The Impact of Spectroscopic Redshift Errors on Cosmological Measurements

TL;DR

Spectroscopic redshift errors can bias cosmological measurements from galaxy surveys, but modeling and correction strategies can mitigate these biases, especially for space-based slitless surveys like Euclid.

Contribution

This work quantifies the impact of redshift errors on cosmological parameters and proposes correction methods for future galaxy surveys.

Findings

Redshift uncertainty causes scale-dependent damping in the power spectrum.

Catastrophic failures suppress the power spectrum amplitude proportionally to their rate.

Correcting for catastrophic failure rate reduces bias but weakens parameter constraints.

Abstract

Spectroscopic redshift errors, including redshift uncertainty and catastrophic failures, can bias cosmological measurements from galaxy redshift surveys at sub-percent level. In this work, we investigate their impact on the full-shape analysis using contaminated mock catalogs. We find that redshift uncertainty introduces a scale-dependent damping effect on the power spectrum, which is absorbed by counterterms in clustering model, keeping parameter biases below $5\%$. Catastrophic failures suppress the power spectrum amplitude by an approximately constant factor that scales with the catastrophic rate $f_c$. While this effect is negligible for DESI galaxy populations ($f_c=1\%$), the slitless-like errors, combining redshift uncertainty with $f_c=5\%$ catastrophics, introduce significant biases in cosmological constraints. In this case, we observe $6\%$ to $16\%$ shifts ($\sim2.2\sigma$ level) in estimating the fractional growth rate $df\equiv f/f^{\rm{fid}}$ and the log primordial amplitude $\ln(10^{10} A_{s})$. Applying the correction factor $(1-f_c)^2$ on the galaxy power spectrum mitigates the bias but weakens the parameter constraints due to new degeneracies. Alternatively, fixing $f_c$ to its expected value restores the constraining power with a modest bias of $1.0\sigma$. Our results indicate that for space-based slitless surveys such as \textit{Euclid}, at minimum accurate estimation of $f_c$ and its incorporation into the clustering model are essential to get unbiased cosmological inference. Extending to evolving dark energy and massive neutrino cosmologies, redshift errors do not bias the dark energy properties parametrized by $w_0$ and $w_a$, but can degrade constraints on the summed neutrino mass $\sum m_\nu$ by up to 80% in the worst case.

Tables

Table 1. Table 1 : Summary of contaminations and mock catalogs used in our study. Mocks include uncontaminated (clean) and those affected by redshift errors. We consider redshift uncertainties for LRG-like Gaussian (dv-LRG), QSO-like Gaussian (dv-QSO-G) and QSO-like Lorentzian (dv-QSO-L) profiles. Redshift catastrophic failures are based on analyses of DESI DR1 ELG repeat observations  [ yu_elg_2025 ] , with f c f_{c} being the contamination rate. We consider f c = 1 % f_{c}=1\% with ELG-like catastrophic pattern (dv-ELGcatas) in DESI survey. For slitless-like errors (dv-slitless), we assume combined errors with LRG-like uncertainty and f c = 5 % f_{c}=5\% ELG-like catastrophics to mimic the expected observing conditions in slitless survey like Euclid . The corresponding redshift error distributions applied in contaminated mocks are shown in Figure ˜ 1 .

Mocks label	clean	dv-LRG	dv-QSO-G	dv-QSO-L	dv-ELGcatas	dv-slitless
${\sigma }_{\Delta v}$/$w_{\Delta v}$ ($\mathrm{km}\cdot s^{-1}$)	-	85.7	300	100	-	85.7
$f_c$	-	-	-	-	1%	5%

Equations

$$\mathcal{N}(\Delta v_{\mathrm{smear}}|\mu ,{\sigma }_{\Delta v})=\frac{1}{\sqrt{2\pi {\sigma }_{\Delta v}}}\exp \left(-\frac{(\Delta v_{\mathrm{smear}}-\mu {)}^2}{2{\sigma }_{\Delta v}}\right)$$

$$\mathcal{L}(\Delta v_{\mathrm{smear}}|p,{\mathrm{w}}_{\Delta v})=\frac{1}{1+((\Delta v_{\mathrm{smear}}-p)/{\mathrm{w}}_{\Delta v}{)}^2}$$

$$\mathcal{L}\mathcal{N}(\Delta {\hat{v}}_{\mathrm{c}}|{\mu }_{\text{ran}},{\sigma }_{\text{ran}}^2,{\hat{v}}_0)=\frac{1}{\sqrt{2\pi }{\sigma }_{\text{ran}}(-\Delta {\hat{v}}_{\mathrm{c}}+{\hat{v}}_0)}\exp \left\{-\frac{{\left[\ln (-\Delta {\hat{v}}_{\mathrm{c}}+{\hat{v}}_0)-{\mu }_{\text{ran}}\right]}^2}{2{\sigma }_{\text{ran}}^2}\right\}$$

$$F(\text{r})=\frac{w(\text{r})\left[n_g(\text{r})-\alpha n_s(\text{r})\right]}{A_g^{1/2}}$$

$$P_{\ell }(k)=(2\ell +1)\int \frac{d{\Omega }_k}{4\pi }\left[\int d{\text{r}}_1\int d{\text{r}}_{2}F({\text{r}}_1)F({\text{r}}_2)\times e^{i\text{k}\cdot ({\text{r}}_1-{\text{r}}_2)}{\mathcal{L}}_{\ell }(\hat{\text{k}}\cdot {\hat{\text{r}}}_h)-{\mathrm{SN}}_{\mathrm{\ell }}\right]$$

$$P_{g,s}^{\text{EFT}}(k,\mu )=P_{g,s}^{\mathrm{PT}}(k,\mu )+({\alpha }_0+{\alpha }_2{\mu }^2+{\alpha }_4{\mu }^4+...)k^2{P}_{g,s}^{\mathrm{L}}(k)+\left({\mathrm{SN}}_0+{\mathrm{SN}}_2{\mathrm{k}}^2{\mu }^2+...\right),$$

$$P_{g,s}^{\mathrm{smear}}(k,\mu )=P_{g,s}(k,\mu )D(k,\mu )$$

$$D(k,\mu )=\{\begin{array}{ll}\exp \left[-(k\mu {\sigma }_{\mathrm{zerr}}{)}^2\right],& \text{Gaussian damping},\\ 1/\left[1+(k\mu {\mathrm{w}}_{\mathrm{zerr}}{)}^2\right],& \text{Lorentzian damping}.\end{array}$$

$$D(k,\mu )\to 1+a_2(k\mu {)}^2+a_4(k\mu {)}^4+\dots$$

$$P_{g,s}^{\text{EFT}+f_c}(k,\mu )\approx (1-f_c{)}^2{P}_{g,s}^{\text{EFT}}(k,\mu )$$

$$\left|\frac{R_{\mathrm{corr}}(k)-R_{\mathrm{mock}}(k)}{R_{\mathrm{mock}}(k)}\right|\lesssim 6\%,\mathrm{For}0.02\le \mathrm{k}\le 0.3\mathrm{h}{\mathrm{Mpc}}^{-1}$$

$${\chi }^2(\Theta )=({\mathcal{D}}_{\mathrm{data}}-{\mathcal{D}}_{\mathrm{model}}(\Theta ){)}^T{\mathcal{C}}^{-1}({\mathcal{D}}_{\mathrm{data}}-{\mathcal{D}}_{\mathrm{model}}(\Theta ))$$

$${\mathcal{C}}_{ij}=\frac{1}{N_d-1}\sum _{n=1}^{N_d}\left({\mathcal{D}}_i^{(n)}-{\overline{\mathcal{D}}}_i\right)\left({\mathcal{D}}_j^{(n)}-{\overline{\mathcal{D}}}_j\right),$$

