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PSF calibration requirements for dark energy from cosmic shear


Astronomy & Astrophysics manuscript no. 9150 April 28, 2008

c ESO 2008

PSF calibration requirements for dark energy from cosmic shear
S. Paulin-Henriksson1 , A. Amara1 , L. Voigt2 , A. Refregier1 and S.L. Bridle2

arXiv:0711.4886v2 [astro-ph] 28 Apr 2008

1 2

Service d’Astrophysique, CEA Saclay, Batiment 709, 91191 Gif–sur–Yvette Cedex, France Department of Physics & Astronomy, University College London, London, WC1E 6BT, U.K.

Received 26 November 2007; accepted 4 April 2008 ABSTRACT Context. The control of systematic e?ects when measuring background galaxy shapes is one of the main challenges for cosmic shear analyses. Aims. Study the fundamental limitations on shear accuracy due to the measurement of the point spread function (PSF) from the ?nite number of stars that are available. We translate the accuracy required for cosmological parameter estimation to the minimum number of stars over which the PSF must be calibrated. Methods. We characterise the error made in the shear arising from errors on the PSF. We consider di?erent PSF models, from a simple elliptical gaussian to various shapelet parametrisations. First we derive our results analytically in the case of in?nitely small pixels (i.e. in?nitely high resolution), then image simulations are used to validate these results and investigate the e?ect of ?nite pixel size in the case of the elliptical gaussian PSF. Results. Our results are expressed in terms of the minimum number of stars required to calibrate the PSF in order to ensure that systematic errors are smaller than statistical errors when estimating the cosmological parameters. On scales smaller than the area containing this minimum number of stars, there is not enough information to model the PSF. This means that these small scales should not be used to constrain cosmology unless the instrument and the observing strategy are optimised to make this variability extremely small. The minimum number of stars varies with the square of the star Signal-to-Noise Ratio, with the complexity of the PSF and with the pixel size. In the case of an elliptical gaussian PSF and in the absence of dithering, 2 pixels per PSF full width at half maximum (FWHM) implies a 20% increase of the minimum number of stars compared to the ideal case of in?nitely small pixels; 0.9 pixels per PSF FWHM implies a factor 100 increase. Conclusions. In the case of a good resolution and a typical Signal-to-Noise Ratio distribution of stars, we ?nd that current surveys need the PSF to be calibrated over a few stars, which may explain residual systematics on scales smaller than a few arcmins. Future all-sky cosmic shear surveys require the PSF to be calibrated over a region containing about 50 stars. Due to the simplicity of our models these results should be interpreted as optimistic and therefore provide a measure of a systematic ‘?oor’ intrinsic to shape measurements. Key words. Gravitational lensing - Cosmology: dark matter - Cosmology: cosmological parameters

1. Introduction
Gravitational lensing by large scale structure (or ‘cosmic shear’) has grown rapidly as a research ?eld over the last decade (Refregier 2003b; Hoekstra 2003; Munshi et al. 2006). In the coming years, we can expect that the ?eld will continue to make important contributions to cosmology. For instance, Albrecht et al. (2006) and Peacock & Schneider (2006) have singled out cosmic shear as potentially the most powerful probe for constraining dark energy. A great deal of work is currently under way to develop techniques that will allow the maximum possible potential to be reached and not limited by systematic measurement errors. This can be done by designing and building in-

struments with weak lensing as the central science driver. This approach leads to a top-down mission design process which begins with science requirements, which are translated into technical requirements and then into an instrument design. This method for tackling the problem of high accuracy weak lensing measurements is fundamentally di?erent to the bottom-up approach that is currently used. In the latter approach an existing telescope is used and the science team is assigned the task of extracting the maximum possible information from their surveys (until they hit the systematic limit of the instrument). The ?eld of gravitational lensing lends itself most naturally to top-down design, since most (but not all) of the systematic errors are not astronomical but are as-

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Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear

