{ "I. INTRODUCTION ": [ "Large-scale structure (LSS) measurements have become an extremely powerful probe of cosmology over the past 30 years. Starting with the pioneering Harvard-CfA survey [(<>)1], all the way to the Sloan Digital Sky Survey [(<>)2] and its extension Baryon Oscillation Sky Survey [(<>)3], Two-degree Field survey [(<>)4], and WiggleZ [(<>)5], the LSS surveys have revolutionized our understanding of the distribution of matter and energy in the cosmos, and helped impose percent-level constraints on the cosmological parameters (e.g. [(<>)6]). ", "A major challenge in current and future imaging and spectroscopic LSS surveys is understanding the sample selection. We define calibration to be the measure of our understanding of the selection of our sample of galaxies, and calibration errors to be any unaccounted-for angular and redshift variations in the selection. The purpose of this paper is to determine how well calibration errors need to be controlled in order to avoid substantial degradation of the information we can extract from the LSS. ", "A particular source of uncertainty is known as photometric calibration. The term refers to the adjustments required to establish a consistent spatial and temporal measurement of flux of the target objects in the different bands of observation throughout the entire photometric survey. This is an enormous problem that all existing and upcoming wide area surveys face. The difficulty comes from the variability of various building blocks of the observational pipeline, which makes it difficult to establish a consistent flux baseline at each band (i.e. the flux ze-ropoints). In other words, because the instrument sensitivity is constantly changing, and so are the sources and intensity of noise, it is difficult to consistently compare ", "the fluxes for objects at different parts of the sky imaged at different times. Some examples of the manifestations of the photometric calibration errors in surveys are: ", "Variabilities in the instrument sensitivity and observing conditions cause an angular variability in the depth of observations that the survey can achieve through each filter. Variations in the depth result in angular variations in the number density and redshift distribution of objects. In addition, because galaxy spectra are not flat, and because the sample selection involves more than one filter, depth variations cause variations in the angular and redshift distribution of galaxy types. ", "This variability in the sample selection can, in principle, be accounted for. This is not always done, however, and it is common, for example, for correlation analyses of current data to assume a constant depth for the entire survey. Indeed, several sources of variability have been accounted for in the analysis of existing data – see in particular the pioneering work on the subject in the modern era of LSS surveys by Scranton et al. [(<>)10] (see also Voge-ley [(<>)11]), and the more recent efforts by Ho et al. [(<>)12] and Ross et al. [(<>)8]. These authors modeled a wide variety of systematic errors, some of which qualify as the calibration errors (e.g. seeing, airmass, calibration offsets). In particular, the latter two papers identified bright stars as the major contaminant which adds significant power to the intrinsic clustering signal at large scales, and they applied two separate successful techniques to subtract this systematic contamination. ", "For the upcoming surveys an even more detailed analysis will be needed, ideally utilizing a formalism that is suited to a wide variety of photometric calibration systematics mentioned above and captures any kind of calibration-related systematic. One would also like to provide guidance on how much calibration error, as a function of scale, can be tolerated in order not to degrade the cosmological parameter inferences. Here we aim to address both of these desiderata. ", "In this paper we set out to study calibration errors in the most general way possible. Our goal is to build an end-to-end pipeline into which we can feed calibration errors (or uncertainties) due to an arbitrary cause, and from which we obtain biases in cosmological parameters inferred from measurements of galaxy clustering in some LSS survey. We then turn the problem around, and estimate how well the calibration errors need to be controlled in order not to appreciably bias the cosmological parameter estimates. ", "To keep the scope of this paper reasonable, we only consider measurements of the galaxy two-point correlation function (i.e. its Fourier transform, the power spectrum), and leave other observable quantities – higher-order correlation functions of galaxies, for example – for future work. We also do not consider the effect of the photometric redshift errors which, while very important, are not expected to change our results in a major way, so we leave the photo-zs for a future analysis. ", "The paper is organized as follows. in Sec. (<>)II we describe our formalism of modeling both the true, underlying galaxy density field and the systematic errors describing variations in the photometric calibration. In ", "Sec. (<>)III we present the formalism to derive cosmological constraints and biases on cosmological parameters. In Sec. (<>)IV we propagate the effects of the systematic errors to calculate the biases in the cosmological parameters. We conclude in Sec. (<>)V. Important technical details regarding various aspects of the computation of the effects of the photometric variation systematics on the observable quantities are relegated to the three Appendices. " ], "II. FORMALISM: DESCRIBING SPATIALLY VARYING CALIBRATION ": [ "In this section we start by defining calibration errors and their field c(nˆ), and proceed to derive the biased galaxy fluctuations in terms of this field in multipole space. " ], "A. Calibration errors: definition and basics ": [ "Let true