Full text

1 Introduction

The large-scale structure (LSS) observed by galaxy redshift surveys has played an important role in advancing our understanding of fundamental physics. Clustering features, such as baryon acoustic oscillations (BAO; [eisenstein_baryonic_1998, eisenstein_detection_2005, weinberg_observational_2013, alam_clustering_2017, desi_collaboration_desi_2025-4]), and the full-shape power spectrum [ivanov_cosmological_2020, damico_cosmological_2020, desi_collaboration_desi_2025-1, desi_collaboration_desi_2025-3], have been used to constrain some key quantities in cosmology, including dark energy equation of state $w$, the matter density $\Omega_{\mathrm{m}}$, the structure growth rate $f\sigma_{8}$, neutrino mass $\sum m_{\nu}$ and possible deviations from general relativity. Current leading experiments, the Dark Energy Spectroscopic Instrument111DESI, desi.lbl.gov (DESI; [desi_collaboration_desi_2016, desi_collaboration_desi_2016-1]), and the ongoing space mission Euclid222Euclid, cosmos.esa.int/web/euclid [laureijs_euclid_2011, euclid_collaboration_euclid_2025-2] are mapping the three-dimensional Universe by measuring redshifts for tens to hundreds of millions galaxies. These surveys will cover over more than 15,000 $\rm deg^{2}$ of the sky, and trace the LSS back to redshift $\sim 3.5$. This large probing volume allows the clustering signal to be measured with minimal cosmic variance, leading to sub-percent precision in cosmological measurements. Given this level of precision, detailed studies on systematic errors are critical to ensure the robustness and reliability of cosmological constraints.

Among various observational systematics, we focus on the impact of spectroscopic redshift errors. They persist in data with secure redshifts, where redshift calibration and algorithmic processing have already been applied [blake_wigglez_2010, bolton_spectral_2012, guy_spectroscopic_2023, euclid_collaboration_euclid_2025-1]. There are two primary types of errors: redshift uncertainty and redshift catastrophic failures. Redshift uncertainty refers to minor shifts ($\Delta z\sim 10^{-4}$) in all redshift measurements, primarily caused by instrumental noise and the intrinsic width of spectral features. In contrast, redshift catastrophic failures result from the misidentification of spectral lines, leading to severely incorrect redshifts ($\Delta z\sim 10^{-2}$) in a small fraction of data. These systematic errors can distort the observed clustering signal and potentially bias the cosmological inference. While previous studies have examined their individual impacts [blake_wigglez_2011, zarrouk_clustering_2018, hou_completed_2020, yu_elg_2025, krolewski_impact_2025, euclid_collaboration_euclid_2025-5], a comprehensive analysis that considers all these effects remains unexplored. This work aims to bridge that gap and explore their combined influence on cosmological inference from full-shape clustering analysis.

The spectroscopic redshift errors can be quantified using repeated redshift observations and incorporated into realistic galaxy mocks (e.g., [smith_completed_2020, yu_model_2022, yu_desi_2023, yu_elg_2025]). By comparing analyses on clean mocks with those contaminated by redshift errors, we can assess their impact on clustering statistics and cosmological constraints. Improved facilities and increased statistics in DESI [desi_collaboration_data_2025, yu_elg_2025] provide a better way to model the redshift errors and study their impact on cosmological analyses. Additionally, space-based surveys such as Euclid use slitless spectroscopy at near-infrared to measure redshifts [euclid_collaboration_euclid_2025-4, euclid_collaboration_euclid_2025-3]. This method requires no pre-selection of targets and is particularly effective for probing high-redshift galaxies. However, it is also more susceptible to redshift errors due to the lower spectrum resolution and the potential overlap of different spectra. These challenges have been addressed in the context of redshift interlopers for Euclid [euclid_collaboration_euclid_2025-5]. In this work, we use hypothetical combined redshift errors to represent the conditions expected in slitless surveys. This approach provides a description of the related systematics and offers practical guidance for interpreting and mitigating their impact in cosmological analyzes for spectroscopic surveys.

The outline of the paper is as follows: section˜2 introduces two galaxy redshift surveys, DESI and Euclid, and the spectroscopic redshift errors. section˜3 describes how we construct our contaminated mock catalogs. section˜4 presents the methodologies used to derive cosmological inferences from the mock data. In section˜5, we present the impact of redshift errors on clustering statistics and how these propagate into the cosmological parameter constraints, including extensions $w_{0}w_{a}\rm{CDM}$ and massive neutrino. Finally, we summarize our findings and conclude in section˜6.

2 Spectroscopic Survey and Redshift Errors

Spectroscopic surveys measure the redshifts of objects with high precision by analyzing their spectral features, such as emission and absorption lines. Two massive ongoing spectroscopic surveys, DESI and Euclid, are measuring galaxy clustering with unprecedented accuracy. In its first three years, the ground-based survey DESI has observed over 14 million galaxies [desi_collaboration_desi_2025-4] and plans to extend its initial plan to 60M galaxies across $18{,}000\,\rm{deg}^{2}$. Its dark matter tracers include Bright Galaxies (BGS) at $0.1<z<0.4$ [hahn_desi_2023], Luminous Red Galaxies (LRG) at $0.4<z<1.1$ [zhou_target_2023], Emission Line Galaxies (ELG, mainly [O ii] emitters) at $0.6<z<1.6$ [raichoor_target_2023], and Quasi-Stellar Objects (QSO) at $0.8<z<3.5$ [chaussidon_target_2023]. In parallel, the Euclid mission is conducting slitless spectroscopic survey from space and aim to observe tens of millions of galaxies over $15{,}000\,\rm{deg}^{2}$. Its primary targets are [$\rm{H\,\alpha}$] emitters in the redshift range $0.9<z<1.8$ [euclid_collaboration_euclid_2025-2]. The Euclid Quick Data Release 1 (QR1) is already publicly available [euclid_collaboration_euclid_2025]. The large cosmic volumes probed by DESI and Euclid enable sub-percent precision in cosmological measurements and require a rigorous understanding of systematic errors, particularly those related to redshift measurements.

In this work, the spectroscopic systematic redshift errors are referred to as redshift errors. The redshift errors exist in both ground-based and space-based galaxy redshift surveys. These include redshift uncertainty, the small statistical fluctuations in redshift measurements, and redshift catastrophic failures, the large errors from the true values. Both types of error exists in secure redshift measurements, i.e., those that have been calibrated and successfully passed the redshift determination pipeline. Consequently, these hidden errors can distort the observed clustering pattern and potentially bias our cosmological analyses. Since both redshift uncertainty and catastrophic failures can be quantified from repeated redshift observations, e.g. $dv_{ij}=c(z_{i}-z_{j})/(1+z_{i})$, it is practical to analyze them together. This combined treatment is also relevant for space missions such as Euclid, which apply slitless spectroscopy to measure redshifts and are more susceptible to both types of contamination. In the following, we will describe how redshift uncertainty and redshift catastrophic failures are characterized and modeled, and what are the redshift errors expected in slitless spectroscopic surveys.