sociated with the instrument and the atmosphere. These can therefore be controlled through instrument design and an optimised survey strategy. For this reason, many of the ambitious future imaging surveys that are currently under development have placed weak lensing as primary science driver, including the Dark UNiverse Explorer1 (DUNE), the SuperNovae Acceleration Probe2 (SNAP), the Panoramic Survey Telescope & Rapid Response System3 (Pan-STARRS), the Dark Energy Survey4 (DES) and the Large Synoptic Survey Telescope5 (LSST). The issue is then to establish the instrumental requirements needed to reach the full statistical potential of the survey. In Amara & Refregier (2007a), a wide range of survey parameters, such as area and depth, were considered. Their impact on the statistical potential of a lensing survey was calculated and summarised in a scaling relation that can be used for survey designs to trade-o? one property of a survey against another. In brief, this work ?nds that, once the median redshift of a survey is su?ciently high (z 0.7), the optimal survey strategy is to make the lensing survey as wide as possible. Similar results have been found by Heavens et al. (2006). In the same spirit, Amara & Refregier (2007b) looked at the requirements an ultra-wide ?eld survey places on the control of systematics and concluded with a set of scaling relations that show the tolerance on residual systematic errors as a function of survey parameters. In particular, two types of shape measurement systematics were considered: multiplicative and additive systematics which are, respectively, correlated and uncorrelated with the lensing signal. Huterer et al. (2006) have also studied the impact of multiplicative and additive errors. In the present study, we link the above systematic requirements to errors associated with measurement of the point spread function (PSF) of the instrument. Indeed, since galaxy images need to be PSF-corrected before shapes can be measured, errors in the estimation of the PSF are propagated into errors on galaxy shapes and can mimic the shear signal. The PSF calibration is driven by 3 factors: 1. The PSF model: The model that is chosen to describe the PSF will never perfectly describe the true PSF hence the choice of PSF model introduces an ‘a priori’ systematic; 2. The interpolation scheme: The PSF will vary over the ?eld which means that the PSF at a galaxy position needs to be interpolated from the PSFs of the nearby stars; 3. The ?nite information available from each star: A star provides an image of the PSF that is noisy and pixelated. Thus, the PSF information that we are able to extract from each star is ?nite. As a result, to reach
1 2 3 4 5

high precision, it is necessary to combine the information from several stars. This paper is devoted to the last factor. We quantify the accuracy of the PSF calibration with analytical predictions in the case of in?nitely small pixels and use image simulations to quantify the pixelation e?ects. Previous work has also studied the impact that PSF errors are likely to induce in cosmic shear measurements. For instance, Stabenau et al. (2007) looked speci?cally at the PSF from the current SNAP design and translated this into systematic errors. The study that we present here sets out to be more general and uses simpli?ed PSFs to try and quantify the systematic ?oor of an instrument. We investigate the extent to which the pixel scale will degrade the amount of information relative to our analytic predictions but we do not explore the optimal methods for combining multiple exposures, such as the one proposed by Jain et al. (2006). High et al. (2007) have studied the impact of the pixel scale on the number of useful galaxies (assuming the PSF is known perfectly). While they ?nd a weak dependence, we expect that the driving factor for pixel scale is shape measurement systematics rather than statistical errors. For a survey with a given statistical potential we use the results of Amara & Refregier (2007b) to ?nd the upper limit on the systematic errors in the shears such that the induced bias in cosmological parameters remain subdominant compared to the statistical (marginalised Fisher matrix) errors when estimating cosmological parameters. We then convert the requirements on shear systematics into the minimum number of stars (N? ) needed to measure the PSF to the level of accuracy required. On scales smaller than the area containing N? , there is not enough information from the stars to calibrate the PSF. Therefore, the systematic errors in the cosmological parameters can be dominant over the statistical ones. This means that these small scales should not be used to constrain cosmology unless the PSF variability is known to be extremely small. N? clearly depends on the Signal-to-Noise Ratio distribution of stars, since bright ones contain more information about the PSF than faint noisy ones. It also depends on the stability and the complexity of the PSF, characterised by the number of degrees of freedom that must be estimated or interpolated from the stars. Each degree of freedom increases N? . Figure 1 shows a graphical example to illustrate the calibration requirement we are considering. We see a central galaxy surrounded by several stars. The grey shaded region shows the area over which the PSF must be calibrated and in this example contains 11 stars. Stars do not provide enough information to model the PSF variations on scales smaller than this area. This paper is organised as follows. Section 2 explains the weak lensing and cosmological context and summarises the issue of accurate PSF calibration. Section 3 shows the analytical predictions of the PSF calibration accuracy, expected in the case of in?nitely small pixels. Section 4 de-

http://www.dune-mission.net http://snap.lbl.gov/ http://pan-starrs.ifa.hawaii.edu http://www.darkenergysurvey.org http://www.lsst.org

Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear

3

the centroid xcen is given by the ?rst order moments divided by the total ?ux: xcen = i 1 Fi = (0) F (0) F
(1)

d2 x xi f (x)

(2)

and the quadrupole moment matrix is given by the second order moments divided by the total ?ux: Qij = Fij
(2)

F (0)

=

1 F (0)

cen d2 x (xi ? xcen i )(xj ? xj ) f (x) .

(3) The square rms radius R2 and the 2 component ellipticity ? = [?1 , ?2 ] are de?ned by: R2 ≡ Q11 + Q22 , 2Q12 Q11 ? Q22 , ?2 ≡ , ?1 ≡ Q11 + Q22 Q11 + Q22 ? ≡
2 ?2 1 + ?2 .

(4) (5) (6)

Fig. 1. Illustration of the required number of stars N? characterised in this paper. When measuring the shape of a galaxy (marked by a red spiral), the PSF needs to be calibrated with at least N? nearby stars (black asterisks) contained in the shaded region. In this example N? = 11. On scales smaller than this there is not enough information coming from the stars to measure the PSF variations.

Consider now we have unbiased estimators of R2 and ? with variations δR2 and δ? around the true values. We adopt the following de?nitions: δR2
2 2

≡ σ 2 [R2 ] , ≡ 2σ 2 [?] .

