The effect of photometric redshift uncertainties on galaxy clustering and baryonic acoustic oscillations

In the upcoming era of high-precision galaxy surveys, it becomes necessary to understand the impact of redshift uncertainties on cosmological observables. In this paper we explore the effect of sub-percent photometric redshift errors (photo-$z$ errors) on galaxy clustering and baryonic acoustic oscillations (BAO). Using analytic expressions and results from $1\,000$ $N$-body simulations, we show how photo-$z$ errors modify the amplitude of moments of the 2D power spectrum, their variances, the amplitude of BAO, and the cosmological information in them. We find that: a) photo-$z$ errors suppress the clustering on small scales, increasing the relative importance of shot noise, and thus reducing the interval of scales available for BAO analyses; b) photo-$z$ errors decrease the smearing of BAO due to non-linear redshift-space distortions (RSD) by giving less weight to line-of-sight modes; and c) photo-$z$ errors (and small-scale RSD) induce a scale dependence on the information encoded in the BAO scale, and that reduces the constraining power on the Hubble parameter. Using these findings, we propose a template that extracts unbiased cosmological information from samples with photo-$z$ errors with respect to cases without them. Finally, we provide analytic expressions to forecast the precision in measuring the BAO scale, showing that spectro-photometric surveys will measure the expansion history of the Universe with a precision competitive to that of spectroscopic surveys.


INTRODUCTION
A new generation of wide-field cosmological galaxy surveys will soon map the spatial distribution of hundreds of millions of galaxies over a wide range of redshifts. With these, it will be possible to measure the expansion history and structure growth in the Universe with exquisite precision. These measurements can then be used to put strong constraints on the contributors to the total energy density of the Universe, the law of gravity on large scales, and perhaps will offer hints to explain the accelerated expansion of the Universe (see Weinberg et al. 2013, for a review).
Some of these future galaxy surveys will employ highresolution spectrographs, which will deliver precise estimates for the redshift of galaxies (e.g. DESI, WEAVE, 4MOST). Other surveys, instead, will rely on either low-resolution spectrographs, linear variable filters, or a system of narrow-band filters (e.g. J-PAS, PAU, EUCLID, SphereX). The advantage of the later is that they allow faster mapping speeds and/or that they avoid any target pre-selection. However, this approach adds non-negligible uncertainties in the measured redshifts.
In configuration space, adding uncertainties to the redshift of the galaxies can be regarded as a smoothing operation on the galaxy field along the line-of-sight. Conversely, in Fourier space, they can be regarded as a reduction in the amplitude of line-of-sight modes. Therefore, there is an effec-tive loss of cosmological information for a survey with a finite number of mass tracers.
The impact of redshift uncertainties on the galaxy clustering and on the baryonic acoustic oscillations (BAO) has been explored by several authors (Seo & Eisenstein 2003;Glazebrook & Blake 2005;Blake & Bridle 2005;Dolney et al. 2006;Cai et al. 2009;Benítez et al. 2009;Sereno et al. 2015). These authors showed that an increment in redshift errors leads to a dilution in the BAO signal on small scales due to a decrement in the clustering signal relative to the discreteness (a.k.a shot) noise. Nevertheless, they showed that the BAO scale can still be measured, although with less precision. For instance, as shown by Cai et al. (2009), the uncertainty on the measured acoustic scale doubles for a redshift error of σz/(1 + z) ∼ 0.3 %. This has motivated surveys such as J-PAS, which aims at delivering a sub-percent redshift accuracy employing a set of 56 contiguous 150Å-wide filters.
It is clear the cosmological potential of future photometric galaxy surveys despite of the redshift errors and of the strong requirement in terms of photometric calibration and control of systematics. However, the observed clustering of galaxies will be sensitive on how the noise properties of the redshift estimator couple to the intrinsic anisotropic galaxy clustering. Hence, a detailed modelling is necessary to guarantee robust cosmological inferences and an accurate extraction of the information encoded in the BAO signal.
In this paper we develop a complete framework for the exploitation of the BAO signal under the presence of redshift errors. First we provide analytic expressions for how the monopole and the quadrupole of the redshift-space power spectrum (together with their covariances) are affected. Our model is able to capture, within few percent and for all wavelengths explored, these quantities as measured in cosmological N -body simulations.
Then, we explore in detail the signal-to-noise ratio of the BAO wiggles and its cosmological constraining power. We show how and why the BAO contrast is enhanced with small redshift uncertainties, and explore in detail how the cosmological information is modulated by the interplay of redshiftspace distortions, redshift errors, and the number density of the sample.
We employ our findings to develop a procedure that can be applied to simulated and/or observed data to extract the BAO scale in an unbiased manner. We present our results for a wide range of galaxy number densities, typical redshift errors, and error distributions. We provide a simple fitting function that captures our numerical results accurately.
Our paper is organized as follows. In §2 we describe the set of cosmological simulations that we use, the way that we compute clustering statistics from these simulations, and how we introduce redshift uncertainties. In §3 we derive analytical expressions for the impact of redshift errors on the shape and variance of the monopole and quadrupole of the power spectrum. Then, in §4, we model how redshift errors alter the BAO feature and the information that they encode. In §5 we build an unbiased model for the BAO wiggles in the power spectrum monopole and apply it to simulated samples with different redshifts errors, large-scale biases, probability density functions, and number densities. We conclude and summarize our most important results in §6.