2.1 Redshift uncertainty

Redshift uncertainty is an inevitable systematic error in spectroscopic redshift measurements. It arises because the spectral lines used to determine redshifts have specific widths rather than being ideal delta functions. These widths can result from a combination of physical (e.g., galaxy thermal motions, turbulence, gravitational broadening) and observational (e.g, instrumental resolution, observing condition) effects. The values of redshift uncertainty are reported for different types of targets within different redshift surveys. For example, in eBOSS galaxy samples, the redshift accuracies, characterized by the root mean square (RMS) of $dv_{ij}$, are about 65.6, 20 and 300$\,\rm{km}\cdot s^{-1}$ for LRG, ELG and QSO [ross_completed_2020, raichoor_completed_2020, lyke_sloan_2020]. From DESI Visual Inspection, the redshift precisions, defined as the median absolute deviation (MAD) values of $dv_{ij}$, reaches about 10, 10, 40 and 100$\,\rm{km}\cdot s^{-1}$ for BGS, ELG, LRG and QSO respectively [lan_desi_2023, alexander_desi_2023].

The effect of redshift uncertainty is equivalent to randomly smearing the peculiar velocity of an observed object along the line-of-sight. This smearing velocity, $\Delta v_{\rm{smear}}$, defined as the same as $dv_{ij}$, is empirically quantified using redshift difference. The distribution of observed $\Delta v_{\rm{smear}}$ can be statistically modeled. For galaxies, this distribution is typically Gaussian $\mathcal{N}(\mu,\,\sigma_{\Delta v})$, with $\sigma_{\Delta v}$ characterizing the redshift uncertainty dispersion [ross_completed_2020, yu_model_2022, yu_desi_2023]:

[TABLE]

For QSO, the distribution of $\Delta v_{\rm{smear}}$ usually exhibits extended wide tails. In such case, a multiple-Gaussian model [lyke_sloan_2020, chaussidon_target_2023] or a Lorentzian distribution [yu_desi_2023] provide a better fit to repeat observations. In our work, we use the Lorentzian profile, $\mathcal{L}(\,p,\,\mathrm{w}_{\Delta v})$, to describe the QSO redshift uncertainty with extended tail:

[TABLE]

Here, $\mathrm{w}_{\Delta v}$ captures the width of the distribution, thus characterizing the redshift uncertainty in the Lorentzian profile. To assess how the presence of long-tailed uncertainty distributions may impact clustering and cosmological measurements, we adopt both Gaussian and Lorentzian profiles to model QSO redshift uncertainty. A comparison between these profiles is shown in the left panel of Figure˜1.

2.2 Redshift catastrophic failures

Redshift catastrophic failures (or redshift catastrophics) arise in a small fraction of redshift measurements, typically identified through significant redshift differences in repeated observations. These errors can result from misidentification of sky residuals or confusion between spectral lines of different galaxies [blake_wigglez_2010, yu_elg_2025, euclid_collaboration_euclid_2025-1]. Faint or ambiguous spectral features can also lead to catastrophically incorrect redshifts. The catastrophic contamination rate, $f_{c}$, and the threshold for identifying it vary in different surveys. For example, the WiggleZ survey reports 5% $f_{c}$ of redshift blunders with $|\Delta z|>400\,\rm{km}\cdot s^{-1}$ (larger than $4\sigma_{\Delta v}$) [blake_wigglez_2010]. In eBOSS, the reported $f_{c}$ is less than 1% for LRGs and ELGs ($|\Delta z|>1000\,\rm{km}\cdot s^{-1}$) and 1.6% for quasars ($|\Delta z|>3000\,\rm{km}\cdot s^{-1}$) [raichoor_completed_2020, ross_completed_2020, lyke_sloan_2020]. For DESI DR1, using the same $|\Delta z|$ thresholds as in eBOSS, the $f_{c}$ is estimated to be less than 1% for all tracers at $z<2.1$ [desi_collaboration_data_2025].

In the study of DESI DR1 repeated ELG observations, redshift catastrophics are classified into two main categories: sky confusion at $z=1.32$ and random catastrophics [yu_elg_2025]. Sky confusion, responsible for 24% of these errors, arises from misidentifying residuals of sky emissions at around $8625\,\text{\AA }$ and can be approximated as a Gaussian distribution centered at the corresponding sky line positions. The remaining 76% are random catastrophic failures that encompass various error patterns. These can be modeled using a log-normal profile defined as follows:

[TABLE]

This function models the absolute log-normal distribution of redshift catastrophics ($\Delta\hat{v}_{\rm{c}}=\log_{10}(|dv_{ij}|)$) with parameters $\mu_{\text{ran}}$, $\sigma_{\text{ran}}^{2}$ and $\hat{v}_{0}$. In our study, we focus on the random pattern while varying the contamination rate $f_{c}$ to generalize catastrophic patterns for different surveys. We note that including sky confusion would not affect significantly the results [yu_elg_2025]. The distributions of catastrophics for $f_{c}=1\%$ and $5\%$ are shown in the right panel of Figure˜1, with the magnitude of $\Delta v_{\rm{c}}$ varying in $10^{3}$–$10^{6}\,\rm{km}\cdot s^{-1}$.

2.3 Combined errors for Slitless redshift surveys

Space missions such as Euclid [euclid_collaboration_euclid_2025-2], Roman [spergel_wide-field_2015] and CSST (scheduled for launch in 2027; [csst_collaboration_introduction_2025]) employ slitless spectroscopy to map the 3D distribution of galaxies over wide areas of sky. Early dataset from Euclid QR1 is now publicly available [euclid_collaboration_euclid_2025] and provides initial evaluations of the performance of space spectroscopic redshift measurement. Unlike traditional ground slit-based spectroscopy that use fibers to capture individual targets, slitless spectroscopy disperses the light from all sources in the field simultaneously, enabling efficient and continuous observations without the need for target pre-selection. This approach allows for high sky coverage and dense sampling of galaxies in the redshift range relevant to LSS studies.

However, the slitless design magnifies the redshift errors. Euclid spectroscopic redshifts are derived from spectra provided by the Near Infrared Spectro-Photometer (NISP; [euclid_collaboration_euclid_2025-4]), which operates with a resolution of $R\simeq 500$ [euclid_collaboration_euclid_2025-3]. In comparison, the ground-based DESI survey uses instruments with resolution $R\sim 2000$ to $5500$ [desi_collaboration_desi_2016]. The lower spectral resolution and slitless spectroscopy pipeline lead to broader spectral lines and consequently increased redshift uncertainty. Additionally, the lack of spatial isolation between sources causes catastrophic redshift errors [euclid_collaboration_euclid_2025-1, euclid_collaboration_euclid_2025-3], typically through cross-contamination and self-contamination. Cross-contamination arises from overlapping spectrograms of neighboring sources, while self-contamination results from the degeneracy between spatial and spectral information along the dispersion direction. These combined effects lead to a higher rate of redshift catastrophic failures in the sample [euclid_collaboration_euclid_2025-5].

To address these challenges, we adopt a hypothetical redshift error that combines both redshift uncertainty and catastrophics to simulate the expected observational systematics in slitless surveys. This model assumes a larger uncertainty than those typical of ELG (i.e., LRG-like, $\sigma_{\Delta v}=85.7\,\rm{km}\cdot s^{-1}$) and a higher rate of catastrophic failures ($f_{c}=5\%$ with $|\Delta v|>1000\,\rm{km}\cdot s^{-1}$). For reference, [euclid_collaboration_euclid_2025-1] reports that the scatter of redshift difference between Euclid and DESI is $\sigma_{dz}=0.8\times 10^{-3}\approx 240\,\rm{km/s}$, and the overall success rate indicates that a catastrophic rate $f_{c}\gtrsim 10\%$ is expected. Therefore, our hypothetical scenario remains relatively optimistic compared to the current expectations from slitless survey.