(7) (8)

scribes the simulations we use to validate these predictions and extend them to ?nite pixels. In section 5 we give the ?nal accuracy of the PSF calibration and derive the number of stars it requires.

|δ?|

2. Weak lensing 2.1. Shear Measurement
Weak gravitational shear is locally estimated using the shapes of background galaxies (for a review, see e.g. Bartelmann & Schneider 2001). Shear estimation methods can be divided into two families: those computing an estimator of the shear from a set of weighted sums over pixel values, for instance the common ‘KSB+’ method (Kaiser et al. 1995; Luppino & Kaiser 1997; Hoekstra et al. 1998), and those ?tting a model to the observed galaxy shape and deriving an estimator from the model. This includes the methods proposed by Kuijken (1999); Bernstein & Jarvis (2002); Refregier et al. (2002) and Voigt & Bridle (2008). In this paper, we need a general formalism to propagate the error on the PSF into an error on the shear estimate. For this purpose, we consider as shape parameters the 2 component ellipticity ? and the squared radius R2 , both de?ned using the unweighted second order moments of the galaxy. For an object with surface brightness f (x1 , x2 ), the total ?ux F (0) is the zeroth order moment of the surface brightness: F (0) = d2 x f (x) , (1)

The factor 2 in equation 8 is due to the fact that ? has 2 components and we de?ne σ [?PSF ] as the standard deviation of one of the components. The weak gravitational shear γ = [γ1 , γ2 ] can then be shown to be estimated using: γ = (P γ )?1 ?gal , where P γ = 2? < |?gal | >≈ 1.84 .
2

(9)

(10)

The symbol ‘ ’ indicates an estimator so that < γ >= γ , Pγ is the shear susceptibility and the subscript ‘gal’ corresponds to values measured on a galaxy. The value of < |?gal | > ≈ 0.4 comes from the typical ellipticity distribution in current data sets. In practice, the measurement of the galaxy ellipticity is uncertain because of the noise in the image. We write the induced error as δ?noise , which has a null average < δ?noise >= 0. There may also be a systematic e?ect δ?sys so that the estimated ellipticity is ?gal = ?gal + δ?sys + δ?noise . (11)
2

In practice δ?sys may be due to several factors. For instance, the STEP collaboration (Heymans et al. 2006; Massey et al. 2007; Rhodes et al. 2008) investigate the contribution to δ?sys from imperfections in the shear measurement methods. In a di?erent spirit, this paper focusses on the contribution to δ?sys from the limited information available on the PSF due to the photon noise in the star

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Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear
2

2 tity σsys examined in Amara & Refregier (2007b), which indicates the variance of the systematic errors 2 σsys ≡ (P γ )?2 |δ?sys |2

images. We identify the variance |δ?sys |

with the quan-

.

(12)

2 We translate an upper limit on σsys into constraints on the number stars that are needed to calibrate the PSF.

2.2. Systematics from PSF calibration
Since a galaxy image needs to be PSF corrected before its shape can be measured, errors in the estimation of the PSF propagate into an error δ?sys in the measured ellipticity. Here, we consider errors in the PSF radius parameter RPSF and in the 2 component PSF ellipticity ?PSF . To ?rst order and for the unweighted moments (de?nitions given by equations 3 to 6), we have (see Appendix A for details): δ?sys
2 δ RPSF ? ? (?gal ? ?PSF ) 2 Rgal

This equation con?rms the intuition that the PSF ellipticity of a cosmic shear survey should be small. We see that 2 the last ellipticity term inside the brackets is |?PSF | and should be reduced when optimising the survey to have ?PSF as small as possible. 2 For instance, to reach the requirements of σsys 10?7 , with a typical well sampled cosmic shear survey with |?gal |2 = 0.16, Rgal ≥ 1.5 RPSF and ?PSF 0.05, would require:
2 σ [RPSF ] 2 RPSF

10?3 , 10?3 .

(16) (17)

σ [?PSF ]

In the following, we discuss how these upper limits translate into requirements on the PSF calibration.

3. Analytical model
In this section, we consider the general problem of ?tting a 2D model of the PSF to a pixelated image of a star with additive uncorrelated gaussian noise using χ2 minimisation. This allows us to derive a number of analytic results in the limit of in?nitely small pixels (i.e. in?nitely high resolution), which will serve as a useful comparison base for the numerical simulation studies presented in section 4.