Numerical Simulations
The first N -body calculation employed in this work is the Millennium XXL (hereafter MXXL). The MXXL simulation followed 6720 3 particles of mass mp = 8.456 × 10 9 M⊙ inside a cubical region of L = 3 h −1 Gpc on a side. The gravitational forces were computed with a lean version of the Gadget code , and the softening length was set to 10 h −1 kpc. The cosmological parameters adopted are identical to those of the Millennium Simulation . Explicitly: Ωm = 0.25, ΩΛ = 0.75, Ω b = 0.045, ns = 1, H0 = 73 km s −1 Mpc −1 , and σ8 = 0.9. We refer the reader to Angulo et al. (2012) for more details. Throughout our paper we will employ a catalogue of stellar-mass selected galaxies with a space density ofn = 10 −2 h 3 Mpc −3 at z = 1, as predicted by a semi-analytic model for galaxy formation carried out on top of the MXXL merger trees (Angulo et al. 2014).
We complement the MXXL results with a suite of 300 simulations with the same volume and cosmological parameters but lower mass resolution. This suite has an aggregated volume of 8100 h −3 Gpc 3 , which is sufficiently large for statistical studies of the BAO signal. For computational efficiency, we carried out these simulations using the Comoving Lagrangian Acceleration (COLA) method (Tassev et al. 2013). The COLA method is able to predict accurately the evolution of the matter density field on intermediate and large scales at a fraction of the computational cost of a full N -body simulation (Howlett et al. 2015;Koda et al. 2015).
Each COLA simulation evolved 1024 3 particles, each one of mass 1.7 × 10 12 h −1 M⊙, from z = 9 down to z = 1 using 10 timesteps. The Gaussian initial conditions were created using 2nd order Lagrangian Perturbation theory, and gravitational forces were computed using a Particle-Mesh algorithm with a Fourier grid of 1024 3 mesh points. Each simulation took 3 CPU hours to complete.
Together, the MXXL and the COLA suite will allow us to investigate the impact of redshift uncertainties on the shape of the power spectrum, its variance, and the information content of the BAO scale. Unless otherwise stated, we will explore the z = 1 outputs of our calculation, which is motivated by the target redshift range of future wide-field surveys.

Power spectrum & covariance measurements
Throughout this paper we consider the power spectrum, P (k), defined by: where indicates an ensemble average, δD() is the Dirac delta function, and δ(k) is the Fourier transform of the density contrast field, δ(x). Operationally, we compute the power spectrum by Fast Fourier Transforming the distribution of objects in our simulations mapped onto a grid of 1024 3 cells using the cloud-in-cell scheme. We then estimate the anisotropic power spectrum as where k ≡ |k| is the modulus of the k wave-vector and µ = k ·ẑ. The above sum runs over the N k wave-vectors ki that lie within of a set of (k, µ) bins, which we define as equally spaced in ∆k = 0.002 hMpc −1 and ∆µ = 0.01. The respective multipoles thus become where P ℓ is the Legendre polynomial of order ℓ. We perform the above integral using the trapezoid rule. Additionally, for the monopole, ℓ = 0, we apply a first order correction to reduce the contribution of shot noise P0(k) → P0(k)−n −1 , wheren is the mean number density of objects considered. Note that such correction assumes that the noise has no frequency dependence and altogether vanishes for the quadrupole, ℓ = 2, as it is does not display an angular dependence. An ensemble of M power spectrum measurements can be used to compute the corresponding covariance matrix: where P m ℓ (ki) is the i-th measurement of the power spectrum multipole at the scale ki, andP ℓ (ki) is the average.
We calculate the precision matrix, C −1 ℓ , by computing the inverse of C ℓ (ki, kj) using an algorithm based on a LU factorization. The expected value of C −1 ℓ when estimated with a finite number of measurement is biased. Following Hartlap et al. (2007), we correct for this as follows: where N bins is the number of k bins. The numerical value of the correction factor for the case of our COLA ensemble (N = 300) and the range of k which we use in §5 is 1.496.

Redshift uncertainties
In our simulations, we model redshift errors and redshift-space distortions in the flat sky approximation, i.e. we perturb the position of objects along theẑ direction, xz: where vz is the physical velocity along the z axis in km s −1 , and H(z) the Hubble parameter at redshift z. δz is a random variable whose probability density function (PDF), Pr(δz), is given by the distribution redshift errors. By default, we will assume that Pr(x) is a Gaussian distribution with zero mean and standard deviation σ = σz(1 + z) c H −1 (z), where σz(1 + z) is the typical error in the units of redshift. However, in reality redshift errors may follow non-Gaussian PDFs. For instance, the comparison of photometric and spectroscopic redshifts in the COSMOS survey showed that Pr(δz) is described by a Lorentzian variate (Ilbert et al. 2009). Additionally, Pr(δz) at low redshifts usually shows a tail towards higher redshifts, which is a natural consequence of imposing z > 0 in otherwise symmetric PDFs. Therefore, in addition to the Gaussian case, we will consider three families of functional forms for Pr(x): where σ is a parameter that controls the width of the distribution, and Γ is the Gamma function. For the second family of distributions, β controls the excess kurtosis. Distributions with β < 2 show extended wings like a Lorentzian, whereas with β > 2 are boxier than a Gaussian. For the third family, κ determines the skewness and the excess kurtosis. Note that we disregard the possibility of interlopers (galaxies systematically assigned to incorrect redshifts due to spectroscopic line misidentification).

CLUSTERING WITH REDSHIFT ERRORS
In this section we derive general analytic expressions for the impact of redshift uncertainties on the shape of the monopole and the quadrupole ( §3.1) of the power spectrum, and their variances ( §3.2). In all cases, we will contrast these predictions with results from numerical simulations.