3 Contaminated Mocks

We use the Quijote $N$-body simulation suite to construct our mock catalogs [villaescusa-navarro_quijote_2020]. The Quijote fiducial simulations assume a flat $\Lambda\rm{CDM}$ universe, with cosmology consistent with the Planck 2018 result [planck_collaboration_planck_2020]: $h=0.6711,\,\Omega_{\mathrm{m}}=0.3175,\,\Omega_{\mathrm{b}}=0.049,\,\sigma_{8}=0.834,\,n_{s}=0.9624,\,\sum m_{\nu}=0.0\,{\rm{eV}},\,w=-1$. To study the impact of redshift errors on neutrino mass measurement, we also include the ‘Mnu_p’ catalog with $\sum m_{\nu}=0.1\,\mathrm{eV}$. In Quijote simulation, each realization contains $512^{3}$ CDM particles with mass $m_{p}\approx 6.5\times 10^{11}\,M_{\odot}$, evolved within a comoving box of volume $1\,h^{-3}\rm{Gpc}^{3}$. To match the redshift range of interest, we use the snapshots at redshift $z=1.0$. A mass selection with $13.1<\log_{10}(M_{h}/(h^{-1}M_{\odot}))<13.5$ is applied to ensure that the truncated halos represent the dominant structures contributing to the galaxy clustering. We use 500 realizations from the Quijote  suite to construct mock catalogs contaminated by different types of redshift errors.

The halo catalogs include information such as the Cartesian positions $X,Y,Z$, the corresponding peculiar velocities $\bm{v}_{\rm{pec}}$, and the halo masses, $M_{h}$. Assuming the line-of-sight is along the $x$-direction, we use the $x$-component of the peculiar velocity, $v_{{\rm{pec}},x}$, to transform halo positions from real space to redshift space at redshift $z$: $X_{\rm{rsd}}=X_{\rm{real}}+v_{{\rm{pec}},x}\,(1+z)/H(z)$. The clean mock catalogs are built using this peculiar velocity without modification. To build contaminated mocks with redshift errors, we add an additional velocity shift $\Delta v_{\rm{error}}$ to $v_{{\rm{pec}},x}$. This $\Delta v_{\rm{error}}$ simulates the line-of-sight velocity dispersion introduced by redshift errors.

To generate the $\Delta v_{\rm{error}}$ distribution, we use different profiles to simulate redshift errors for different surveys. For redshift uncertainty, we use the magnitudes reported from eBOSS and DESI Early Data Release (EDR) [zarrouk_clustering_2018, yu_desi_2023]. Specifically, we apply Gaussian profiles in Eq.˜2.1 with $\sigma_{\Delta v}=85.7\,\rm{km}\cdot s^{-1}$ to simulate LRG-like smearing and $\sigma_{\Delta v}=300\,\rm{km}\cdot s^{-1}$ for QSO-like smearing. We also consider a Lorentzian profile in Eq.˜2.2 to mimic the extended tails observed in QSO redshift uncertainty. The Lorentzian profile is parametrized with $\mathrm{w}_{\Delta v}=100\,\rm{km}\cdot s^{-1}$ and truncated by $\Delta v_{\rm{smear}}<3000\,\rm{km}\cdot s^{-1}$. Using these values, we have almost the same RMS of $\Delta v_{\rm{smear}}$ for the QSO Gaussian and Lorentzian smearing. The redshift smearing effects for ELG are small ($\sigma_{\Delta v}\sim 10\,\rm{km}\cdot s^{-1}$; [yu_elg_2025]) and we neglect them in this study. A summary of the configurations for the contaminated mocks is in Table˜1, with the comparison of distributions shown in the left panel of Figure˜1. In all cases, we assume the redshift errors are symmetric to zero (i.e., $\mu=0$ and $p=0$).

For redshift catastrophics, we adopt a model based on DESI DR1 ELG repeat observations [yu_elg_2025]. The velocity shifts caused by catastrophics follow a symmetric distribution centered at zero, with each half modeled by the profile $\mathcal{LN}(\Delta\hat{v}_{c}\,|\,0.64,0.25^{2},6.62)$ defined in Eq.˜2.3. We set $f_{c}=1\%$ to represent ELG-like catastrophics, corresponding to the contamination rate in eBOSS and DESI galaxies. As discussed in section˜2.3, to mimic redshift errors expected in slitless surveys, we consider a scenario combining LRG-like Gaussian smearing ($\sigma_{\Delta v}=85.7\,\rm{km}\cdot s^{-1}$) with $f_{c}=5\%$ ELG-like catastrophics. The contaminated mocks are summarized in Table˜1 and the distributions of ELG-like catastrophics and sliltless-like errors are shown in the right panel of Figure˜1.

4 Methods

4.1 Power spectrum estimator

We estimate the power spectrum of mock catalogs using the weighted galaxy density field with a convenient normalization [feldman_power_1994]:

[TABLE]

where $w(\textbf{r})$ is the weight function. $n_{g}$ and $n_{s}$ are the observed number densities of galaxy and random catalogs, respectively. The factor $\alpha=N_{g}/N_{s}$ normalizes the contribution from the random, and the normalization factor $A$ is defined with the expected mean space density of galaxies $\bar{n}(\textbf{r})$ as $A_{g}=\int d\textbf{r}\left[\bar{n}(\textbf{r})w(\textbf{r})\right]^{2}$. The estimator of power spectrum multipoles $P_{\ell}$ is given by [feldman_power_1994, hand_optimal_2017]:

[TABLE]

Here, $\mathcal{L}_{\ell}$ is the Legendre polynomial and $\textbf{r}_{h}=(\textbf{r}_{1}+\textbf{r}_{2})/2$. The shot noise term $\rm{SN_{\ell}}$ is expressed as ${\rm{SN}}_{\ell}(\textbf{k})=(1+\alpha)\int d\textbf{r}\,\bar{n}(\textbf{r})w^{2}(\textbf{r})\mathcal{L}_{\ell}(\hat{\textbf{k}}\cdot\hat{\textbf{r}})$, which only contributes to monopole $(\ell=0)$ and is negligible relative to $P_{\ell}$ for $\ell>0$. In practice, we use the pypower111https://github.com/cosmodesi/pypower [hand_optimal_2017] to compute the power spectrum multipoles from our mock catalogs.