galaxy counts on the sky be denoted by N(nˆ), where nˆ is an arbitrary spatial direction. The survey mean is given by N¯ ≡ N(nˆ)sky, where the average here is taken over the observed sky. These true fluctuations in the galaxy counts can be expanded into harmonic co-efficients am as ", "(1) ", "Consider a survey where a deterministic calibration error c(nˆ) biases galaxy counts. In other words, given the true galaxy number counts in some direction N(nˆ), the observed number is ", "(2) ", "which implicitly defines the calibration field c(nˆ). We can expand the calibration field relative to its fiducial value of zero (corresponding to no error) ", "(3) ", "where hereafter we assume that the calibration error dominates on large scales, and persists only out to some maximum multipole calib,max, corresponding to the minimal angular scale of π/calib,max radians. ", "The statistical properties of the two galaxy number-density field, and the calibration-error field are, respectively ", "(5) ", "Throughout the paper, angular brackets · indicate ensemble averages, that is, averages over different realizations of the Universe. To reiterate, N(nˆ) is the Gaussian ", "random, isotropic field as predicted by inflation, while c(nˆ) is a deterministic function given by calibration errors in the survey at hand. ", "In the remainder of this paper, we use the following definition: Calibration variations (or errors) are departures of c(nˆ), or its harmonic coefficients cm, from zero. Our goal is to estimate how accurately those variations have to be known in order not to bias the cosmological parameter estimates. ", "Notice that we do not lose any generality by assuming that the c-field is fixed, rather than stochastic like the true galaxy density field. In this paper, we are effectively asking how much does this fixed systematic error bias the usual cosmological constraints. We are, of course, free to iterate over a number of specific incarnations of this ’fixed’ error. A more specific example would be to ask how much does the Galactic dust pattern – its direction and amplitude fixed for the moment – bias some cosmological inference if unaccounted for perfectly, and then to repeat the analysis for a number of dust pattern realizations, or even for several different dust models. " ], "B. Galaxy clustering and calibration errors: general case ": [ "We now derive the main results regarding the effect of the calibration errors on the observed clustering of galaxies. Let us first calculate the observed density contrast of galaxies: ", "where Ym ≡ Ym(nˆ), and we expanded the photometric calibration variation field c(nˆ) into spherical harmonics. Here  is the relative bias in the measured mean number of galaxies: ", "(7) ", "(the sky denotes sky average), so that ", "(8) ", "The quantity  can be evaluated directly in real space as above when given the calibration error map, or in harmonic space, combining Eq. ((<>)8) and Eq. ((<>)1) ", "(9) ", "where we used the identity (−1)ma(−m) = a ∗m and the orthogonality relation for spherical harmonics. In cases where fsky < 1, the orthogonality relation does not hold, but Eq. ((<>)9) still does if the coefficients am are interpreted as the cut-sky harmonics of the density field. ", "The observed galaxy overdensity field can also be expanded in terms of the harmonic basis ", "(10) ", "Equating this to the last expression in Eq. ((<>)6) and inverting by multiplying with Ym ∗ and using the orthogonality relation, we obtain the harmonic coefficients of the observed galaxy overdensity field tm in terms the true galaxy fluctuation field am and the calibration field cm ", "where√to obtain the last term in the last line we used 1 = 4πY00. Here we define the coupling matrix R in terms of Wigner 3j symbols ", "Calculating the two-point correlation of tm is now straightforward, and things are simplified because all terms proportional to a single power of am (or its conjugate) vanish – recall that cm are just some numbers here. Moreover, we can ignore the term proportional to δ0 – last term in Eq. ((<>)11) – since it only affects the monopole which is not used in cosmological constraints. The ensemble average of the multipole moments becomes, after some algebra ", "(13) ", "where we defined ", "(14) ", "which is a function that depends on the Wigner 3j symbols as well as the calibration-field coefficients cm. ", "Equation ((<>)13) is the key result in this paper. As the label in the equation shows, the observed galaxy density field t(nˆ) exhibits broken statistical isotropy. In particular, the variance of t is not rotationally invariant any more (i.e. it depends on m), and covariance between the different  modes is not zero any more. We can now utilize this formula and consider the isotropically measured power (i.e. assuming  =   and averaging over m = m ) and estimate how accurately any given systematic, described by the full set of cm, needs to be understood in order not to degrade the accuracy in measuring the cosmological parameters including non-Gaussianity 1 (<>). " ], "C. Galaxy clustering and calibration errors: isotropic power case ": [ "We usually – essentially always, in fact! – assume that the field is isotropic, and then we use the data to calculate the correlation function, power spectrum, etc. Let us see how the assumed-isotropic angular power spectrum is biased in terms of an arbitrary contamination field. ", "Setting  =   and m = m  in Eq. ((<>)13), we get ", "To assume statistical isotropy, we not only set  =   and m = m  but further average over the 2 + 1 values of m for a fixed . Then we obtain the prediction for the angular power spectrum that one would measure assuming statistical isotropy even when the systematics break it: ", "(16) ", "where |U|2 ≡ UU∗ , and where the sum over 2 goes in principle over all multipoles (though only those from the range [− calib,max,+ calib,max] are nonzero), while m2 goes