RPSF Rgal

2

δ?PSF (13)

where Rgal and RPSF are the radius parameters of the galaxy and the PSF respectively. The ?rst two terms show that systematics due to an error on the PSF size are proportional to the ellipticities of the galaxy and of the PSF. In the following, we show that the latter should be optimised to be as small as possible. The last term shows that systematics due to an error on the PSF ellipticity are proportional to the squared ratio between the PSF and galaxy sizes. Combining equations 12 and 13 gives the propagation 2 of PSF errors into σsys . For this purpose we make the following simpli?cations: 1. The galaxy is not correlated with the PSF (i.e. the crossed-terms < ?gal .?PSF > and < ?gal .δ?PSF > are equal to 0). 2. The error on the PSF ellipticity (δ?PSF ) and the PSF ellipticity itself (?PSF ) are not correlated. This is warranted by the fact that, in the assumed small ellipticity regime, δ?PSF does not have any preferred direction, implying < ?PSF .δ?PSF >= 0. 3. We assume that the ellipticity and the inverse squared radius of the galaxy are also uncorrelated. More exactly we assume: ?gal 2 Rgal
2

3.1. General 2D ?t
Consider the PSF surface brightness as a function of position on the image x to be described by a model m(x; p) parameterised by parameters p. The observed surface brightness of a star is f (x) = m(x; p(PSF)) + n(x) where p(PSF) is the true values of the parameters, n(x) is the noise, which is assumed to be uncorrelated (from pixel to pixel) and gaussian with < n(x) >= 0 and 2 < n(x)2 >= σn is assumed constant across the image. The 2 usual χ -functional is given by χ2 (p) =
k ?2 σn [f (xk ) ? m(xk ; p)] , 2

(18)

? |?gal |

2

2 1/Rgal

2

.

(14)

This assumption is reasonable for our work on the PSF calibration presented in this paper. With these simpli?cations, we obtain:
2 σsys = (P γ )?2 4

the sum being over all pixels k in the image. The usual ? is constructed by requiring that dχ2 /dp = 0 estimator p ? . The covariance matrix of the estiwhen evaluated at p mated parameters is given by the inverse of the Fisher matrix: cov[pi , pj ] ? (F ?1 )ij (19) with
?2 Fij = σn k

RPSF Rgal
2

?m(xk ; p) ?m(xk ; p) ?pi ?pj

(20)

× 2σ 2 [?PSF ]
2 2 σ [RPSF ] 2 RPSF 2

+

|?gal |

+ |?PSF |

.

(15)

and the variance of a function P (p) of the ?tted parameters is: ?P ?P cov[pi , pj ] . (21) σ 2 [P ] ? ?p i ?pj i,j

Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear

5

In the following, we consider 2 such functions of the parameters p: the rms radius squared R2 and the 2 component ellipticity ? (as de?ned in equations 4 to 6), in the case of a simple elliptical gaussian PSF (section 3.2) and in the case of more complex PSFs de?ned with shapelets (section 3.3). We de?ne the associated dimensionless complexity factors ψR2 and ψ? such as: R2 ψR2 , S 1 ψ? σ [?] ≡ S where S is the Signal-to-Noise Ratio de?ned as: σ [R2 ] ≡ S= F . σ [F (0) ]
(0)

This parametrisation is particularly convenient because for in?nitely small pixels the Fisher matrix of the parameters, given by equation 20, is diagonal with diagonal elements: Fii = S 2
2 2 1 1 2 1 1 (a2 1 ? a2 ) , , , , , 2 2 2 a2 a2 a2 2 a2 1 a2 A 1 2 1 a2

.

(33)

(22) (23)

From equations 31, 32 and 34, it follows that: (24) σ [R2 ] = 2 4 a4 1 + a2 , S √ 2 a2 1 a2 4 2 . σ [?] = R4 S

Consequently, with equation 21, the errors on the major and minor axes are: ai . (34) σ [ai ] = S (35) (36)

σ [F (0) ] is the standard deviation of the total ?ux. Basically, ψR2 and ψ? characterise the numbers of degrees of freedom in the PSF model associated with R2 and ?. In the limit of in?nitely small pixels, the sum over pixels in equations 18 and 20 can be replaced by a continuous integral over the object. A number of analytical results can be derived. In section 3.2 we derive these for an elliptical gaussian PSF and in section 3.3 we study more complex PSFs described with shapelets.

3.2. Elliptical gaussian
We ?rst consider a 2D elliptical gaussian model parameterised as: 1 A exp ? (x ? xa )T Q?1 (x ? xa ) (25) , m(x; p) = √ 2 π a1 a2 2 p = (xa , A, a1 , a2 , α) (26) where a1 and a2 are the rms major and minor axes of the gaussian, respectively, A is a parameter which controls the amplitude, xa is the (true) centroid, and T stands for the transpose operator. The total ?ux (as de?ned in equation 1) is: √ (27) F (0) = A a1 a2 , the centroid (as de?ned in equation 2) is: xcen = xa i i a2 1 0 0 a2 2 (28) and the quadrupole moment (as de?ned in equation 3) is: Q = R(α)T R(α) (29)

We generalise this simple elliptical gaussian model to more complex PSFs in the following section and test these equations for ?nite pixels using simulations in section 4. Under the approximation of small ellipticity (? 0.1), the dimensionless complexity factors ψR2 and ψ? (de?ned by equations 22 and 23) are constant (i.e. do not depend on the object). Indeed, ? 0.1 implies a1 ? a2 and there√ 2 4 fore in equations 35 and 36, we have a4 1 + a2 ? R / 2 2 4 and 4a2 1 a2 ? R . This leads to: √ (37) ψR2 (gauss) = ψ? (gauss) = 2 .

