dependence of our best-fitting estimation on the range of scales used in the fit. We find that a fit performed on scales $\ell_{min} = 100$ and $\ell_{max} = 200$ leads to changes in the bias estimates of a few percent, which would not significantly affect the theory prediction for the dipole and its covariance.

The magnification bias for the bright populations is estimated from the cumulative luminosity function of the bright population. Special care has to be taken, as the flux threshold that separates bright and faint populations is given by a redshift-dependent function. Therefore, we estimate the values of magnification bias in the 20 sub-bins, where the redshift-dependent flux threshold is computed for a given ratio of bright and faint source. In each sub-bin, the magnification bias is given by the slope of the cumulative flux distribution at the flux threshold. From these 20 values $s_B^i$, we compute an effective value for the magnification bias as follows:

$$s_B = \frac{\sum N_B^i s_B^i}{\sum N_B^i}, \qquad (B2)$$

where $N_B^i$ is the number of bright objects in the sub-bin $i$, and the sum runs over the 20 sub-bins.

The error of this estimate is computed as the square root of the variance of a weighted mean

$$(\Delta s_B)^2 = \frac{\sum N_B^i}{(\sum N_B^i)^2 - (\sum N_B^i)^2} \sum N_B^i (s_B^i - s_B)^2. \qquad (B3)$$

The magnification bias of the faint population is computed from the values of magnification bias of the full catalogue and the bright catalogue, as in equation (14). Its error is estimated from the error on $s_B$ using the linear propagation of the uncertainty, that is

$$\Delta s_F = \frac{\bar{N}_B(z)}{\bar{N}_F(z)} \Delta s_B. \qquad (B4)$$

## APPENDIX C: IMPACT OF EVOLUTION BIAS ON THE FLUX-LIMITED CATALOGUE

In our analysis, we do not attempt to model the redshift distribution within the redshift bin used for dipole measurements. As a consequence, the evolution bias of our catalogues cannot provide a realistic estimate of the evolution bias for the DESI

<figure><img src="image_11.png" /><figcaption>**Figure C1.** Redshift distribution in the three redshift bins, for the bright populations and the equal split. Coloured histograms show the redshift distribution for our baseline case, black dashed histograms show the redshift distribution for the flux-limited catalogues.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure C2.** Impact of evolution bias on the theory prediction for the flux-limited catalogues at $\bar{z} = 0.25$. Top and bottom panels refer to the 10–90 per cent and 50–50 per cent splits between faint and bright populations, respectively. Dashed lines show the theory prediction used in our analysis, which sets the evolution bias to zero. Continuous lines includes the evolution bias, modelled as described in the text. The measurements of the dipole are highlighted in red. The shaded region represents the uncertainty of the theoretical prediction due to errors in the bias measurements (we assume here the evolution bias to be perfectly known). The grey vertical dotted line marks the scale cut used in our analysis, corresponding to the bin $d = 27 \text{ Mpc h}^{-1}$.</figcaption></figure>

BGS sample. For this reason, we neglect this effect in our fiducial analysis. However, we can still attempt to model this effect for our catalogue.

In the case without a flux limit, estimating the evolution bias is straightforward. Our selection of bright/faint populations corresponds roughly to a sharp luminosity threshold. Therefore, we can evaluate the evolution bias from the slope of the redshift distribution of bright/faint objects, which is by construction the same for the two populations. As shown in Fig. C1, the slope is roughly a constant within the redshift bin. We correct for a geometrical factor, as shown in equation (15), and find that the evolution bias of the bright and faint populations is identical, with values ranging between -1 and 1 in the redshift bins we consider. The impact of the evolution bias on the baseline catalogues is completely negligible as a result.

Modelling the evolution bias for the flux-limited catalogue is less straightforward. By applying a fixed flux limit in each redshift bin, we are removing the faintest objects in the bin, preferentially at high-redshift, leading to a redshift distribution that is not necessarily smooth (see dashed histograms in Fig. C1). Furthermore, since we are not applying a sharp luminosity threshold, the evolution bias of the faint sample also depends on the slope of the luminosity function, i.e. the magnification bias, as in equation (17). In order to take into account these two effects, we estimate the evolution bias of the bright/faint populations $f_{B/F}^{\text{evo}}(z)$ as a function of redshift inside the bin, modelling the change of slope at a transition redshift $z_t$ whose value is computed through a fit. The selection of the bright population is not directly affected by the flux limit. Therefore, we estimate the redshift-dependent evolution bias from equation (17), that is

$$f_{B}^{\text{evo}}(z) = - \left. \frac{\partial \ln \bar{N}_{B}}{\partial \ln(1+z)} \right|_{\text{sel}} + \frac{\dot{H}}{H^2} + \frac{2}{rH} - 1, \qquad (C1)$$