from −2 to 2. Note that the term linear in U seen in Eq. ((<>)15) dramatically simplified in the expression for T (Eq. (<>)16) after we used the summation relation ", "For a pure monopole calibration error (i.e. a pure c00 term), one can verify that the effects of the term and the ", "c00 term in Eq. ((<>)16) exactly cancel and T is unchanged. This makes intuitive sense, as a shift in the monopole changes the mean counts on the sky but does not affect the density fluctuations. ", "One can intuitively understand the individual terms on the right-hand side of Eq. ((<>)16): ", "the number of evaluations of (,1,2,m,m1,m2) and tabulating the coefficients U so that the number of operations is only of order 108 (for max = 1000 binned in ∼ 30 multipole bins and considering the calibration variations out to calib,max = 20), and is thus feasible. We plot the biased power spectra T further below in the next section. ", "From the structure of Eq. ((<>)16), it is clear that calculating and storing the coefficients U is challenging. Naively, the problem requires evaluation of roughly 1018 coeffi-cients. Appendix (<>)C describes our approach of limiting ", "Finally, it is worth writing down the observed angular cross-correlation power spectrum between fluctuations tm (i) and tm ∗(j) in two different tomographic redshift bins i and j; it follows straightforwardly from Eq. ((<>)16) that: ", "(18) ", "where and are all evaluated in the redshift bin i (and same for j), and where C(ij) are the true galaxy cross-correlation power spectra. While physical sources of calibration error are typically local and thus redshift-independent, in Sec. (<>)IV we demonstrate that converting from the magnitude error to the calibration field c(nˆ) ≡ (δN/N)(nˆ) depends on the faint-end slope of the luminosity function, which typically is redshift-dependent, hence making the harmonic coeffi-cients of c(nˆ) also z-dependent and thereby potentially introducing couplings between the different redshift bins. " ], "D. Additive and multiplicative systematics ": [ "Before we obtain numerical results on how some realistic calibration errors affect the observed power spectra, it pays to consider qualitatively how the angular power spectrum of galaxies is affected. ", "It is often useful to divide the effect of systematic errors into additive (those whose field is added to the true field observed on the sky), and multiplicative (those whose field multiplies the true field); see e.g. [(<>)13] where his nomenclature has been previously employed in the cosmic microwave background (CMB) context, and [(<>)14] and [(<>)15] who considered the additive and multiplicative systematic errors in weak lensing measurements. These terms refer to the systematic error that either adds to the true galaxy fluctuations, or else multiplies it and modulates the true signal; see Eq. ((<>)6) for the real-space and Eq. ((<>)11) for the harmonic-space picture. For example, on the right-hand side of Eq. ((<>)11) the term am corresponds to the true density field, cm represents the additive effect of the systematic error, while the term containing cmam term together with the geometric factor R and the appropriate sum represents the multiplicative effect ", "of the systematics2 (<>). ", "Additive and multiplicative errors in the counts translate into additive and multiplicative contributions from the calibration field to the observed galaxy power spec∗ trum, see Eq. ((<>)13): the additive error is the term cmc m while the multiplicative error are all terms involving the coupling matrix U. The two kinds of errors produce qualitatively different effects: additive error at some multi-pole 1 only affects power at that multipole, while the multiplicative error affects power at a range of multi-poles; in particular, true power at an arbitrary multipole  would leak to all multipoles in the range ", "The additive terms dominate the error budget on the largest scales, but are subdominant at smaller scales [(<>)11]. This can be understood qualitatively as follows: modulo geometric coupling terms, both additive and multiplicative terms are proportional to the square of coefficients cm, but the multiplicative terms are further multiplied by the fiducial angular power C. Given that C  1 at all  and any redshift 3 (<>), the multiplicative terms are suppressed relative to the additive terms. At higher mul-tipoles, on the other hand, there are more ways in which power from other scales can leak into that  so that the sums associated with multiplicative terms make them the dominant systematic contribution. ", "While it is often assumed for simplicity that the systematic errors, calibration or other, are represented by purely additive errors, we just demonstrated that both kinds of errors are important. In fact, in the plausible scenario where largest-scale information in the survey is ignored to avoid the systematic contamination, the multiplicative errors dominate. In what follows we use the full expressions containing both additive and multiplicative terms. " ], "III. POWER SPECTRA AND THEIR BIASES ": [ "In this section we propagate the effect of calibration errors to estimate biases in the cosmological parameters describing dark energy and primordial non-Gaussianity. " ], "A. Fiducial cosmological model ": [ "We consider a set of cosmological parameters with the following fiducial values: matter density relative to critical dark energy equation of state parameter today its variation with scale factor spectral index and amplitude of the matter power spectrum ln A where (corresponding to at scale scale Note that we hold fixed the Hubble constant km/s/Mpc) (or equivalently, physical matter density ΩM h2), the physical baryon density ΩBh2 , and we assumed a flat universe. On the other hand, we do not assume any other prior information, such as the CMB information from WMAP and Planck. In practice, this prior information