4.2 Galaxy clustering models

4.2.1 Effective-field theory of LSS

Effective field theory (EFT) has become a powerful tool to model the LSS of the Universe. By studying small-scale nonlinear physics, the EFT of LSS extends the standard perturbation theory to scales beyond the breakdown of the fluid and single-stream approximations [baumann_cosmological_2012, carrasco_effective_2012, vlah_lagrangian_2015]. With loop corrections and improved modeling of biased tracers with redshift-space distortions, EFT enables reliable predictions of the galaxy power spectrum multipoles up to $k\approx 0.25\,h\,\rm{Mpc}^{-1}$ in redshift space [mcdonald_clustering_2009, senatore_redshift_2014, senatore_bias_2015, chen_redshift-space_2021, aviles_redshift_2021]. In general, the galaxy redshift-space power spectrum in EFT framework is expressed as [desi_collaboration_desi_2025-1]:

[TABLE]

Here, $\mu$ is the cosine of the angle between the wavenumber vector k and light-of-sight. $P_{g,\,s}^{\rm{PT}}$ is the galaxy redshift-space power spectrum based on perturbation theory (Standard, Euclidian or Lagrangian…). It is derived from the linear matter power spectrum, incorporating loop corrections and galaxy bias parameters, such as $b_{1}$, $b_{2}$ and $b_{s}$. The second terms are the general EFT counterterms parameterized by $\alpha_{i}$, which models the non-perturbative physics and the nonlinear mapping between real and redshift space. The last term corresponds to the stochastic contributions characterized by the shot noise parameters $\rm{SN}_{i}$. The EFT of LSS approach has been successfully applied in recent analyses of BOSS, eBOSS and DESI, leading to direct constraints on cosmological parameters using the full-shape of power spectrum [damico_cosmological_2020, desi_collaboration_desi_2025-3]. Several EFT codes are publicly available [chen_consistent_2020, damico_limits_2021, noriega_fast_2022], and have demonstrated consistent performance in the full-shape analysis [maus_comparison_2024]. In our work, we use the FOLPS$\nu$222https://github.com/henoriega/FOLPS-nu [noriega_fast_2022] code to calculate the EFT power spectrum.

4.2.2 Full-Shape fitting approaches

We adopt two fitting strategies to perform full-shape cosmological inference: ShapeFit compression [brieden_shapefit_2021] and Full-Modeling. These approaches have been extensively tested and validated in analyses of the BOSS and DESI surveys [maus_comparison_2023, noriega_comparing_2024, maus_analysis_2024, lai_comparison_2024, desi_collaboration_desi_2025-1].

Compression analysis focuses on fitting a set of compressed physical parameters that describe the shape and the anisotropies of the galaxy power spectrum. These include the scaling parameters $\alpha_{\parallel}$, $\alpha_{\perp}$ and the growth rate $f\sigma_{8}$. The scaling parameters are often reparameterized into the isotropic dilation parameter $\alpha_{\mathrm{iso}}\equiv\alpha_{\parallel}^{1/3}\alpha_{\perp}^{2/3}$ and the Alcock-Paczynski (AP) parameter $\alpha_{\mathrm{ap}}\equiv\alpha_{\parallel}\alpha_{\perp}^{-1}$. ShapeFit further introduces two shape-related parameters, $m$ and $n$, to capture information from the early-time transfer function. In this work, we vary only one shape parameter $m$, while keeping $n$ fixed [brieden_shapefit_2021, desi_collaboration_desi_2025-1]. We report ShapeFit results relative to a reference cosmology, specifically using $dm\equiv m-1$ and $df\equiv f/f^{\rm{fid}}$.

Another fitting approach, known as Full-Modeling analysis, avoids the compression steps and directly fits cosmological models to the power spectrum signals. In this way, the parameter dependence of the linear power spectrum along and geometrical information are not kept separate. This allows for a direct comparison between theoretical predictions and observational data and thus gives direct constraints on parameters. In our baseline fitting procedure, we consider five cosmological parameters: $h$, $\omega_{\rm cdm}$, $\omega_{b}$, $\ln(10^{10}A_{s})$, $n_{s}$. These parameters are directly used in theoretical models, which facilitates the calculation and a direct comparison of cosmological biases introduced by redshift errors.

In the context of redshift error studies, ShapeFit provides a direct view of how redshift errors distort the shape of observed power spectrum, as reflected by their impact on compressed physical parameters. This makes the interpretation of redshift errors effect straightforward. The Full-Modeling approach, on the other hand, gives cosmological constraints, allowing us to quantify the biases introduced by redshift errors in final cosmological constraints.

4.3 Clustering effect of redshift errors

4.3.1 Redshift uncertainty damping

As pointed out in the eBOSS QSO analyses [smith_completed_2020, hou_completed_2020], an approach to model the impact of redshift uncertainty in power spectrum is to introduce an additional damping term $D(k,\mu)$:

[TABLE]

[TABLE]

where the redshift error parameters $\sigma_{\rm{zerr}}$ or $\mathrm{w}_{\rm{zerr}}$ quantify the scale of redshift uncertainty. For the Gaussian smearing case, $\sigma_{\rm{zerr}}$ is related to the smearing value with $\sigma_{\rm{zerr}}=\sigma_{\Delta v}*{(1+z)}/{H(z)}$, and, as shown in the left panels of Figure˜2, this smearing effects are well captured by the Gaussian damping factor up to $k\sim 0.3\,h\,\rm{Mpc}^{-1}$. In the Lorentzian smearing case, the parametrization implies an additional factor of two, i.e. $\mathrm{w}_{\rm{zerr}}=2\,\mathrm{w}_{\Delta v}*{(1+z)}/{H(z)}$. With Lorentzian damping, we see deviations compared to our mock analyses (orange lines in Figure˜2). This is possibly due to the velocity cut we applied when constructing the Lorentzian smearing profile. However, as we will demonstrate in the following, these deviations can be effectively absorbed within the more flexible EFT framework.

To compare the damping factor with EFT model, we can apply the Taylor expansion on the damping terms. For both Gaussian and Lorentzian smearing, we have:

[TABLE]

We notice that the second and third terms scale as $P(k)(k\mu)^{2}$ and $P(k)(k\mu)^{4}$, respectively. These terms have similar structures to those of EFT counterterms in Eq.˜4.3. In the EFT of LSS, the general form of such counterterms can be derived from a moment expansion of the redshift-space power spectrum [senatore_redshift_2014, aviles_redshift_2021, damico_taming_2024], as well as from comparisons with phenomenological models [ivanov_cosmological_2020]. Consistently, [simon_cosmological_2023] has shown that the leading corrections coming from redshift uncertainty are degenerate with EFT counterterms, particularly those parameterized by $\alpha_{2}$ and $\alpha_{4}$. In section˜5.2.1, we demonstrate equivalently that the redshift uncertainty effect can be absorbed by the EFT nuisance parameters. Therefore, when we do the cosmological fitting, we use the EFT-based clustering model without including the additional damping factor.

4.3.2 Redshift catastrophics correction

Redshift catastrophics have been addressed in theoretical models for redshift surveys like WiggleZ [blake_wigglez_2010] and more recently Euclid [euclid_collaboration_euclid_2025-5]. Assuming that the redshift catastrophic contamination rate is $f_{c}$, an approximate correction is to multiply the EFT galaxy power spectrum by a factor of $(1-f_{c})^{2}$:

[TABLE]

The validity of this approximation depends both on $f_{c}$ and the displacement caused by catastrophics. In Figure˜2, based on our mock analyses with contamination $f_{c}=1\%$ and $5\%$, we find that the relative difference between the correction factor, $R_{\rm{corr}}\equiv(1-f_{c})^{2}$, and the ratio from the mock statistics, $R_{\rm{mock}}\equiv P_{\text{dv},\,f_{c}}/P_{\rm{clean}}$, satisfies:

[TABLE]

This suggests that the approximation is sufficiently accurate for full-shape power spectrum analysis, especially considering that additional nuisance parameters in the EFT can absorb residual mismatches. In DESI, the $f_{c}$ is typically below 1%. Their overall impact is expected to be small [yu_elg_2025] and such correction is not included in the standard EFT model. However, in slitless surveys such as Euclid, where $f_{c}$ is significantly higher, we will show in section˜5.2.2 that incorporating this correction will be necessary to obtain unbiased cosmological constraints.