where we denote with $\left. \frac{\partial \ln \bar{N}_B}{\partial \ln(1+z)} \right|_{\text{sel}}$ the logarithmic slope of the redshift distribution for the bright population. For the faint population, we
<table><caption>**Table C1.** Values of the evolution bias for the flux-limited catalogue, estimated with the method described in the text.</caption><thead><tr><th rowspan="2">z̄</th><th colspan="2">50 per cent faint, 50 per cent bright</th><th colspan="2">10 per cent faint, 90 per cent bright</th></tr><tr><th>Bright</th><th>Faint</th><th>Bright</th><th>Faint</th></tr></thead><tbody><tr><td>0.25</td><td>7</td><td>2</td><td>7</td><td>-17</td></tr><tr><td>0.35</td><td>4</td><td>3</td><td>4</td><td>-2</td></tr></tbody></table>

see that the flux limit affects the redshift distribution of objects with redshift above $z_t$. Therefore, we estimate the evolution bias of the faint population,

$$f_{\text{F}}^{\text{evo}}(z) = \begin{cases} -\left. \frac{\partial \ln \bar{N}_{\text{F}}}{\partial \ln(1+z)} \right|_{\text{sel}} + \frac{\dot{H}_t}{H^2} + \frac{2}{rH} - 1, & \text{for } z \le z_t \\ -\left. \frac{\partial \ln \bar{N}_{\text{F}}}{\partial \ln(1+z)} \right|_{\text{F}_c} + \frac{\dot{H}_t}{H^2} + \frac{2-5s_{\text{lim}}}{rH} - 5s_{\text{lim}} - 1, & \text{for } z > z_t \end{cases}$$

where we define $s_{\text{lim}} \equiv s(z, F_{\text{lim}}) \frac{\dot{N}(z)}{N_F(z)}$. From these redshift-dependent functions $f_{\text{B/F}}^{\text{evo}}(z)$, we compute an effective value for the evolution bias,

$$f_{\text{B/F}}^{\text{evo}} = \frac{\int (dN/dz) f_{\text{B/F}}^{\text{evo}}(z) dz}{\int (dN/dz) dz}, \quad (C2)$$

where $dN/dz$ is the smoothed redshift distribution of the objects inside the bin.

In Table C1, we report the values of the evolution bias in the bins $z = 0.25, 0.35$, where the dipole signal was detected with high statistical significance. We find large differences in evolution bias at $z = 0.25$, for the 10–90 per cent split.

In Fig. C2, we compare the baseline theory predictions at $z = 0.25$ (dashed lines) to a model that accounts for the values of evolution bias reported in Table C1 (continuous lines). Including the evolution bias provides a better model for the measurements of the dipole on large scales. Therefore, the small discrepancy between theory and measurements discussed in Section 5.4 for the 10–90 per cent case is most likely due to our choice of neglecting evolution bias in our modelling. Note that there remains a small disagreement at small scales, below 40 Mpc $h^{-1}$, even after correcting for the evolution bias. Since this disagreement is not there for the 50–50 per cent case, it is probably not due to non-linearities, but rather to some systematic effect that could affect the 90–10 per cent catalogue with flux cut. In particular, this is the catalogue with the smallest number of faint objects.

## APPENDIX D: DIPOLE MEASUREMENTS AT $z = 0.45$

In Figs D1 and D2, we present the measurements of the relativistic cross-correlation dipole for $z = 0.45$, respectively for the case with no flux limit and for the case where a flux limit is imposed. A theory prediction for the signal is not given for the case where the catalogue is split into 90 per cent bright galaxies and 10 per cent faint galaxies, because the measurement of the bias is not reliable due to the small number of objects.

<figure><img src="image_11.png" /><figcaption>**Figure D1.** The measured 2PCF dipole of the $z = 0.45$ catalogue without flux limit for the two different splitting ratios (50 per cent bright and 50 per cent faint, 90 per cent bright and 10 per cent faint) compared to the prediction from linear theory. The shaded region covers the uncertainty of the theoretical prediction due to errors in the bias measurements. The grey vertical dotted line marks the bin with $d = 27$ Mpc $h^{-1}$. The error bars are derived from the jackknife estimate described in Section 5.2.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure D2.** The measured 2PCF dipole of the $z = 0.45$ catalogue with flux limit for the two different splitting ratios (50 per cent bright and 50 per cent faint, 90 per cent bright and 10 per cent faint) compared to the prediction from linear theory. The shaded region covers the uncertainty of the theoretical prediction due to errors in the bias measurements. The grey vertical dotted line marks the bin with $d = 27$ Mpc $h^{-1}$. The error bars are derived from the jackknife estimate described in Section 5.2.</figcaption></figure>