General expressions
Let us consider a set of galaxies with a real-space density contrast field δr(k) discretely sampling a Gaussian field of covariance P (k) and whose redshifts are measured through a noisy but unbiased estimator. The observed redshifts are thus z → z + δz. Assuming that the PDF of δz, Pr(δz), is identical for every galaxy, we can write the redshift-space overdensity field within the Gaussian dispersion model (Kaiser 1987): where β ≡ b −1 d log D(a)/d log a, b is the large-scale bias of the sample, D(a) is the linear growth factor, a = 1/(1 + z) is the cosmological scale factor, σv is a velocity dispersion induced by non-linear dynamics, F (µk) is the Fourier transform of the Pr(δz), and µ =k ·ẑ is the director cosine along the z-axis. Hereafter, for brevity we will not write explicitly the dependence of F on µ and k.
The relation between P r 0 and P0 (the monopoles in real and in redshift space, respectively) is: here, and in the remainder of this paper, ... k brackets will denote angular averages (i.e. averages over µ) . Similarly, we can write the quadrupole in redshift space, P2, in terms of the monopole in real space: It is straightforward to see how redshift errors create an anisotropic clustering, P2 = 0, even if the underlying galaxy field is isotropic. In the general case, redshift errors will couple with the intrinsic redshift-space anisotropies of the galaxy field.

The Gaussian case
In the case of a Gaussian Pr(δz), F (kµ) = exp{−0.5(µkσ) 2 }, with σ = σz (1 + z) c H −1 (z), there are analytic expressions for F 2 k and µ 2 F 2 k : where Hn(x) = n i=0 2 i (2i+1)!! x 2i , !! denotes the double factorial, x = k σ eff , and σ eff = √ σ 2 z + σ 2 v (1+z)c H −1 (z). To obtain these expressions we have assumed that x > 0. 1 . The equivalent expressions in real space can be trivially obtained by setting β = 0. In that case we recover the expression provided in Cai et al. (2009) for the monopole. It is straightforward to see that redshift errors suppress P0 and P2 on all scales, with a stronger suppression for larger wavelengths. In real space, P0 is suppressed by 1.08 at kσ = 0.5, whereas redshift errors induce a negative P2 which is 6.67 times P r 0 on the same scale. In redshift space, redshift errors couple with RSD, thus their net impact depends on β. In general, RSD increase the clustering along the line-of-sight for kσ eff < 1 and suppress the clustering on smaller scales. The suppression due to redshift errors is smaller than in real space for scales where kσ eff > 1. For instance, considering a sample with β = 1, the suppression of the power spectrum is 1.77 at kσ eff = 2, whereas in real space is 2.27 at kσ = 2. However, on large scales the net effect of RSD and redshift errors is an increment in the amplitude of the monopole, e.g. it is increased by 1.66 at kσ eff = 0.5. Additionally, the smallscale clustering suppression due to redshift errors inverts the sign of the quadrupole on scales kσ eff > 1.

Comparison with numerical simulations
In the left panel of Fig. 1 we display the average redshift-space monopole of the DM particles in our COLA ensemble at z = 1. We have perturbed those particles following the procedure described in §2.3 to mimic the impact of Gaussian redshift errors. Symbols display three cases with different typical redshift uncertainties, as indicated by the figure. As expected, redshift errors suppress the clustering on small scales, while leaving large scales unaffected. Additionally, the suppression implies that the larger the redshift error, the larger the scale where the contribution of shot noise becomes important. The predictions of Eq. 12, shown by solid lines, quantitatively capture the relevant effects. The accuracy of the model, quantified in the bottom panel, is within 5 % on all scales shown. Note that we employ the measured real-space power spectrum to build this model and we use σv = 3 × 10 −4 , which is obtained by finding the value that fits the best the redshift-space monopole power spectrum without redshift errors.
In the right panel of Fig. 1 we show the redshift-space quadrupole power spectrum. Symbols display three cases with different redshift uncertainties, as indicated by the legend. The predictions of Eqs. 11 and 13, shown by solid lines when the quadrupole is positive and by dashed lines when is negative, capture the relevant effects. The accuracy of the model, quantified in the bottom panel, is the same as for the monopole.

General expressions
Redshift errors modify the shape of the power spectrum as well as its variance. The variance of the power spectrum is defined as the diagonal elements of the power spectrum covariance matrix, i.e: where ... denote ensemble averages over multiple realizations/universes. Note the factor of two appears because only half of the modes are independent owing to the reality of δ(x).
The above expression reduces to: in the Gaussian limit (i.e. assuming that δR(k) and δI (k) - the real and imaginary parts of δ(k), respectively -are Gaussian random variables with zero mean and standard deviation P (k)/2). Combining Eqs. 9 and 14 we obtain an expression for the variance of P0 under the presence of shot noise and redshift errors: We note that in real space and without redshift errors, F 2 k = F 4 k = 1, and our expression for the variance reduces to that provided by Colombi et al. (2009).
Similarly, we can compute the variance of P2: The variance of P2 differs to the variance of P0 because the terms associated to the shot noise are zero, as we stated in §2.2. Note that P2 and σ[P2] are both zero in real space without redshift errors [ (3µ 2 −1)F 2 k = (3µ 2 −1) 2 F 4 k = 0]. Redshift errors do not modify the covariance structure of the measured density contrast, so in particular if the realspace correlation matrix is diagonal, then so it is in redshift space with or without redshift errors. Hence, there is no extra coupling in Fourier space induced by redshift errors (or linear RSD).

The Gaussian case
For a Gaussian Pr(δz), F 4 k , µ 2 F 4 k , and µ 4 F 4 k have analytic expressions: where x = √ 2 kσ eff and we assume that x > 0 to derive these expression 2 . Using them, it is straightforward to construct analytically the variance of P0 and P2.
From the above expressions we can see that redshift errors also reduce the variance of the power spectrum: the values of the first three terms in brackets in the RHS of Eq. 16 are always smaller than their σ eff = 0 counterparts since F (µk) is always less than unity. On the other hand, the last term in brackets remains unchanged, thus, at a fixed scale, the shot noise contribution progressively dominates as redshift errors increase. Figure 3. Ratio of the S/N of the redshift-space power spectrum monopole, P 0 /σ[P 0 ], to that of a case with no redshift uncertainties in real space, P r 0 /σ[P r 0 ]. Symbols display this quantity computed from our COLA ensemble whereas lines do so for the analytic model of Eqs. 9 and 16. In each panel, black, red, and blue colours represent the cases where σz/(1 + z) = 0, 0.3 %, and 0.5 %, respectively. For comparison, the horizontal dashed line shows the prediction for the S/N whenn → ∞ and σv = 0, i.e. without shot noise and only large-scale RSD.