3.3. Shapelet model
To explore a wide variety of possible PSFs we consider a basis set that allows for complexity. In this framework, the shapelet basis sets are particularly convenient because: (i) shapelets provide some orthonormal basis sets that have already been studied in number of publications (Refregier 2003a; Refregier & Bacon 2003; Massey & Refregier 2005), tested on the STEP2 and STEP3 simulated data (Massey et al. 2007; Rhodes et al. 2008) and used for the weak lensing analysis of the Canada-FranceHawai-Telescope-Legacy-Survey (CFHTLS) data set in the framework of a comparison with X-ray surveys (Berg? e et al. 2007); (ii) as we show in the following, in the case of simple objects the standard deviations of the squared radius σ [R2 ] and ellipticity σ [?] can be written as proportional to some complexity factors containing all the information about the basis. A shapelet basis set is characterised by 2 parameters: nmax , the maximum order of the functions in the basis and the scale parameter β . The surface brightness f (x) of an object is described by:
nmax n

where α is the position angle of the major axis counterclockwise from the x-axis and: cos α sin α R(α) = (30) ? sin α cos α is the rotation matrix which aligns the coordinate system with the major axis. The ellipticity and the squared radius (as de?ned in equations 4 to 6) are:
2 R 2 = a2 1 + a2 , a2 ? a2 ? = 1 2 2 . R

f (x) =
n=0 m=?n

fn,m χn,m (x, β )

(38)

(31) (32)

with χn,m the polar shapelet functions and fn,m the (complex) polar shapelet coe?cients of the object. The scale of

6

Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear

the oscillations described by the function χn,m is proportional to 1/(n + |m|). It is often convenient to impose a lower limit to the scales which are described. This corresponds to setting the coe?cients with |m| > nmax ? n to 0. This con?guration is called ‘diamond’ and is only de?ned for an even nmax . For simple objects with reasonable substructures and tails, the number of coe?cients required is small when R2 ? 2β 2 . This relation is exact for a circular gaussian represented with nmax = 0. We show in Appendix B that, under the approximation of small ellipticity (? 0.1) and for stars described with a judicious scale parameter where R2 = 2β 2 , the dimensionless complexity factors de?ned by equations 22 and 23 (σ [R2 ] ≡ R2 ψR2 /S and σ [?] ≡ ψ? /S ) depend only on the basis (i.e. not on the object itself) and are given by: ψR2 ? ψ? ? ψ? ? N (N + 1) , 3 N (N + 4) without diamond , 3 N (N ? 2) with diamond 3 (39) (40) (41)

4.1. Elliptical gaussian PSF
Figure 2 shows the measured standard deviations on R2 and ? as a function of S , for a range of pixel scales (from 0.9 to 3.9 pixels per PSF FWHM). The black diamond symbols correpond to the highest resolution simulations (with 3.9 pixels per PSF FWHM). These points are close to the predictions (black curve) which are calculated analytically for the idealised case with in?nitly small pixels (i.e. ini?nitly high resolution, see section 3.2). The upper and bottom panels show di?erent representations of the same curves: on upper panels the y-axis shows σ , while on the bottom panels, it shows the relative di?erences with the in?nite resolution case. As the pixel size increases (i.e. when the resolution decreases), the standard deviation increases monotonically and moves away from the analytical predictions. This shows that the analytic model can describe a high resolution case while simulations are required to take into account pixelation e?ects. Our results in this paper are based on analytical predictions and assume the optimistic case where pixelation is good enough not to degrade the results. From Figure 2 we see that, for the smallest pixel scale we consider (3.9 pixels per PSF FWHM), the results from the analytic calculations and the simulations agree to better than 10%. As the size of the pixels is increased to 0.9 pixels per PSF FWHM we see a dramatic increase of the standard deviation of a factor ? 10. Some of this pixel scale e?ect will be mitigated through the use of dithering, but more simulations are needed to fully quantify the extent to which this will help.

where N is the largest even integer lower than or equal to nmax . Figure 3 illustrates equations 22, 23 and 39 to 41 by showing σ [?] and σ [R2 ] as a function of nmax . ψ? and ψR2 are given in table 2 for 16 bases with nmax between 4 and 16 and for the elliptical gaussian model presented in the previous section.