would largely serve to fix h, ΩM h2 , ΩBh2 , and curvature ΩK . Note that this rather restricted set of assumptions about the set of cosmological parameters and external information about them is sufficient for our analysis: we are primarily concerned about the effect of the calibration systematics on the measured power spectra, and on the biases in the dark energy and non-Gaussianity parameters. Given that the systematics strongly depend on the properties of the galaxy sample and the survey (as we discuss further below), it is not necessary to model the up-to-date knowledge about the cosmological parameters in great detail. ", "We assume a survey covering 5,000 square degrees (so fsky  0.12) with information out to zmax = 1, corresponding roughly to the Dark Energy Survey (DES). We assume that the number density of the galaxies is n(z) ∝ z2 exp(−z/z0) with z0 = 0.3, and that photometric redshifts enable splitting the sample in five tomographic bins centered at z = 0.1, 0.3, 0.5, 0.7 and 0.9; see the left panel of Fig. (<>)1. The fiducial statistical constraints on the dark energy parameters are σ(w0) = 0.06 and σ(wa) = 0.24. ", "Instead of the power spectrum in wavenumber P(k), we consider measurements of the angular power spectrum of galaxy fluctuations. In the Limber approximation, which ", "is valid on intermediate to small angular scales, the angular power is given as (e.g. [(<>)16, (<>)17]) ", "(19) ", "where i and j are referring to one of the five redshift bins 4 (<>)and b(k, z) is the bias defined below. The full, beyond-Limber expression, as well as the definition of the W(z) terms, are given in Appendix (<>)A. The power spectrum is calculated using the transfer function output by CAMB, and its nonlinearities are modeled with the Smith et al. [(<>)18] formulae that were based on a halo model and fit to simulations. Appendix (<>)A has all of the details of how we calculate the power spectrum. ", "We consider information from multipoles 1 ≤  ≤ 1000, corresponding to spatial scales from about 10 arcmin to 180 degrees. To obtain accurate constraints on the parameter fNL which come from large angular scales, we use every individual multipole between  = 1 and calib,max = 20; beyond this we use ten more widely separated bins with Δ  100. Therefore, we use a total of 30 bins in ; for the low- ones we do not assume the Limber approximation and use Eqs. ((<>)A4), while for the higher multipoles we use the Limber approximation, Eq. ((<>)19) above. ", "Finally, we also allow for the presence of primordial non-Gaussianity. We adopt the widely studied ‘local’ model of non-Gaussianity ", "(20) ", "(where Φ is the primordial Newtonian gravitational potential, ΦG is its Gaussian component, and fNL is a dimensionless parameter), the bias becomes scale-dependent, with a new term that goes as k−2 [(<>)19] ", "(21) ", "where b0 is the usual Gaussian bias (on large scales, where it is constant), δc ≈ 1.686 is the collapse threshold, a is ", "FIG. 1. Left panel: density distribution of galaxies assumed in this paper, and boundaries of the five redshift bins. We ignore information at z > 1, thus roughly modeling the difficulties with establishing accurate photometric redshifts at that range (for the DES). Right panel: Angular power spectra C(ii) for five redshift bins [Cross-correlations between the bins, while used in the analysis, are very small and not important nor shown in the figure.]. For the first and fifth bin we show, at the low multipole end, the full expression that we use at large scales (see Eq. ((<>)A4)) and, at the high-multipole end, the linear power spectrum for reference. For the first and fifth redshift bin we also show the cosmic variance errors plus shot noise. ", "the scale factor, ΩM is the matter density relative to critical, H0 is the Hubble constant, k is the wavenumber, T(k) is the transfer function, g(a) is the growth suppression factor, and c is the speed of light 5 (<>). We assume the fiducial model with the Gaussian bias b0 = 2 and zero non-Gaussianity, fNL = 0. ", "The full set of cosmological parameters that we use is therefore ", "(22) ", "The cosmological constraints can then be computed from the Fisher matrix ", "where α and β stand for all pairs of bin indices (i, j) with i ≤ j (since C(ji) ≡ C(ij)). The observed power spectrum is equal to the raw power plus shot noise ", "(24) ", "where Nisr is the number of galaxies per steradian in the tomographic bin i. Moreover, Cov−1 in Eq. ((<>)23) is the inverse of the covariance matrix between the observed power spectra; assuming observations in the linear (and ", "therefore Gaussian) regime only, the covariance matrix follows directly from Wick’s theorem: ", "(25) ", "The minimal error in the i-th cosmological parameter is, by the Cram´er-Rao inequality, σ(pi)  (F−1)ii. ", "Finally, we would like to estimate the bias in the cosmological parameters, δpa, given an arbitrary systematic error in the power-spectrum, δC. The bias can be estimated using the Fisher matrix formalism as follows: " ], "B. Biases in the observed power spectra ": [ "We would like to fairly compare the biases as a function of 1, so we choose to adopt coefficients describing the uncertainties in the calibration field c1m1 that lead to a fixed variance in the calibration pattern on the sky c(nˆ) (and corresponding to, as we will shortly see, fixed variance in the angular variations of the magnitude limits of the survey). Thus, we have ", "(27) ", "since is the angular power spectrum of the systematics (and really just the sum of their ", "coefficients squared) and where the reader is reminded that, in this particular calculation we are “turning on” one (1, m1) pair at a time. ", "To consider the calibration variation in a single multi-pole (1, m1) that leads to a fixed variance in c(nˆ), we therefore make the choice ", "(28) ", "where in the m = 0 case both real and imaginary part of the c1m1 