4.4 Cosmological inference and Covariance matrices

Cosmological inference employs Bayesian statistics to infer constraints on physical or cosmological parameters. Assuming the evidence from the data is normalized to one, the posterior probability is expressed as $P(\Theta|\mathcal{D})\propto\mathcal{L}(\mathcal{D}|\Theta)P(\Theta)$, with $\Theta$ being the set of input parameters, including compressed parameters $\left\{\alpha_{\mathrm{iso}},\;\alpha_{\mathrm{ap}},\;df,\;dm\right\}$ by ShapeFit or cosmological parameters $\left\{h,\;\omega_{cdm},\;\omega_{b},\;\rm{ln}(10^{10}A_{s}),\;n_{s}\right\}$ by Full-Modeling. It also contains nuisance parameters considered in the EFT model: $\left\{b_{1},\;b_{2},\;b_{s},\;\alpha_{0},\;\alpha_{2},\;\alpha_{4},\;\rm{SN}_{0},\;\rm{SN}_{2}\right\}$. Assuming that we have Gaussian-distributed data, the likelihood function can be expressed as $\mathcal{L}(\Theta)\propto e^{-\chi^{2}(\Theta)/2}$. The $\chi^{2}$ statistic quantifies the goodness of fitting and is computed based on the difference between the observed data and the model predictions, which reads as:

[TABLE]

In our study, the vector $\mathcal{D}$ includes a combination of the power spectrum monopole $P_{0}(k)$ and quadrupole $P_{2}(k)$. The subscript ’data’ and ’model’ refers to the measurements from observations (here mocks) and theoretical model predictions. For each type of redshift-error, we have constructed 500 corresponding contaminated mocks and use them to build the data vector and covariance matrix. The covariance matrix $\mathcal{C}$ is computed and normalized to the number of the datasets, $N_{d}$, with the formulation:

[TABLE]

The $\mathcal{D}_{i}^{(n)}$ denotes the $i$-th bin of the data vector from the $n$-th mock realization and $\overline{\mathcal{D}}_{i}$ is the ensemble mean. In Quijote suite, the volume of each realization is $V_{1}=1\,h^{-3}\rm{Gpc}^{3}$. To model the reduced cosmic variance expected from galaxy surveys, we use the rescaled covariance as: $\mathcal{C^{\prime}}=({V_{1}}/{V_{n}})\,\mathcal{C}$, where $V_{n}$ is the target survey volume. In our analyses, we rescale the covariance to an effective volume of $V_{25}=25\,h^{-3}\rm{Gpc}^{3}$ to match the expected survey volumes of DESI and Euclid.

5 Results

5.1 Impact of redshift errors on clustering statistics

The impact of redshift errors on the clustering statistics is presented in Figure˜3 and Figure˜4. For redshift uncertainty, it introduces a smearing effect that systematically suppresses the power spectrum amplitude, with the suppression becoming stronger at smaller scales (i.e., larger $k$). The lower panels of Figure˜3 show the residual plot in units of the clean mocks error bar $\sigma$. For an effective survey volume of $V_{25}$, the LRG-like smearing leads to minimal impacts on the clustering, with power spectrum difference $\Delta P<3\sigma$. For QSO-like redshift uncertainty, both Lorentzian and Gaussian smearing lead to $\Delta P>3\sigma$ for $k\gtrsim 0.09\,h\,\rm{Mpc}^{-1}$ in the monopole and quadrupole. Notably, the QSO Lorentzian smearing, with the same RMS value as the Gaussian case, shows a comparatively flatter suppression at large $k$.

Redshift catastrophics affect only a small fraction of objects but cause significant shifts on their positions along the line-of-sight. Section˜4.3.2 shows that this distortion leads to an overall reduction on the power spectrum, scaling approximately as $(1-f_{c})^{2}$. As illustrated in Figure˜4, the ELG-like catastrophics with $f_{c}=1\%$ leads to a 2% reduction of the clustering amplitude, with maximum deviations around $3\sigma$. However, slitless-like errors (i.e., $\sigma_{v}=85.7\,\rm{km}\cdot s^{-1}$ and $f_{c}=5\%$) introduce an overall suppression of the power spectrum amplitude by 10%-12%, resulting in a deviation $>3\sigma$ for $k\gtrsim 0.02\,h\,\rm{Mpc}^{-1}$ in the monopole and $k\gtrsim 0.12\,h\,\rm{Mpc}^{-1}$ in the quadrupole. Such $3\sigma$ deviations, if uncorrected, could bias cosmological inferences. In the following subsections, we will describe their impact when running the inference in different cosmological cases: $\Lambda\rm{CDM}$, $w_{0}w_{a}\rm{CDM}$ and massive neutrino.

5.2 Baseline Fitting

Our baseline full-shape analysis fits the power spectrum monopole and quadrupole over the range $0.02\leq k\leq 0.2\,h\,\rm{Mpc}^{-1}$, using the standard EFT model without including the damping term from redshift uncertainty or the correction factor from redshift catastrophics. We adopt two fitting strategies, ShapeFit and Full-Modeling, to extract cosmological information. The results are presented within the standard flat-$\Lambda\rm{CDM}$ model assuming a BBN prior on baryon abundance $\Omega_{\mathrm{b}}$ and wide Gaussian prior on spectral index $n_{s}$. The cosmological inference is performed using the desilike333https://github.com/cosmodesi/desilike package and parameter priors are set to the same as the those in DESI DR1 full-shape analysis [desi_collaboration_desi_2025-1]. The parameter space is sampled with the code emcee [foreman-mackey_emcee_2013]. All plots and confidence intervals are generated using the GetDist  package [lewis_getdist_2025]. Beside the baseline full-shape analysis, we also evaluate the impact of redshift errors on BAO fitting. These results are discussed in Appendix˜A.

5.2.1 Cosmological influence

In this section, we assess the impact of redshift errors on cosmological inference. These results are based on 200 independent fits to the Quijote mocks catalogs in Table˜1. For each fit, we use the mean power spectrum derived from 25 randomly selected mocks (drawn from 500), with a covariance matrix rescaled to the volume $V_{25}$. Comparisons of the best-fit parameters between analyses from clean and contaminated mocks are presented in Figure˜5 for ShapeFit and in Figure˜6 for Full-Modeling. The corresponding $\chi^{2}$ comparison, provided in Appendix˜B, confirms that all our fits achieved comparable goodness-of-fit.

In Figure˜5 and Figure˜6, the upper panels illustrate the impact of redshift uncertainty on cosmological inference. The fitting results from both ShapeFit and Full-Modelling indicate that no significant bias (<5%) in cosmological parameter estimates. As discussed in section˜4.3.1, the suppression of the power spectrum caused by redshift uncertainty is expected to be absorbed by EFT counterterms. Figure˜7 plots the posterior distributions of all nuisance parameters. These posteriors reveal negative shifts in $\alpha_{2}$ and $\alpha_{4}$, consistent with the expected absorption of the damping effect. Another parameter that shows variation is $\mathrm{SN}_{2}$, which represents the second-order stochastic term and scales as $(k\mu)^{2}$. Its positive shift in the Lorentzian smearing case could be led by the flatter suppression pattern of the Lorentzian profile at large $k$.