4. Simulations
In this section, we use image simulations to test and validate the analytic predictions of the previous section. These simulations allow us to investigate the e?ects of pixelation in the case of an elliptical gaussian PSF. In this simple case we show that pixelation degrades the accuracy of the shape measurements and thus, if the pixel scale is increased, the number of stars needed to calibrate the PSF also increases. Star images are simulated for: (i) a simple elliptical gaussian PSF and (ii) complex PSFs derived from the STEP III simulations (Rhodes et al. 2008). In all cases the pixelated images are produced by distributing the star centroids randomly and uniformly over the central pixel. Additive gaussian noise is then added to each image and the model ?tting is done using χ2 minimisation (for the work presented here, we have not added Poisson noise, which arises from the object itself). gaussian stars are simulated at high resolution (each pixel is considered as the sum of 7 × 7 sub-pixels), while shapelet stars are simulated using the analytic pixel integration scheme proposed in the online IDL pipeline6 by Massey & Refregier (2005).
6

4.2. Shapelet PSF
We now turn our attention to the expected standard deviations of R2 and ? (predicted by equations 22 and 23) for more complicated PSFs. We still work under the setup that the model ?tted to the data can exactly describe the truth. With the notation used in section 3.1, this means that when we ?t the parameters p of a model m(x, p) there exists a solution p(PSF) that exactly describes the PSF. We model the PSF by decomposing PSF-D of STEP III (Rhodes et al. 2008) into shapelets. We create 16 PSF models with increasing levels of complexity by decomposing the STEP PSF into basis sets with increasing nmax values. The ?tted models are listed in table 2. This gives 16 slightly di?erent PSFs close to PSF-D (?i vary between -0.02 and 0.02) which are shown in ?gure 4. The e?ect of the level of complexity of the PSF model on the standard deviation of a measurement of the size and ellipticity of the PSF is found by generating ? 105 realisations in the interval 10 ≤ S ≤ 1000 for each of the 16 PSF shapelet basis sets and by ?tting with the corresponding shapelet model. We compare the analytical predictions given by equations 22, 23 and 39 to 41 with simulations in ?gure 3. We see they provide a good description, although they some-

http://www.astro.caltech.edu/erjm/shapelets

Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear

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Fig. 2. Solid lines: analytical predictions in the case of in?nitly small pixels for σ [?] and σ [R2 ] (see equations 35 and 36). Data points joined by straight lines: simulation with ?nite pixel scales. All these results are for an elliptical gaussian PSF with an ellipticity ? ? 0.1. Each data point corresponds to about 104 realisations. Black diamonds, red horizontal-vertical crosses, orange stars, green triangles, blue squares and purple diagonal crosses correspond to sizes a1 = 1.7, 1, 0.7, 0.6, 0.5 and 0.4 pixels respectively. This corresponds to about 3.9, 2.3, 1.6, 1.4, 1.2 and 0.9 pixels respectively per PSF FWHM. Each data point corresponds to about 104 realisations. On the upper panels the y-axis shows σ [R2 ]/R2 and σ? , while on the bottom panels it shows the relative di?erences as compared to the in?nite resolution case. The Signal-to-Noise Ratio is noted ‘SNR’ rather than S in the text. times underestimate σ [R2 ] and σ [?] by a few percent. The results are similar whether or not the diamond con?guration is used. We have de?ned an e?ective signal-to-noise of the stars Se? as the expected rms signal-to-noise given the maximum and minimum limiting values of S used for PSF calibration (Smax and Smin ) and the number density of stars per unit of S : dn/dS , in the data set:
Smax 2 Se?

5. Requirements
In this section, we estimate the number of stars N? required to estimate the PSF to su?cient accuracy for a given survey. First, each star is an independent realisation of the PSF. Thus the PSF size and ellipticity can be estimated by:
2 RPSF = 2 2 2 k Rk /σ [Rk ] = 2 2 k 1/σ [Rk ] 2 k ?k /σ [?k ] = 2 k 1/σ [?k ] 2 Sk 2 k Sk 2 k ?k Sk 2 k Sk 2 k Rk

=
Smin

dn dS dS

?1

Smax Smin

dn 2 S dS . dS

(46)

We then de?ne the e?ective number of stars N? by
2 N? Se? = 2 Sk

(47)

,

(42) (43)

k

?PSF =

where the sums are made on all the stars. These estimators are the minimum variance weighted averages if we assume the probability distribution in each of R2 and the two components of ? is a gaussian for each individual star. Second, variances are given by equations 22 and 23 and can be written: σ
2 2 [RPSF ]

where the sum is over the stars available around the galaxy. Therefore, the e?ective number of stars N? is equal to the actual number of stars if all the stars have the same S. Substituting into equation 15 gives the requirement on the number of stars needed to calibrate the PSF. If the PSF ellipticity is at most a few percent then we can 2 neglect the |?PSF | term leaving: N?
lim P γ σsys 2 Se? ?2

=

σ 2 [?PSF ] =

4 2 RPSF ψR 2 2 S ? ?i 2 ψ? 2 ≡ ? S?i

R4 ψ 2 2 ≡ PSF 2R N? Se? 2 ψ? 2 . N? Se?