have the given value. ", "Figure (<>)2 shows the difference between the observed isotropic part of the power spectrum T (see Eq. ((<>)16)) and the fiducial C, divided by the statistical error (cosmic variance plus shot noise) for our assumed DES-type survey with fsky  1/8. We assume a constant calibration error with rms 6 (<>)of 0.01 or 0.001 per each multipole 1 separately. While a fixed 1 of the systematic errors affects all multipoles , it affects  = 1 the most, so in this graph we only plot the effect on the observed power spectrum at the same multipole at which the systematic errors occurs. In other words, Figure (<>)2 shows the maximally affected multipole , for a fixed calibration variation error. The error in the measured tomographic angular power spectra decreases with multipole , which can be understood easily as follows: to a good approximation, the additional terms in the observed isotropic power spectrum, Eq. ((<>)16), are dominated by the additive term |cm|2/(2+1). We find this is the case even for the lowest-redshift tomographic bin where C is the highest and thus helps boost the multiplicative terms that contain the U coefficients. Moreover, generally we find that   0. Therefore, ", "(29) ", "where C ≡ C(ii) is referring to the power spectrum in some redshift bin i, and same for T and cm. For a fixed variance in the photometric variations, cm √is independent of , so that this expression goes as 1/[ 2 + 1C], decreasing with  approximately as −1.3 , at least out to  = 10 plotted in this Figure. ", "Figure (<>)2 further shows that higher redshift bins are affected more than the lower redshifts. This is easy to understand: at higher redshift, the cosmological signal is smaller, given that it averages over more large-scale structure along the line of sight, and therefore it is more susceptible to the (redshift-independent) calibration bias. ", "FIG. 2. Difference between the the observed isotropic part of the power spectrum T (see Eq. ((<>)16)) and the fiducial C, divided by the statistical error (cosmic variance plus shot noise) for our assumed DES-type survey with fsky  1/8. We assume a constant rms photometric variation of 0.01 (dashed curves) or 0.001 (solid curves) per each multipole . Note that higher redshift bins are affected more than the lower redshifts. The fall-off with  can be understood analytically; see text for details. ", "The one important thing to take away from Fig. (<>)2 is therefore that the calibration error is expected to be most damaging to the deepest surveys (or highest-redshift slices of a survey). And it is precisely those highest redshifts that are most valuable in providing information about dark energy and primordial non-Gaussianity. " ], "IV. SENSITIVITY TO CALIBRATION ERRORS ": [ "Let us now consider a few specific examples. First, we will study the sensitivities to an arbitrary systematic bias in calibration at each multipole separately, i.e. one (1, m1) pair at a time in the c1m1 . Then we study two concrete examples of physical effects that cause calibration biases: corrections to the dust extinction maps, and variable survey depth. " ], "A. From calibration to galaxy counts ": [ "Consider the observed angular density of galaxies in some direction in the sky N(nˆ) ≡ n(z, nˆ)dz, where n(z, nˆ) is the galaxy density in that direction and at redshift z. Calibration errors correspond to variations in the magnitude limit of the survey δmmax(nˆ). The observed density of galaxies changes since the galaxy density is a strong function of the survey depth (i.e. the magnitude ", "FIG. 3. Bias divided by marginalized statistical error in the cosmological parameters for the fixed magnitude root-mean-squared variation of 0.001 (solid curves) or 0.01 (dashed curves), as a function of multipole at which the systematics are introduced. We show the bias/error ratio for the non-Gaussianity parameter fNL, constant equation of state of dark energy w, and the square root of the DETF figure of merit, FoM1/2 , which serves to gauge any additional dependence brought forth by the temporal variation in the equation of state wa. To convert the magnitude variation to the δN/N error, we used Eq. ((<>)31) and the best-fit faint-end slope of the luminosity function, s(z), estimated from simulations of [(<>)20]; see text for details. The error bars show dependence on which of the m1-values, for a fixed 1, contains the calibration error; this dependence is small at the largest scales where the calibration error clearly has the largest effect. As discussed in the text, the monopole 1 = 0 has no effect on the biases by definition. The dashed horizontal line denotes a fixed bias/error ratio of 0.3, which is approximately the upper limit of how much effect a systematic error should have on the cosmological parameters without seriously affecting the overall constraints in a survey. ", "where mmax is the maximal apparent magnitude observed in that direction in some waveband. It follows that the systematic bias in the observed fluctuations is ", "(31) ", "Often we have information about the selective extinction EB−V ; the relation to magnitude extinction is δm ≡ δA = δ[R EB−V ] where R is the ratio of total to selective extinction and A is the alternative notation sometimes used for extinction. Assuming that R is known perfectly 7 (<>), δm  R δ(EB−V ), and thus (restoring the di-", "rection nˆ explicitly) ", "While s(z) is galaxy-population dependent, we can still estimate (δN/N)sys to be very roughly of order δ(EB−V )(nˆ), given that s(z) is of order 0.1-1 while R takes values between about 1 and 5 depending on the band; see e.g. the appendix of Schlegel et al. [(<>)21] and important updates given in the Table 6 of Schlafly and Finkbeiner [(<>)22]. ", "FIG. 4. Same as Fig. (<>)3, except now the fixed magnitude error of 0.001 is shared equally among all multipoles in the range 1 ≤ 1 ≤ 1,max. This is perhaps a more