Comparison with simulations
In the left panel of Fig. 2 we display the variance of P0 for different redshift uncertainties, as shown by the legend. Symbols indicate measurements from our COLA ensemble for the same samples shown in Fig. 1, whereas solid lines show the prediction of Eq. 16. As for the monopole of the power spectrum, the agreement between our model and the numerical results is remarkable, showing a discrepancy always at or below the 5 % level.
In the right panel of Fig. 2 we show the variance of P2 using the same labelling as the left panel. Symbols show measurements from our COLA ensemble for the same samples shown in Fig. 1, whereas solid lines show the predictions of Eq. 17. In this case, the agreement between our model and the numerical results is within 10 % for k < 0.3 hMpc −1 .

Signal-to-noise ratio for the monopole
Let us now consider the signal-to-noise ratio (S/N) of the redshift-space power spectrum monopole, P0/σ[P0]. The net effect of redshift errors on the S/N is the result of a balance between the suppression induced by redshift errors in the power spectrum monopole and its variance.

Comparison with simulations
We now compare our analytical expressions (i.e. those derived in the previous two subsections) with the results from our ensemble 300 N -body simulations. Specifically, in Fig. 3 we show the S/N relative to that computed in real space without redshift errors. We display results for two number densities and three different redshift uncertainties, as indicated by the legend. In all cases we can see that our model, indicated by the lines, reproduces fairly well the numerical data, displayed by symbols.
In all three cases and in both the simulated and the analytic results, the S/N converge to the case without redshift errors and non-linear RSD (such case is indicated by the horizontal dashed line) as kσ eff → 0. Interestingly, this limiting value is ∼ 10 % lower than that in real space (indicated by the horizontal dotted line). This implies that, despite the clustering enhancement due to RSD, the S/N in redshift space is lower than in real space in the regime where shot noise is subdominant. This is a consequence of the increment in the variance arising from the average of modes with a wider range of amplitudes than in real space.
Additionally, in all cases we appreciate an increase in the S/N relative to the case without redshift errors at a scale of kσ eff ∼ 1, and a decrease at higher wave-numbers. Interestingly, for σz/(1 + z) = 0.3 %, the enhancement occurs at k ∼ 0.15hMpc −1 , the range of scales where BAO are located and thus, it contains relevant cosmological information. As BAO are heavily suppressed by non-linearities and RSD at k 0.3hMpc −1 , this could imply that stronger cosmological constraints are derived after a smoothing operation. We will return to this in the next section. Overall, these results indicate that, for scales kσ eff ∼ 1, the S/N with redshift errors is larger than without errors.

Toy model
To quantitatively understand the modifications in the S/N of the redshift-space power spectrum monopole due to redshift errors, we build the following toy model: In this expression, the terms in parenthesis provide the contribution of the angular integral at only two µ-values (µ1, µ2). The symbolP r 0 (k) denotes the measured power spectrum monopole in real space and η(µ1, 2) describes the contribution of RSD in the two µ bins. We will assume that µ1 < µ2 and thus η(µ1) < η(µ2) since on linear and quasi-linear scales η(µ) should be a monotonically increasing function of µ, and for µ = 0, η(µ) should adopt its real space value, η(µ = 0) = 1.
For an ensemble average over a given k bin we have that , and the S/N per radial k mode reads: with ∆µ 2 = µ 2 2 − µ 2 1 and η21 = η(µ2)/η(µ1). From this expression, we shall consider three different cases: • No redshift errors, kσ = 0. In this case the S/N is always below √ 2, which is the value corresponding to real space, given by η(µ) = 1: • Very large redshift errors, kσ → ∞. Now, the S/N = 1 since all information is lost along the parallel modes.
• Small redshift errors, kσ → 0. In this case we obtain, to first order in (kσ) 2 : That is, in this limit the S/N increases with respect to the case where there are no redshift errors since η21 > 1 in redshift space. This behaviour must thus yield a local maximum in the S/N since for larger kσ values we must recover the second case just considered above. This reflects the fact that in this limit the redshift errors affect more the variance than the amplitude of the power spectrum monopole, thus slightly increasing the ratio of these two quantities.

BARYONIC ACOUSTIC OSCILLATIONS WITH REDSHIFT ERRORS
We now investigate the effect of redshift errors on the BAO in Fourier space and the cosmological information they encode.