,

(44) (45)

RPSF Rgal

4

2 2 |?gal |2 + 2ψ? ψR 2

(48) lim where σsys is the upper limit acceptable for σsys . The number of stars N? is a central requirement of cosmic shear

8

Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear

Fig. 3. Lines: analytical predictions in the case of in?nitely small pixels for σ [R2 ] and σ [?] (see equations 22 and 23). Data points: simulation with small pixels (i.e. 10 pixels per PSF FWHM). All these results are for stars imaged through the complex PSFs shown in ?gure 4 and with 10 < S < 1000. Each data point corresponds to about 1.5 × 104 realisations. Triangles and diamonds correspond to basis sets with and without the diamond con?guration respectively. For σ [R2 ] the solid line corresponds to the theoretical prediction (which does depend on the diamond option). For σ [?], solid and dashed lines correspond to theoretical predictions with and without the diamond option respectively. Without the diamond option, the complexity factors depend on the largest even integer lower than or equal to nmax (see equations 39 and 40), this is why theoretical predictions look like a ‘staircase’. On the other hand this behaviour does not appear with the diamond con?guration because the latter is de?ned only for even values of nmax . We see that analytical predictions provide a good description, although they sometimes underestimate σ [R2 ] and σ [?] by a few percent. The Signal-to-Noise Ratio is noted ‘SNR’ rather than S in the text. surveys: if we need to measure the PSF so that the systematic e?ects of the shear power spectrum stay below lim. σsys , we need to combine the information from at least N? stars. Thus, for a given survey, one needs to estimate N? in order to optimise the instrument and mission design. This is a requirement on the PSF stability. Indeed the star density must be taken into account to give the minimum area containing N? stars, over which the PSF must be stable. Low S stars do not contribute a great deal of information to the PSF calibration. For instance if the density of stars scales roughly as dn/dS (S ) ∝ 1/S , which is consistent with the simple star count model of Bahcall & Soneira (1980), equation 46 shows that the bright stars strongly dominate. In fact, not only do low S stars bring little information, they may also induce a bias in the PSF calibration. This bias will be studied in forthcoming work. For the moment, we avoid this regime by adopting an arbitrary lower limit of Smin = 100. On the other hand, the high S cut o? Smax and the star density distribution dn/dS , depend on the properties of the surveys. They depend on a number of factors including: the line of sight in the Milky Way, the instrumental con?guration and the observing strategy. Equation 48, therefore, can not be computed for a general case, but it can be simpli?ed and scaled to typical values, to give easily readable requirements on the star population in a data set. First, since the galaxy size distribution is steep, it is pessimistic but reasonable to approximate: Rgal RPSF
4



Rgal RPSF

4

(49)
min

where (Rgal /RPSF)min is the minimum value that this ratio can reach, typically about 1.5. Second, we note that the 2 variance |?gal | is around 0.16, as stated in section 2, thus for usual PSF models where the complexity factor of the squared size ψR2 is of the same order as the complexity factor of the ellipticity ψ? the expression inside the brack2 ets in equation 48 is driven by ψ? and we can neglect ψR2 . For example, for non-diamond shapelets with nmax = 4 or for diamond shapelets with nmax = 6, the ellipticity complexity factor is ψ? ? 3 (see table 2). Third, as mentioned in section 2, Amara & Refregier (2007b) ?nd for a DUNE-

Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear
lim. 2 like cosmic shear experiment (σsys ) = 10?7 . Fourth, we take for Se? a typical value of 500, which is roughly S of an AB magnitude 20 star when imaged with a DUNE-like telescope (1500 seconds in a broad RIZ band). With these central values, we obtain:

9

N?

Se? 50 500

?2

(Rgal /RPSF)min 1.5

?4

ψ? 3

2

lim 2 (σsys ) 10?7

?1

.

(50) Therefore the PSF of a DUNE-like survey will need to be stable over a region containing about 50 stars. The scaling relation given by equation 50 allows one to study di?erent survey con?gurations. This is illustrated by table 1 that gives N? for 6 typical con?gurations comparable to: the Canada-France-Hawai-Telescope Legacy Survey (CFHTLS), the Kilo-Degree Survey7 (KIDS), the Large Synoptic Survey Telescope (LSST), the SuperNovae Acceleration Probe (SNAP) and the Dark UNiverse Explorer (DUNE). To build this table, we have estimated the accuracy σ [w0 ] and σ [wa ] that can be achieved according to the surface covered, the median redshift zm and the galactic surface density ng in the case where systematics are lower than statistical errors and when ?tting the 7 cosmological parameters: ?m , ?b , σ8 , h, w0 , wa and n. This estimation is made through the scaling relations proposed by Amara & Refregier (2007a). We have lim. 2 then estimated the value of (σsys ) required to achieve this assumption of sub-dominant systematics, according to Amara & Refregier (2007b). Finally we have computed N? according to our scaling relation (eq. 50). 1. The two ?rst lines show surveys comparable to the current largest data sets optimised for cosmic shear. For instance the CFHTLS which currently covers 50 deg2 (Fu et al. 2007) and will eventually cover 170 deg2 . The relatively poor constraints got on w0 with such surveys (and the absence of constraint on wa ) illustrate the fact that cosmic shear surveys of the current generation do not constraint signi?cantly the dark energy. They rather aim to constraint the combination σ8 × ?α M. lim. However this does not change the requirement on σsys since Amara & Refregier (2007b) show that this requirement does not depend on the considered cosmological parameter. One can see that these surveys need to calibrate their PSF over few stars. Assuming a typical star density of ? 1.arcmin?2 at these magnitudes, this shows that at scales smaller than few arcmin the PSF correction may introduce signi?cant systematics. This is consistent with the results of Benjamin et al. (2007) who describe a joint analysis of 100 deg2 from several surveys and ?nd signi?cant B modes on these scales. 2. The third line corresponds to a cosmic shear survey of the next generation, such as KIDS/VIKING. 3. The fourth line shows an ambitious deep survey from the ground covering 15, 000 deg2, which is the largest
7 http://www.eso.org/sci/observing/policies/cSurveys/sciencecSurveys.html