The lower panels of Figure˜5 and Figure˜6 assess the impact of redshift catastrophics and hypothetical slitless-like errors on cosmological inference. The ELG-like catastrophics with $f_{c}=1\%$ show negligible bias in parameter estimates (with the largest discrepancy $<6\%$), consistent with the previous findings in [yu_elg_2025]. In contrast, slitless-like errors with $f_{c}=5\%$ show noticeable biases, particularly in estimating the fractional growth rate parameter $df$ in ShapeFit and the log primordial amplitude parameter $\ln(10^{10}A_{s})$ in Full-Modeling. Both parameters are systematically underestimated by $6\%{-}16\%$ compared to clean case. We identify that this negative shift arises mainly from the $5\%$ catastrophic contamination in slitless-like errors, which significantly suppresses the power spectrum amplitude (see Figure˜3). If not adequately modeled in theory, this effect could bias the estimates of amplitude-related parameters. Nevertheless, in section˜4.3.2, an approximate correction by multiplying the power spectrum by a factor of $(1-f_{c})^{2}$ can mitigate this suppression. In the following section, we incorporate this correction into the cosmological analyses to further investigate the influence of redshift catastrophics.

5.2.2 Catastrophics-correction for slitless survey

In this test, we present Full-Modeling fitting results based on the mean power spectrum calculated from all 500 mocks, with the covariance matrix rescaled to volume $V_{25}$. By averaging over 500 mocks, we minimize the impact of cosmic variance, so any observed deviations in parameter estimates can be more confidently attributed to redshift errors. The left panel of Figure˜8 shows the posterior distributions of parameters using the baseline EFT fit for three contamination cases: clean, ELG-like catastrophics ($f_{c}=1\%$), and slitless-like errors (i.e., $\sigma_{v}=85.7\rm{km}\cdot s^{-1}$ and $f_{c}=5\%$). The corresponding mean values of $\ln(10^{10}A_{s})$ with 68% confidence intervals are summarized in Table˜2. Comparing the $\ln(10^{10}A_{s})$ constraint from clean case, we find a small shift of $0.4\sigma$ by ELG-like catastrophics and a significant one of $2.2\sigma$ by slitless-like errors. These results are consistent with our earlier findings in section˜5.2.1. The linear bias parameter $b_{1}$ is also known to be related to the overall amplitude of galaxy power spectrum. In Figure˜8, we find a shift of $2.7\sigma$ in its mean value for the slitless-like errors case using baseline EFT. We note that $b_{1}$ is highly degenerate with $f_{c}$ and $\ln(10^{10}A_{s})$, but the shift in $b_{1}$ is insufficient to compensate for the bias induced by slitless-like errors on $\ln(10^{10}A_{s})$.

To incorporate the effect of catastrophics in the clustering model, we extend the baseline EFT model by adding an additional parameter, catastrophic rate $f_{c}$, and multiplying the factor $(1-f_{c})^{2}$ to the galaxy power spectrum as described in Eq.˜4.7. When freeing $f_{c}$ parameter during sampling with a flat prior $f_{c}\in[0,1]$, we refer to the $\texttt{EFT}+{\rm{free}}\,f_{c}$ model. The posterior distributions, including the variation of $f_{c}$, are shown in the right panel of Figure˜8, with the corresponding constraint on $\ln(10^{10}A_{s})$ reported in Table˜2. The $\texttt{EFT}+{\rm{free}}\,f_{c}$ model succeed in recovering an unbiased mean value of $\ln(10^{10}A_{s})$ for slitless-like errors, with a discrepancy below $0.1\sigma$ compared to the clean case. However, this improvement comes at the cost of loss in constraining power: we measure a $\sim 67\%$ degradation in the $\ln(10^{10}A_{s})$ constraint. This loss is due to the strong degeneracy between $f_{c}$ and $\ln(10^{10}A_{s})$, as illustrated in the Figure˜8 $\ln(10^{10}A_{s}){-}f_{c}$ plane. When fixing the parameter $f_{c}$ to its expected value, referred to the $\texttt{EFT}+{\rm{fixed}}\,f_{c}$ model, we successfully recover the constraining power but find a slightly larger deviation ($\sim 1.0\sigma$) in the estimated mean value. This deviation is possibly due to the combined effects of redshift uncertainty in the slitless-like errors and the approximation applied in the correction model.

This test is informative for slitless spectroscopy survey like Euclid, which aims to constrain cosmological parameters via galaxy power spectra. As we have shown that this hypothetical redshift errors can bias cosmological inference, an accurate estimation of the catastrophics rate and the implementation of the catastrophics-corrected model, either with $f_{c}$ fixed or varied under a narrow Gaussian prior, are necessary to get unbiased constraints. Since our slitless-like errors scenario is optimistic and slitless survey may have other additional complex systematics, further investigations on modeling redshift errors are still required.

5.3 Beyond $\Lambda\rm{CDM}$ cosmologies

Recent spectroscopic surveys such as DESI have extended full-shape power spectrum analysis beyond the standard $\Lambda\rm{CDM}$ framework to test cosmological models with evolving dark energy ($w_{0}w_{a}\rm{CDM}$) and massive neutrinos. Assuming a flat Universe and combining the full-shape and BAO analyses with external datasets, the DESI DR2 finds a mild preference for a dynamical dark energy model with $w_{0}>-1$ and $w_{a}<0$  [desi_collaboration_desi_2025-4] and places a tight constraint on the total neutrino mass, $\sum m_{\nu}<0.0642\,\rm{eV}\ (95\%CL)$ [elbers_constraints_2025]. This section investigates how redshift errors affect the measurements of dark energy equation of state $w_{0}$-$w_{a}$ and $\sum m_{\nu}$ through the full-shape analysis. In the following, we use the baseline EFT fitting setup and present results obtained by fitting the mean power spectrum of 500 mocks, with the covariance matrix rescaled to the effective volume $V_{25}$.

5.3.1 Evolving dark energy

We assess the influence of redshift errors on the measurement of evolving dark energy using the Chevallier-Polarski-Linder (CPL) parametrization $w(a)=w_{0}+w_{a}(1-a)$ [chevallier_accelerating_2001, linder_exploring_2003]. The cosmological fits are performed using the Quijote fiducial mocks described in section˜3, which assume a $\Lambda\rm{CDM}$ model with $w_{0}=-1$ and $w_{a}=0$. The resulting parameter constraints are shown as the horizontal lines in Figure˜9. From the comparisons, we find that redshift errors do not introduce significant bias in the measurement of dark energy properties. However, we know that using full-shape analysis alone to constraint $w_{0}w_{a}\rm{CDM}$ model suffer from projection effects [desi_collaboration_desi_2025-1]. To study this aspect, we use iminuit [james_minuit_1975] to estimate the maximum a posteriori (MAP) value, starting from the maximum log-posterior points found in MCMC chains. In Figure˜9, we find offsets between the mean of the marginalised posteriors (solid circles) and MAP values (open circles) for parameters $h$, $w_{0}$ and $w_{a}$, with the offsets being most pronounced for mocks contaminated by QSO-like smearing. Since the redshift errors impact the clustering pattern similarly with the behavior of certain parameters (cosmological or nuisance), their presence could complicate the parameter degeneracies and thus amplify projection effects. Such impact should be investigated in more detail for redshift surveys aiming to constrain dark energy by full-shape analysis.