realistic assumption than the one shown in Fig. (<>)3 where all of the error comes from a single multipole. The biases now appear larger because the contributions from the largest scales dominate the budget at a fixed 1,max. " ], "B. Calibration Bias per Multipole ": [ "Let us consider biases in our six cosmological parameters as a function of bias in a single multipole c1m1 . Following the prescription in the previous section, we assume that the variance of the calibration field is fixed and constant separately at each multipole 1; see Eq. ((<>)28). ", "We now propagate the calibration variation in a given (1, m1), separately for 1 ≤ 1 ≤ 20 and −1 ≤ m1 ≤ 1, and for magnitude given in the above equations, to the observed angular power spectra via Eq. ((<>)16). ", "Figure (<>)3 shows the bias divided by the statistical error in cosmological parameters for the fixed magnitude rms variation per multipole of δmmax 2 sky 1/2 = 0.01 or 0.001. We use Eq. ((<>)32) to translate this to the calibration-field variation, and then a modified version of Eq. ((<>)28) to calculate the harmonic coefficients c1m1 that enter the calculation: ", "(34) ", "where, relative to Eq. ((<>)28), we have an additional term of (21+1)−1/2 to keep the variance in each 1 fixed since we are now distributing power over all m1-modes. We adopt the fiducial redshift-dependent faint-end slope of the lu-", "(35) ", "estimated from the simulations of [(<>)20], assuming a DES i-band magnitude limit of 24. This functional form roughly describes the trend that the highest redshifts are most affected by variations in the survey depth. We emphasize that this form for s(z) is meant purely for illustration, as different galaxy samples will have different s(z). We consider biases in the non-Gaussianity parameter fNL, the (constant) equation of state of dark energy w, and the square root of the dark energy figure-of-merit, which is the inverse area of the 95% contour in the w0-wa plane [(<>)23, (<>)24]. Note that the latter quantity takes into account the temporal variation of DE, and the square root serves to compare it fairly to the bias in constant w; the two quantities, σ(w) and FoM1/2 , show very similar behavior in these results. The error bar at each 1 shows the rms dispersion of the (21 +1) values of m1 into which we put the systematics. So, for example, at 1 = 6 and for either one of the rms values for the calibration error, the error bars show the dispersion in the bias/error ratios for 13 different values of m1. ", "Figure (<>)3 clearly indicates that systematic errors have the largest impact at largest angular scales assuming a fixed contribution to the variance from each multipole. Bias in the non-Gaussianity parameter fNL is larger than that for the dark energy parameters, which is expected ", "because most of the information on fNL comes from large angular scales which are particularly susceptible to calibration errors. For the calibration systematics at smaller scales – 1 beyond six or so corresponding to variation at scales less than about 30 degrees on the sky – the effect of the systematics asymptotes to a smaller value. The minimum in the FoM1/2 and w curves around 1  6 is due to the transition from the dominance of the additive errors at larger scales to multiplicative errors at smaller scales. Since calibration errors at large angular scales are the most damaging, it is sufficient to consider only those, and our choice of the maximum multipole of where the error enters, 1 ≤ calib,max = 20 is therefore sufficient. ", "Figure (<>)4 is similar to Fig. (<>)3 except now we show the effects when the magnitude error is split equally among all multipoles less or equal to 1,max (instead of all of it being lumped in a single multipole as in Fig. (<>)3). Given that the effect of the systematic error decreases with 1, the biases in the cosmological parameters are larger than in the previous figure. Qualitatively, the two figures paint a consistent picture of the potentially deleterious effects of the calibration variations even at a level corresponding to O(0.001–0.01) magnitudes. " ], "C. Example I: Corrections to dust maps ": [ "We now study a specific scenario of calibration systematics: corrections to the SFD [(<>)21] dust extinction maps. Dust in our Galaxy causes extinction, which in turn alters the observed galaxy fluctuations across the sky. While the Galactic dust has been mapped out reasonably accurately, we ask how accurately it needs to be mapped out in order not to bias the cosmological parameters. ", "To start, we need a model for the variations in the SFD map. We adopt results from the work of Peek and Graves [(<>)25] (hereafter PG10) who used ’standard crayons’ – objects of known color – in the SDSS to correct the SFD maps over the north galactic cap region (for a related work, see Schlafly and Finkbeiner [(<>)22]). Schematically, therefore ", "The SFD map is shown in the top left panel of Fig (<>)5, while the PG10 correction is displayed in the top right panel. To convert this EB−V map to δN/N fluctuations (see Eq. ((<>)33)), we assume DES observations in the i-band, for which R = 1.595 [(<>)22]. ", "The bottom left panel in Fig. (<>)5 shows the angular power spectrum extracted using Polspice software package [(<>)26]. As explained in Appendix (<>)B at length, the most reliable way of modeling the calibration errors was to first extract the power from the map, then generate a full-sky realization consistent with that power (using the isynfast routine in HEALPix). We explicitly verified the intuitive expectation that the results do not depend much on the realization. ", "The bottom right of Fig. (<>)5 shows the resulting biases in fNL and the square root of the dark energy FoM, as a function of the faint-end slope of the luminosity function s(z); the w = const case is not shown here or in the following Figure since it gives very similar results as the