The shape of the BAO signal
Let us begin by considering the following quantity: where P sm 0 is a smoothed version of P0, i.e. a power spectrum that displays the same broadband shape but no oscillatory features. Therefore, B(k) is insensitive to the overall shape of the observed power spectrum, and isolates the BAO wiggles.
Let us now write a theoretical model for B(k). Motivated by Renormalized Perturbation Theory (Crocce & Scoccimarro 2008) 3 , the non-linear redshift-space power spectrum can be written as: where b is the large-scale bias, P 0,lin (k) is the linear theory power spectrum monopole in real space, Pmc(k) is the contribution of mode coupling, and G(k, µ) is a propagator which is well approximated by a 2D exponential function: where σ and σ ⊥ are parameters that control the loss of information due to non-linearities along and perpendicular to the line-of-sight, respectively. Note that σ ⊥ < σ , i.e. BAO are smeared further along the line-of-sight as a result of nonlinear RSD (e.g. Sánchez et al. 2008).
We then obtain: where we have assumed b 2 F 2 k P sm 0,lin ≈ b 2 F 2 k (P sm 0,lin G + P sm mc ), see Crocce & Scoccimarro (2008). We recall that brackets denote angular integrals (see §3.1.1). Therefore, BAO are diluted by the weighted average of G(k, µ) over µ, where the weights are set by the relative suppression of line-of-sight modes caused by redshift errors. We display G eff in the left panel of Fig. 4 for different Gaussian redshift errors assuming σ = 10 Mpc, σ ⊥ = 5 Mpc, b = 1, and the value for β expected the cosmology of our N -body simulations. For comparison, we also show G(k, µ = 0) and G(k, µ = 1) as dotted and dashed lines, respectively. As we can see, the greater the value of the redshift error, the smaller the contribution of G(k, µ ∼ 1) and G eff → G(k, µ ∼ 0).
This has an interesting consequence. Since G(k, µ ∼ 1) < G(k, µ ∼ 0) owing to non-linear RSD, redshift errors make BAO appear sharper in the power spectrum monopole. We explicitly show this in the right panel of Fig. 4, where we display B(k) measured in our COLA ensemble in real and redshift space. In real space, the BAO are the same with and without redshift errors. In redshift space, we find a different behaviour; the BAO wiggles are sharper for the simulated catalogues including redshift errors. Consistently with our previous discussion, we can see that the larger the redshift errors, the sharper the BAO.

The Gaussian case
In the case of Gaussian redshift errors, Eq. 29 has an analytic expression: where u = k σ 2 eff + 1 2 (σ 2 − σ 2 ⊥ ) and we assume u > 0 4 . It is useful to take the limit of G eff when σ ∼ σ ⊥ (real space): where the smoothing of the BAO feature is independent of redshift errors. However, in redshift space σ > σ ⊥ and the exponential is weighted by a factor that is greater for larger redshift errors as we can see in the left panel of Fig. 4.

Cosmological information on the BAO scale
We now explore how the modifications introduced by RSD and redshift errors in the BAO and in P0 affect the cosmological information encoded in the position of the BAO feature. Following Ross et al. (2015), let us consider a given scale in the power spectrum, k = k 2 + k 2 ⊥ . The observed scale when assuming a fiducial cosmology will be k ′ = k 2 α −2 + k 2 ⊥ α −2 ⊥ , where α ≡ H fid (z)/H(z) and α ⊥ ≡ DA(z)/D fid A (z). In the above expressions, DA(z) is the angular diameter distance, H(z) is the Hubble parameter, and the fid superscripts denote these quantities computed in the fiducial cosmology.
The observed monopole is thus P0(k ′ ) = P0(k/α), where Expanding the solution to first order, we obtain α = α m α n ⊥ , where m and n are given by: where the denominators ensure that m(k) + n(k) = 1. Note that the higher the value of m, the more sensitive α is to the Hubble parameter. For the case of Gaussian redshift errors, m(k) and n(k) have analytic expressions given by Eqs. 12-13. Note the known case m = 1/3 and n = 2/3 (Eisenstein et al. 2005) is recovered only in real-space (when β = σv = σ = 0). In redshift space, there is a dependence of m and n on β even if σ = σv = 0. In such case, our expressions reduce to the scale-independent formulae presented in Ross et al. (2015). However, as we can see in the general case presented here, non-linear RSD break the scale independence even in the case without redshift errors.
In general, redshift errors and non-linear RSD decrease the sensitivity of the measured P0 on H(z), whereas large-scale RSD increase its sensitivity as a consequence of the line-of-  sight clustering enhancement. Therefore, because the relative strength of these effects depends on the scale considered, the magnitude of the redshift error, and the clustering properties of the underlying sample; the exact degeneracy between α and α ⊥ (and thus the whole information content of P0 under redshift errors) will depend on those details. In Table 1 we provide the value of m at different scales for samples with several combinations of large-scale bias and redshift errors. We see that the lower the large-scale bias of the sample, the stronger the constraints on the line-of-sight component of the BAO and thus, on the Hubble parameter. For samples with redshift errors, we find that on large scales (kσ eff < 1) these constraints are similar to those in the absence of errors, however, they are considerably weaker as smaller scales are considered.
Following a similar procedure as the one employed to compute the dependence of α on the perpendicular and radial components, we can compute which can be used to relate the precision in measuring a scale We draw these ellipses using Eq. 36, compute the values of m eff with and Eq. 37, and use the values of σα from Table 3.
in the power spectrum to the precision in the radial and perpendicular component σα and σα ⊥ , respectively:

The scale-dependence of cosmological information
The scale dependence of m and n introduces a complication while extracting information from BAO analyses, which combine a relatively wide range of scales (typically from k ∼ 0.01 hMpc −1 to k ∼ 0.3 hMpc −1 ). We estimate the effective cosmological dependence as a S/N weighted average of m(k). Explicitly, where the lower limit of the integrals are set to 0.05 hMpc −1 and the upper limit to 0.3 hMpc −1 . Finally, note that the factor G eff captures the suppression in amplitude of the BAO and thus the loss of constraining power of high wavenumbers. We display in Fig. 5 the degeneracy between σα and σα ⊥ in redshift space. From top to bottom, the panels displays results forn = 10 −2 h 3 Mpc −3 ,n = 10 −3 h 3 Mpc −3 , and n = 10 −4 h 3 Mpc −3 . Different colours indicate different redshift errors. The area of the ellipses are set by the total error in α in each case.
In the case of no redshift errors, there is almost no scale dependence in m and n 5 . Thus the ratio of the major and minor semi-axes of the ellipses are identical in all panels. In the case with redshift errors, the shape of the ellipses is modified in a way that also depends on the number density. For a low number densities, there is only cosmological information on large scales, which are also less affected by redshift errors. Hence the shape of the ellipses approach that of the no redshift error case. As we consider larger number densities, smaller scales can be exploited. These scales, in turn, are more affected by redshift errors and thus, there is a loss of information regarding the line-of-sight clustering. Consequently, ellipses become more elongated on the parallel direction.
The above highlights the need for a sophisticated an accurate modelling of the relevant physical and observational effects when interpreting BAO constraints in photometric galaxy surveys.