5.3.2 Massive neutrinos

For the massive neutrino case, we use the Quijote $Mnu\_p$ catalogs with $\sum m_{\nu}=0.1\,\rm{eV}$ to build our clean and contaminated mock catalogs. During the fitting, we allow $\sum m_{\nu}$ to vary with a flat prior of $\,\mathcal{U}(0.0,5.0)\,\rm{eV}$ in the $\Lambda\rm{CDM}$ framework. The upper limits of $\sum m_{\nu}$ at 95% confidence level for all mocks analyses are presented in Table˜3. We first note that using full-shape analysis alone to constrain the $\sum m_{\nu}$ has slow convergence and large uncertainties, particularly for the low neutrino mass case. Nevertheless, we find that redshift uncertainty introduces extended tails in the $\sum m_{\nu}$ posterior distribution, leading to a weaker constraint on the neutrino mass. Compared to the clean case ($<0.78\,\rm{eV}$), the 95% upper limits increase by up to $\sim 80\%$ when QSO Gaussian smearing is included. This degradation occurs because both massive neutrinos and redshift uncertainty have similar suppression on small-scale power spectrum. Massive neutrinos do so through their thermal velocities and free-streaming, while redshift uncertainty smears the clustering structure along the line of sight. This degeneracy complicates the convergence and leads to weaker constraints on neutrino mass. In contrast, we find no noticeable bias in the $\sum m_{\nu}$ measurements due to ELG-like catastrophic failures or slitles-like redshift errors.

6 Conclusions

In this paper, we investigate the impact of spectroscopic redshift errors on cosmological measurements. section˜2 introduces the redshift errors, which can affect redshift measurements in both ground-based (fiber-fed) and space-based (slitless) spectroscopic surveys [ross_completed_2020, yu_elg_2025, euclid_collaboration_euclid_2025-1]. There are two main types of redshift errors. Redshift uncertainty refer to small errors present in all redshift measurements, and redshift catastrophic failures correspond to large deviations from the true values in a small fraction of the measurements. Both type of errors distort the observed clustering pattern and can lead to biases in the inferred cosmological results.

The effect of redshift errors can be quantified and modeled using repeat observations. In section˜3 we use the Quijote simulation suite to construct our contaminated mock catalogs. For redshift uncertainty, we consider Gaussian or Lorentzian profiles for LRG-like and QSO-like smearing [zarrouk_clustering_2018, yu_desi_2023]. For redshift catastrophics, we use a log-normal distribution [yu_elg_2025] with contamination rate $f_{c}=1\%$ to model ELG-like catastrophics. The slitless-like errors assume a hypothetical scenario combining $f_{c}=5\%$ catastrophics and LRG-like uncertainty to simulate the observing condition in slitless survey. The following section˜4 describes the methodologies to perform the full-shape power spectrum analysis. In our baseline fitting, we use clustering models based on the effective field theory (EFT) of LSS, without including the damping term or the multiplicative factor. We implement both ShapeFit and Full-Modeling fitting approach to extract cosmological information. The analysis is targeted on DESI- and Euclid-like surveys with effective volume $V_{25}=25\,h^{-3}\rm{Gpc}^{3}$. Our key findings are summarized below:

Redshift uncertainty introduces diffusive effect in the clustering, leading to a scale-dependent suppression to the observed power spectrum. The EFT cournterterms, parameterized by $\alpha_{2}$ and $\alpha_{4}$, capture this smearing effect, resulting in compressed and cosmological parameter estimates difference $<5\%$ compared to the uncontaminated case. In contrast, redshift catastrophics cause an overall reduction on the power spectrum amplitude that approximately scales as $(1-f_{c})^{2}$. While this effect is negligible (difference $<6\%$) for ELG-like catastrophics, it significantly biases the estimation of amplitude-related parameters for slitless-like errors, namely $df$ in ShapeFit and $\ln(10^{10}A_{s})$ in the Full-Modeling. In that case, we observe shifts of $6\%{-}16\%$, corresponding to a $\sim 2.2\sigma$ deviation relative to the clean case. To model this effect, we use the correction factor $(1-f_{c})^{2}$ applied to the power spectrum in Eq.˜4.7. When freeing $f_{c}$, the extended model mitigates the bias in $\ln(10^{10}A_{s})$ to below $0.1\sigma$, but leads to $67\%$ degradation in its precision due to the strong degeneracy between $f_{c}$ and $\ln(10^{10}A_{s})$. Fixing $f_{c}$ to its expected value in the fit restores the constraining power with a modest $1.0\sigma$ deviation. This highlights the importance of accurately estimating the catastrophic failure rate $f_{c}$ and applying the appropriate correction model to ensure unbiased cosmological results in slitless spectroscopic surveys.

Beyond the vanilla $\Lambda$CDM model, we explore the impact of redshift errors on the measurement of evolving dark energy and massive neutrinos. For the $w_{0}w_{a}\rm{CDM}$ model, we find that redshift errors do not bias the estimates of $w_{0}$ and $w_{a}$, although they can potentially enhance projection effects. In the massive neutrino case, redshift errors weaken the constraint on $\sum m_{\nu}$ up to 80% when considering QSO-like Gaussian smearing.

For future work, we plan to apply our framework to the DESI DR2 analysis. The DESI DR2 dataset provides a large sample of repeat observations well-suited for characterizing redshift errors. Using this dataset, we aim to jointly model redshift uncertainty and catastrophic failures for each tracer population across a wide range of redshift. We also plan to conduct more detailed investigations of redshift errors, including lightcone effects, redshift evolution, impact on all various clustering statistics and so on.

Acknowledgments

SH, AR, JY and JPK, acknowledge support from the Swiss National Science Foundation (SNF) ‘Cosmology with 3D Maps of the Universe” research grant 200020_207379. This research was essentially supported by computational resources provided by the High Performance Computing (HPC) facility of University of Geneva (UNIGE), especially for the Bamboo cluster.

Appendix A BAO fitting results

BAO serves as a standard ruler in cosmology and is generally considered robust against a wide range of systematic effects. In this work, we also test whether BAO measurements remain unbiased under the spectroscopic redshift errors. BAO analysis focuses on constraining two dilation parameters, $\alpha_{\parallel}$ and $\alpha_{\perp}$, which are reparameterized into two scaling parameters, $\alpha_{\mathrm{iso}}$ and $\alpha_{\mathrm{ap}}$, respectively. These two parameters provide information about the isotropic BAO scale and AP effect. We adopt the BAO modeling framework implemented in DESI DR1 analysis [desi_collaboration_desi_2025-2] using the desilike package. The results are obtained by fitting the monopole and quadrupole of the two-point correlation function from the mean of 25 randomly selected mocks out of 500 realizations without BAO reconstruction. The covariance matrix is built from 500 mocks and rescaled to an effective volume of $V_{25}=25\,h^{-3}\rm{Gpc}^{3}$.

Table˜4 summarizes the mean and standard deviation for $\alpha_{\mathrm{iso}}$ and $\alpha_{\mathrm{ap}}$ best-fit values and $\chi^{2}$ obtained from 200 independent fits. The inferred $\alpha_{\mathrm{iso}}$ and $\alpha_{\mathrm{ap}}$ values show only minor differences from the clean case, both in their means and standard deviations. The best-fit $\chi^{2}$ values also show no increase after including redshift errors, indicating a same level of goodness of fit. Therefore, we conclude that BAO measurements remain unbiased and robust against spectroscopic redshift errors at the current level of precision.

Appendix B $\chi^{2}$ results for baseline analysis

Figure˜10 presents the distribution of best-fit $\chi^{2}$ values for the baseline analysis. The left two panels show results for the ShapeFit analysis, corresponding to the scatter plot in Figure˜5, while the right two panels show the Full-Modeling results for Figure˜6. From the $\chi^{2}$ distributions, we do not observe significant differences (<$15\%$) or systematic shifts between the clean and contaminated mocks analyses. This indicates that the theoretical model provides a same level of good prediction with the observational data, even potential deviations in the parameter estimates may arise under contamination.