FoM1/2 . We assume calibration variations are given by the PG10 corrections, and that s(z) is constant in redshift (we nevertheless allow for the redshift-dependent s(z) in all equations). As mentioned around Eq. ((<>)31), the biases are very sensitive to s(z), scaling very nearly as s(z)2 . This can be easily understood: in Sec. (<>)II C we mentioned that the bias in the power spectrum is dominated by the added calibration power ∝ |cm|2 , while the (real-space) calibration field is linear in s(z) (Eq. (<>)32); hence ", "(36) ", "In other words, in the case where the additive calibration errors dominate so that calibration simply adds power (|cm|2 term), the biases in the cosmological parameters are proportional to this added power, and hence to the square of the faint-end slope of the luminosity function. Therefore, the faint-end slope of the luminosity function is a key factor relating the photometric magnitude variations to the cosmological parameter biases. Steep faint-end slopes will lead to particularly stringent requirements on our understanding of the large-angle photometric variations in the survey. Regardless of the value of the faint-end slope, however, the bottom right panel of Fig. (<>)5 shows that the effects of the imperfectly estimated Galactic dust on the cosmological parameters can be very significant. " ], "D. Example II: variability of survey depth ": [ "Our second example is based on the expectations of photometric depth variations of the Dark Energy survey. We use a map (Jim Annis, private communication) simulating observations over 525 night of observation spread over five years; see the top panel of Fig. (<>)6. The observing conditions on the site are based on historical atmospheric data of the CTIO site between 2005 and 2010. The tiling strategy uses multiple massive overlaps to generate a survey that is as homogeneous as possible. Each part of the sky is imaged ten times in each of the five DES fil-ters (grizY). For simplicity, we only focus on the i-band survey-depth map. ", "The effect of the unaccounted-for variability in the survey depth is the same as that of the photometric calibration error. However, the variability is large and expected to be taken into account; therefore, we (arbitrarily) adopt the final calibration error to be equal to one-tenth of the depth-variation map (i.e. 1/10 of its amplitude shown in the top panel of Fig. (<>)6). In other words, we assume ", "FIG. 5. Top left: Schlegel et al. [(<>)21] SFD extinction map, EB−V (nˆ), in Galactic coordinates with the 10 degrees Galactic plane cut. Top right: corrections to the SFD map from the work of Peek and Graves [(<>)25]. Bottom left: angular power spectrum of the PG10 map extracted by Polspice and shown without the usual ( + 1)/(2π) term so that the relative contribution of different multipoles can be more easily seen. Bottom right: bias/error ratios for fNL and the square root of the DETF FoM assuming PG10 map represents the calibration error, as a function of the faint-end slope of the luminosity function s. Note that the biases increase very sharply with s, roughly scaling as s 2 . The desired bias/error limit (horizontal dashed line) is exceeded already for s  0.3 for fNL and s  0.8 for the dark energy equation of state. ", "We follow the same procedure as with the dust example above, and calculate the power spectrum of the depth variability map using Polspice; see the bottom left panel of Fig. (<>)6. The variability of the survey depth will of course be accounted for in the data analysis – if it were not, it would lead to large biases in cosmological parameter estimates (as we easily verified using our formalism). The question is, then, to what accuracy do these variations need to be understood? ", "We answer that question by plotting, in the bottom right panel of Fig. (<>)6, the bias in the (square root of the) DE FoM, and non-Gaussianity parameter fNL, as a function of the faint-end slope of the luminosity function s(z). As in the previous example of the corrections to the SFD dust maps, we find that the biases in the cosmological parameters are significant, and that they strongly depend on the faint-end slope of the luminosity function. In fact, even assuming that only 10% of the variability in the survey depth is the “calibration error” – the case shown in the bottom right panel of the Figure – the bias/error ", "ratios are still large if s(z)  O(1). " ], "V. CONCLUSIONS ": [ "In this paper we made a first fully general study of the effect of the photometric calibration variations on the measured galaxy clustering angular power spectra. We derived a general formula for how a calibration variation with arbitrary spatial dependence affects the measured galaxy angular power spectrum. We illustrated the results assuming the standard set of cosmological parameters (including fNL), DES-type dataset with five tomographic bins out to zmax = 1, and two specific examples of real-world photometric calibrations. We now summarize our findings. ", "Photometric variations modulate the observed angular distribution of galaxy counts according to Eq. ((<>)2). This modulation translates into additive and multiplicative changes to the observed density fluctuation field, cf. ", "FIG. 6. Top panel: i-band magnitude limits estimated for the upcoming observations of the Dark Energy Camera at CTIO as a function of angular position. The pattern of variations in the magnitude limits are set by the variations in the observing conditions and the survey tiling strategy over the five years of the survey. Bottom left: angular power spectrum of the magnitude limi map, extracted using Polspice and shown without the usual (+1)/(2π) term so that the relative contribution of different multipoles can be more easily seen. Bottom right: biases in the cosmological parameters vs. the faint-end slope of the luminosity function s(z) assuming calibration error maps