This paper has been typeset from a TeX/LATEX file prepared by the author.
<table><caption>**Table C1.** Values of the evolution bias for the flux-limited catalogue, estimated with the method described in the text.</caption><thead><tr><th rowspan="2">z̄</th><th colspan="2">50 per cent faint, 50 per cent bright</th><th colspan="2">10 per cent faint, 90 per cent bright</th></tr><tr><th>Bright</th><th>Faint</th><th>Bright</th><th>Faint</th></tr></thead><tbody><tr><td>0.25</td><td>7</td><td>2</td><td>7</td><td>-17</td></tr><tr><td>0.35</td><td>4</td><td>3</td><td>4</td><td>-2</td></tr></tbody></table>

see that the flux limit affects the redshift distribution of objects with redshift above $z_t$. Therefore, we estimate the evolution bias of the faint population,

$$f_{\text{F}}^{\text{evo}}(z) = \begin{cases} -\left. \frac{\partial \ln \bar{N}_{\text{F}}}{\partial \ln(1+z)} \right|_{\text{sel}} + \frac{\dot{H}_t}{H^2} + \frac{2}{rH} - 1, & \text{for } z \le z_t \\ -\left. \frac{\partial \ln \bar{N}_{\text{F}}}{\partial \ln(1+z)} \right|_{\text{F}_c} + \frac{\dot{H}_t}{H^2} + \frac{2-5s_{\text{lim}}}{rH} - 5s_{\text{lim}} - 1, & \text{for } z > z_t \end{cases}$$

where we define $s_{\text{lim}} \equiv s(z, F_{\text{lim}}) \frac{\dot{N}(z)}{N_F(z)}$. From these redshift-dependent functions $f_{\text{B/F}}^{\text{evo}}(z)$, we compute an effective value for the evolution bias,

$$f_{\text{B/F}}^{\text{evo}} = \frac{\int (dN/dz) f_{\text{B/F}}^{\text{evo}}(z) dz}{\int (dN/dz) dz}, \quad (C2)$$

where $dN/dz$ is the smoothed redshift distribution of the objects inside the bin.

In Table C1, we report the values of the evolution bias in the bins $z = 0.25, 0.35$, where the dipole signal was detected with high statistical significance. We find large differences in evolution bias at $z = 0.25$, for the 10–90 per cent split.

In Fig. C2, we compare the baseline theory predictions at $z = 0.25$ (dashed lines) to a model that accounts for the values of evolution bias reported in Table C1 (continuous lines). Including the evolution bias provides a better model for the measurements of the dipole on large scales. Therefore, the small discrepancy between theory and measurements discussed in Section 5.4 for the 10–90 per cent case is most likely due to our choice of neglecting evolution bias in our modelling. Note that there remains a small disagreement at small scales, below 40 Mpc $h^{-1}$, even after correcting for the evolution bias. Since this disagreement is not there for the 50–50 per cent case, it is probably not due to non-linearities, but rather to some systematic effect that could affect the 90–10 per cent catalogue with flux cut. In particular, this is the catalogue with the smallest number of faint objects.

## APPENDIX D: DIPOLE MEASUREMENTS AT $z = 0.45$

In Figs D1 and D2, we present the measurements of the relativistic cross-correlation dipole for $z = 0.45$, respectively for the case with no flux limit and for the case where a flux limit is imposed. A theory prediction for the signal is not given for the case where the catalogue is split into 90 per cent bright galaxies and 10 per cent faint galaxies, because the measurement of the bias is not reliable due to the small number of objects.

<figure><img src="image_13.png" /><figcaption>**Figure D1.** The measured 2PCF dipole of the $z = 0.45$ catalogue without flux limit for the two different splitting ratios (50 per cent bright and 50 per cent faint, 90 per cent bright and 10 per cent faint) compared to the prediction from linear theory. The shaded region covers the uncertainty of the theoretical prediction due to errors in the bias measurements. The grey vertical dotted line marks the bin with $d = 27$ Mpc $h^{-1}$. The error bars are derived from the jackknife estimate described in Section 5.2.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure D2.** The measured 2PCF dipole of the $z = 0.45$ catalogue with flux limit for the two different splitting ratios (50 per cent bright and 50 per cent faint, 90 per cent bright and 10 per cent faint) compared to the prediction from linear theory. The shaded region covers the uncertainty of the theoretical prediction due to errors in the bias measurements. The grey vertical dotted line marks the bin with $d = 27$ Mpc $h^{-1}$. The error bars are derived from the jackknife estimate described in Section 5.2.</figcaption></figure>

This paper has been typeset from a TeX/LATEX file prepared by the author.