EXTRACTING THE BAO INFORMATION
In the previous section we showed how redshift errors modify the shape of P0(k), its variance, and the BAO feature. We now employ that information to create a model to extract the BAO scale from observational and/or simulated data, even under the presence of redshift errors. We discuss our model and its motivation in §5.1. We then describe our fitting procedure in §5.2, and in §5.3 we present and discuss the results of applying it to our simulated catalogues.

Modelling the monopole of the power spectrum
Based in the expressions provided in §3 and §4.1, we can write the following model for the observed power spectrum monopole under the presence of non-linearities, RSD, and redshift uncertainties: where α, which is introduced in §4.2, is equal to unity only if the length scale encoded in B(k) matches that of the fiducial cosmology 6 . In §4 we showed that over the range of scales where we expect most of the BAO signal to reside (kσ eff < 1), G eff can be approximated by a Gaussian function with an effective suppression scale (see Eq. 31). Thus we model where k * is a combination of the BAO dilution factors due to redshift errors, non-linearities, and RSD. Note that this expression is independent of redshift errors (they appear just in k * ) and thus, it is similar to the template employed in the analysis of SDSS data (e.g. Percival et al. 2007Percival et al. , 2010Anderson et al. 2012). Furthermore, the free parameters of the model only enter in the expression for B(k). Therefore, these parameters are only constrained by BAO information, i.e. the model PT extracts the BAO scale regardless of the overall power spectrum shape.

Parameter Likelihood Calculation
We assume that the probability of observing d = P 0,obs (k) is given by a multivariate Gaussian distribution: where π = {α, k * } are the parameters of the model PT given by Eq. 38. The priors on these parameters are assumed to be flat over the range: α ∈ [0.93, 1.07] and k * ∈ [0.05, 0.8]. C −1 is the data precision matrix 7 , which we compute from our COLA measurements as described in §2.2. The range of scales considered is k = (0.05 − 0.30) h Mpc −1 . We do not employ smaller scales since BAO wiggles are practically washed out at higher k values due to non-linearities and shot noise. We sample the posterior probability distribution function of π employing the publicly available code emcee 7 We show in §B we obtain the same results using analytical precision matrices computed from the inverse of Eq. 16. (Foreman-Mackey et al. 2013). This code is an affine invariant MCMC ensemble sampler which has been widely tested and used in multiple scientific studies. We configure the code to analyse the monopole of the power spectrum with a chain of 100 random walkers with 5 000 steps each and a burning phase of 500 steps. We check that this burning phase is sufficient to obtain well-behaved chains.
Additionally, we have checked that the standard deviations of the best fit values from the COLA ensemble are compatible with the uncertainties estimated from the MCMC analysis for each one of our simulated catalogues.

Extracting the BAO scale from simulated catalogues
In Fig. 6 we show the quality of the best fit model when applied to the COLA ensemble. Symbols show the average measured B(k) in redshift space, whereas lines show the average best fit model. We display three cases for different σz values, which have been offset from one to another for clarity. Shaded areas indicate the 1σ region computed from the ensemble of COLA simulations. In all cases, the typical deviations between the data and the best fit model are statistically insignificant. Therefore, the model of Eq. 39 is indeed a very good description of the measured power spectra, as can be best seen in the bottom panel of this figure. We now explore quantitatively the results from the best fit in a wide range of conditions.