is consistent with a fixed fraction of 10% of amplitude (or 1% of power) of the magnitude-limit map shown in the top (bottom left) panel. The desired bias/error limit (horizontal dashed line) is exceeded for s(z)  1. ", "Eqs. ((<>)6) and ((<>)11), which in turn generate additive and multiplicative changes to the observed power spectrum. ", "As shown in Eq. ((<>)13), photometric variations across the survey masquerade as apparent violations of statistical isotropy. Hence, explicit tests of statistical isotropy could provide a useful way to identify unaccounted-for variations in the photometry. In this paper, we focused on the effects in the angle-averaged power-spectrum, cf. Eq. ((<>)16). We found that large-angle modulations of power (dipole, quadrupole, etc), are particularly damaging to cosmological analysis. We demonstrate this explicitly (cf. Eq. ((<>)29) and Fig. (<>)2) for the case where the variance in the photometric calibration error field is concentrated in one multipole 1 at a time. Note that the spatially uniform photometric decrement or increment across the sky (i.e. the monopole, 1 = 0) is unobservable since it only affects the mean number of galaxies in the survey. ", "Specializing in the angle-averaged power spectrum as done in Eq. ((<>)16), one can explicitly show that largest-angle fluctuations are dominant (for a fixed induced vari-", "ance on the calibration error field c(nˆ)); see Fig. (<>)2. Moreover, highest-redshift clustering measurements are most susceptible to the photometric variations, essentially because their angular power is the smallest and is therefore most affected by the photometric variation. ", "Less obviously, we find that the additive errors (e.g. term proportional to |cm|2 in Eq. ((<>)16)) are typically dominant over the multiplicative biases (terms proportional to the coefficients U) for all redshift bins and at large angular scales. The reason is simple: because they couple different multipoles, multiplicative terms are suppressed relative to the additive ones by the fiducial angular power spectrum C factor; see the term with C2 in Eq. ((<>)16). Since C  1 even at low-z (and all ), the additive terms dominate the error budget if all  modes are used in the analysis. However, at slightly smaller angular scales (  10) the multiplicative error terms dominate the error budget and can significantly bias the cosmological constraints, as discussed in Sec. (<>)IV. Therefore, it is important to include both multiplicative and additive aspects of the calibration error to accurately model biases ", "in cosmological analyses. ", "The photometric variation calibration errors affect the galaxy clustering signal at large spatial scales, and lead to biases in the inferred cosmological parameters. Parameters describing dark energy and, especially, primordial non-Gaussianity are particularly affected since they imprint signatures in the clustering of galaxies precisely at these large scales. Figures (<>)3 and (<>)4, the principal plots in this paper, show these cosmological parameter biases for a fixed contribution of the calibration error at each multipole separately and for a range of multipoles, respectively. In Sec. (<>)IV we further give two specific real-world examples of what the photometric variations could occur: errors in mapping the dust in our Galaxy, and variations in survey depth. We find that these calibration errors lead to potentially large cosmological biases, especially if the faint-end slope of the luminosity function s = d log N/dm|mmax is steep. In particular, in the Fisher matrix approximation, the cosmological parameter biases scale as s2 . ", "As a by-product of this work, we developed a reasonably fast algorithm, and provide a code 8 (<>), to calculate the full biased angular power spectrum of galaxies given an arbitrary photometric calibration variation map on the sky. As mentioned above, this calculation would be rather trivial if we could assume that the power spectrum errors are purely additive; however we demonstrated that multiplicative errors are important and may in fact dominate, necessitating the more numerically intensive calculation for surveys of interest. ", "From our analyses, it appears that the total rms of the calibration error has to be kept at the level at somewhere between 0.001 and 0.01 magnitudes, depending on how large a scale one wants to consider in order to maximize extraction of cosmological information, in order not to bias the cosmological parameters appreciably. This is a very stringent requirement! Achieving it, however, may not be as difficult as it sounds given that this is the time-averaged error to be tolerated at the end of the whole survey. Moreover, there are several other tools that we have at our disposal that we did not consider in this preliminary work. For example, one could use the survey itself to internally determine (“self-calibrate”) the photometric-variation errors, similarly to what weak lensing, cluster count, or type Ia supernova surveys are doing or planning to do with their systematic errors. One could also use measurements of the higher-point correlation functions to help determine these nuisance parameters (in our language, the c1m1 ). We leave these promising avenues for future study. " ], "ACKNOWLEDGEMENTS ": [ "We thank Jim Annis for providing the expectations of photometric depth variations for the DES, and Joshua Frieman, Enrique Gaztanaga, Andrew Hearin, Shirley Ho, Will Percival, Ashley Ross, Eddie Schlafly and Daniel Shafer for useful discussions. We also thank the anonymous referee for a number of useful suggestions. CC and DH are supported by the DOE OJI grant under contract DE-FG02-95ER40899. CC is also supported by a Kavli Fellowship at Stanford University. DH is additionally supported by NSF under contract AST-0807564, and NASA under contract NNX09AC89G. WF is supported by NASA under contract NNX12AC99G. " ] }