The impact of redshift errors
We start by presenting the distribution of best fit values, (α, k * ), for 300 independent catalogues of dark matter par- Table 2. Result of the MCMC analysis of the ensemble of COLA simulations for n = 10 −2 h 3 Mpc −3 . We show the mean value of α and its error, the average σα, and the mean value of k * ant its error. ticles with n = 10 −2 h 3 Mpc −3 extracted from our COLA ensemble. We provide these results in Table 2. In Fig. 7 we show histograms of the distribution of best fit values. Top panels show the results in real space whereas bottom panels do so for redshift space. Different colours indicate different redshift errors, as indicated by the legend. The left panels of Fig. 7 show that the mean of the best fit α values is statistically compatible between the cases with different redshift uncertainties. This implies that our estimator is unbiased relative to the case without redshift errors. The mean α value, however, is different than unity by ∼ 0.2%. Average values areᾱ ∼ 1.0018 andᾱ ∼ 1.0023 in real and redshift space, respectively. This small shift is caused by mode coupling Figure 8. Error in α from the analysis of 300 COLA simulations with n = 10 −2 h 3 Mpc −3 in real space (left panel) and redshift space (right panel). We use the same colour coding as Fig. 7. In real space, σα increases with the redshift error. On the other hand, in redshift space it is smaller for the samples with σz/(1+z) = 0.3 % than for the samples without redshift errors (see text).
induced by the non-linear gravitational evolution of the mass density field (e.g. Angulo et al. 2008;Crocce & Scoccimarro 2008;Smith et al. 2008;Padmanabhan & White 2009). However, it can in principle be corrected for with reconstruction algorithms (e.g. Eisenstein et al. 2007;Schmittfull et al. 2015) or with a recalibration of the α estimator.
In real space, the distribution of best fit k * values is compatible across catalogues with different redshift errors. In redshift space, however, k * depends strongly on redshift errors with larger values for larger redshift errors, i.e. the larger the redshift errors the shaper the BAO wiggles. This can be understood by our analytic discussion presented in §4.1. Redshift errors effectively suppress the contribution of the BAO signal along the line-of-sight. In real space, BAO are isotropic, thus redshift errors do not change the value of k * , whereas in redshift space, BAO are more diluted along the line-of-sight, thus k * prefers larger values when redshift errors are added.
We now consider the accuracy in the estimation of α. We display our results in Fig. 8. In real space (left panel), the total error in α increases with increasing redshift error. For σz/(1 + z) = 0.3%, α is estimated with a ∼ 30% less certainty than the σz = 0 case. Since the BAO shape is independent of the redshift errors, the total S/N is reduced by an increase in the noise owing to the shot noise contribution becoming comparatively more important (see §3). In redshift space, the uncertainty in α is a non-monotonic function of the redshift error. For σz/(1 + z) = 0.3% the error decreases relative to the no redshift error case, whereas for σz/(1 + z) = 0.5 %, the error increases. This can be understood as a balance of two effects. The overall S/N increases at scales kσ ∼ 1 and decreases on higher wavenumbers due to the increased shot noise contribution. Therefore, a larger or smaller uncertainty depends on the wavelength where kσ ∼ 1 relative to the BAO scales. For σz/(1 + z) = 0.3 %, this occurs at k ∼ 0.2 -the relevant BAO scales -thus we expect stronger constraints on α. On the other hand, for σz/(1 + z) = 0.5 % this occurs at k ∼ 0.1, which implies that a considerable fraction of the BAO signal is measured with lower S/N.
In addition to the above effect, in redshift space the BAO signal becomes sharper with redshift errors, increasing the constraining power. However, we have checked that this effect is less important.
We present our results in Table 3. First, we see that the mean value of α is independent on the number density of the sample, which supports the robustness of our analysis procedure. Second, there is a trend with the bias in α being smaller for larger redshift errors. This is likely a consequence of larger redshift errors implying that the constrains on α come from larger, more linear, scales.
The uncertainty in α is shown in the left panel of Fig. 9. For high number densities, the relation between σα and σz is non monotonic as discussed previously. As we consider lower number densities, the uncertainty in α increases for all values of σz but it does so more rapidly for larger values of σz. At the typical densities of spectroscopic redshift surveys (i.e. SDSS-III/BOSS),n ∼ 3 × 10 −4 h 3 Mpc −3 , the error in α increases monotonically with the redshift errors -∼ 20(45)% larger for σz/(1 + z) = 0.3(0.5)%, respectively. Conversely, to reach the same accuracy on α, the number density should be 50 % larger for σz/(1 + z) = 0.3 % and 2.5 times larger for σz/(1 + z) = 0.5 %.
On the other hand, the uncertainty is different for the angular diameter distance, which depends on the power spectrum modes perpendicular to the line-of-sight, than for the Hubble parameter, which depends on the power spectrum modes parallel to the line-of-sight. We can see this in the right panels of Fig. 9, which show the dependence of the accuracy of the radial and perpendicular components of α on the number density and the redshift errors. Note that for constructing this figure we have employed Eq. 37, where we have adopted the effective values of m and n corresponding to each case.
We can see that a modest increase in the number density already delivers constrains comparable to those of the no redshift error case. Atn ∼ 3 × 10 −4 h 3 Mpc −3 , the constrains on α ⊥ for all the cases shown are almost identical. On the other hand, a considerable increase inn is necessary to reach comparable constrains on α ; the number density should be ∼ 3(20) times larger for σz/(1 + z) = 0.2(0.3)%.
All the above considerations should be taken into account for the optimal design of a survey or a target sample. For instance, the redshift error of a given galaxy sample might not only depend on the hardware employed, but also on the intrinsic galaxy properties (e.g. brighter objects having more accurate redshift estimates). In such case, the sample that delivers the strongest constrains is not necessarily that with the smallest redshift error.

The impact of biased tracers
The effect of the large-scale bias in real space is straightforward: it increases the amplitude of the power spectrum in all scales, thus reducing the relative contribution of the shot noise. The picture is somewhat more complicated in redshift space, since the RSD enhancement on linear scales depends on β, and there is a decrease on small scales. Additionally, we expect biased tracers to display a different BAO signal than the mass field (Angulo et al. 2012;Prada et al. 2016).
We study the effect of the large-scale bias by analysing several samples drawn from the MXXL simulation with different number density, large-scale bias, and redshift errors. We gather the results of the analysis in Table 4. The difference between the large-scale bias of the different samples is small but allow us to extract some conclusions. The first is that the shift in α is compatible with zero at the 1σ level for all the samples. Moreover, it is compatible with the bias obtained from the COLA mocks and thus, the fact that we are studying samples of DM haloes instead of DM particles like in COLA does not introduce any systematic bias in α.
As we commented before, the parameter of our model which controls the suppression of the BAO feature is k * . We find that the value of k * is the smallest for the samples with the greatest large-scale bias, where a higher value of k * means a smaller suppression of the BAO feature. This confirms the analytical predictions of Eq. 29.
We find that σα is slightly smaller for samples with greater large-scale bias and redshift errors, being the difference more important for the samples with the smallest number density. This is because a higher large-scale bias reduces the rel-ative contribution of the shot-noise to the power spectrum monopole.
The large-scale bias also modifies the cosmological information encoded in α, since a higher bias decreases the dependence of the BAO scale on the Hubble parameter (see Table 1).

Analytical estimation of σα
In the previous subsections we explored the dependence of σα on the redshift errors, the number density, and the large-scale bias b. We showed that their net effect can be understood in terms of their impact on the overall amplitude of the power spectrum with respect to the shot noise. Therefore, we expect that a good predictor for the error recovering the BAO scale be proportional to the total signal-to-noise encoded in a particular configuration. We thus writẽ similarly to Eq. 37, the lower limit of the integral is set to 0.05 hMpc −1 and the upper limit to 0.3 hMpc −1 . We compute the value of the normalization constant A = 6.5 hMpc −1 by fitting the values retrieved by our previous MCMC analyses. We recall that we provided analytical expressions for the power spectrum monopole, P0, and its variance, σ[P0], in §3.1 and §3.2, respectively. We can appreciate the performance of this model in Fig. 9, whose predictions are displayed as coloured lines. We can see that our model captures qualitatively the behaviour of the measured uncertainties in α.

Effect of the PDF of redshift errors
In the previous sections we have modelled redshift errors as Gaussian distributions. However, this might not be necessarily a good approximation to reality under some circumstances. Therefore, to finalize our paper we explore the performance of our fitting procedure when considering different probability distribution functions with varying levels of skewness and kurtosis (see §2.3). Fig. 10 shows the best fit values of α from the COLA ensemble with different PDFs for redshift errors and forn = 10 −2 h 3 Mpc −3 . Shaded regions denote the 68 % and 95 % confidence levels for a Gaussian PDF with σz/(1+z) = 0.3%. The x-axis indicates the excess kurtosis of each distribution. Note that we plot the Cauchy distribution at a excess kurtosis of 9 (the actual value is not defined). Note also that the skewness Figure 10. Shift of the BAO scale computed from the average power spectrum of the COLA ensemble with redshift errors following different PDFs as indicated by the legend. The difference between the 84th and 16th percentile of the different PDFs is 2σz, where σz/(1 + z) = 0.3 %. The coloured areas encloses the 1σ and 2σ regions for redshift errors following a Gaussian distribution. We introduce these families of distributions in §2.3.
is zero for the family Norm1. For the family Norm2, it goes from 1.73 for κ = 0.5 to 416.9 for κ = 2.0.
We can see that the recovered BAO scale is largely insensitive to the actual PDF shape at the statistical level of our simulated catalogues -all but one case is compatible with the Gaussian case at the 2σ level. In light of the discussion presented in §4.1, we expect that the effect of different PDFs be to change the weighted average of G(k, µ). For extreme PDFs, this could in principle introduce systematic errors in the estimation of α. In practice, however, we expect that a reasonable estimate of the redshift error PDF would allow the construction of an unbiased estimator.

CONCLUSIONS
We have presented a detailed study of the effect of redshift uncertainties on the galaxy clustering, with an emphasis on the BAO signal. Our main findings can be summarized as follows: • We provide analytical expressions for the impact of redshift errors on the shape and variance of the redshift-space power spectrum monopole and quadrupole. In Figs. 1 and 2, we show that these capture the relevant modifications and that they agree with results from N -body simulations within 5 % up to k = 0.6 h Mpc −1 .
• We show analytically and with simulations that redshift errors in redshift space make the BAO feature sharper (Fig. 4). This is because redshift uncertainties reduce the weight of Fourier modes parallel to the line-of-sight, which display a more diluted BAO signal due to non-linear RSD.
• We compute explicitly the dependence of the cosmological information encoded on the BAO on redshift errors, redshiftspace distortions, and number density of the galaxy sample considered. We show in Fig. 5 the degeneracy between the error in H(z) and DA(z) for samples with different configurations.
• Based on those findings, we build a model for extracting the BAO information from the spherically-averaged power spectrum. We then apply this model to simulated galaxy catalogues with different levels of shot noise, large-scale bias, and redshift errors. For all of them we find that redshift errors do not significantly shift the BAO with respect to the case without redshift errors.
• We find that the error in α depends on the sharpness of the BAO and the relative amplitude of the power spectrum monopole with respect to the shot noise, showing that the error in α is smaller for samples with σz/(1 + z) < 0.4 % and n > 3 × 10 −3 h 3 Mpc −3 than for samples with no redshift errors.
• In §5.5 we consider different PDF the redshift errors. We display in Fig. 10 our findings which suggest that our BAO estimator is robust against different PDFs of redshift errors.
• Finally, in Eq. 41 we provide a quick method to forecast the uncertainty on the BAO scale based on analytical expressions provided in this paper.
We have probed that it is crucial a profound understanding of the effect of redshift errors on the galaxy clustering to extract correctly the BAO scale and to understand what combination of cosmological parameters this scale is constraining. We have also demonstrated that samples with smaller errors are not always the best ones to measure the BAO scale, it depends on the number density and the cosmological parameter that we want to constrain. Therefore, it will be very important for photometric surveys to define carefully the galaxy sample employed to compute α.

APPENDIX B: EFFECT OF NON-DIAGONAL TERMS IN COVARIANCE MATRICES
In §3 we derive analytically the covariance matrix of the power spectrum monopole, where non-diagonal terms are zero because we assume that the matter density field is Gaussian. However, this approximation breaks on non-linear scales, where non-linearities couples different k-modes. In this section we estimate the validity of this assumption using N -body simulations.
We employ in §5 precision matrices estimated from ensembles of 300 COLA simulations. We correct those precision matrices by a prefactor introduced by Hartlap et al. (2007), as we mention in §2.2. We check that the relative difference between the diagonal terms of the corrected covariance matrices and the diagonal terms of the precision matrices computed inverting Eq. 16 is within 5 %.
We check whether non-diagonal terms modify α or its error in §5 using analytical precision matrices. We find that the relative difference between the new and old values is within 4% and thus, non-diagonal terms at z = 1 are not important in the range of scales that we are using. Therefore, this motivates the use of analytical precision matrices, since they can be computed instantly for different combinations of redshift errors, large-scale biases, number densities, and cosmological parameters.