cosmic shear survey that can be perfomed from Mauna Kea with a small airmass8. This illustrates one of the main limitation of doing cosmic shear from the ground: although the survey is rather deep, the galaxy surface density ng and the median redshift zm are limited. This is due to the fact that faint galaxies are too small: they have sizes comparable to the PSF and can not be included in a cosmic shear analysis. This is included in the scaling relation 50 through the minimum dilution factor (Rgal /RPSF )min . Note that it is not possible to change signi?cantly the reference value of (Rgal /RPSF)min = 1.5 adopted in equation 50 because it is at the power 4. For instance, a change from 1.5 to 1.2 to increase ng would also increase N? by a factor 2.5. Such a survey is similar to the LSST (Kahn & LSST Collaboration 2006) except that the LSST collaboration is developing a speci?c observing strategy that aims to drastically reduce ψ? . This strategy is based on stacking many very short exposures taken with di?erent orientations of the camera in order to circularise the PSF. Another e?ect of this strategy is that the e?ective Signal-to-Noise Ration Se? of stars may be much larger than 300, as assumed in table 1. But this strategy also has a fundamental limit: the more ψ? is reduced, the more ψR2 increases (for instance because of the astrometric errors while stacking the low Signal-to-Noise Ratio exposures). For such a LSST-like strategy, estimating N? through equation 50 is not relevant because in this equation we assume that ψ? and ψR2 are of the same order. N? must be estimated through equation 48. 4. The ?fth line corresponds to a deep space survey covering 1000 deg2. This is similar to SNAP as described by Massey et al. (2004) and Refregier et al. (2004). 5. The last line corresponds to a DUNE-like survey (full sky from space with a medium depth and a reasonable galaxy density) adopted as the reference in the scaling relation (eq. 50).

6. Conclusions
We have studied the PSF calibration requirements for cosmic shear to measure cosmological parameters. We connect the ?nite information that we are able to extract from stars to the statistical error of the PSF calibration, then to the error on the galaxy ellipticity estimation. We express our results in terms of the minimum number of stars (N? ) that are needed to calibrate the PSF in order to keep the systematic errors below statistical errors for cosmological
On one hand, half of the sky is available for cosmic shear, corresponding to ?20,000 deg2 . The other half is masked by the Milky Way. This is why one uses to call ‘full-sky’ the surveys covering 20,000 deg2 . On the other hand, in order to maintain the air mass in a reasonable range, we make the optimistic assumption that the minimum elevation of ground observations is 50 deg in local horizon coordinates. With this constraint, a telescope located at Mauna Kea (i.e. Hawaii) can cover about 15, 000 deg2 during the year
8

10

Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear surface (deg2 ) 50 170 1,500 15,000 1,000 20,000 zm 0.8 0.8 0.8 0.9 1.2 0.9 ng (.arcmin?2 ) 10 10 20 30 50 40 σ [w0 ] (σ [wa ]) 0.9 (nr) 0.5 (nr) 0.1 (0.6) 0.03 (0.1) 0.07 (0.3) 0.02 (0.1)
lim. 2 (σsys )

Ground surveys (Se? = 300) Space surveys (Se? = 500)

4 × 10?6 2 × 10?6 6 × 10?7 10?7 3 × 10?7 10?7

N? /(ψ? /3)2 3 6 25 104 15 50

Table 1. The required number of stars N? for PSF calibration of typical surveys, as predicted by our scaling relation (equation 50) in function of the e?ective Signal-to-Noise Ratio of stars Se? (de?ned in equation 46) and of the scienti?c lim 2 requirement given in terms of (σsys ) as de?ned by Amara & Refregier (2007b) (i.e. the upper limit on the variance of systematics to ensure that systematics are sub-dominant compared to statistical errors when ?tting cosmological parameters). The estimations of σ [w0 ] and σ [wa ] are made through the scaling relations proposed by Amara & Refregier (2007a). The mention ‘nr’ means that the constraint is ‘not relevant’. Corresponding surveys (similar to the current lim. 2 generation of cosmic shear surveys) do not constraint the dark energy but rather constraint σ8 . However, (σsys ) does not depend on the considered cosmological parameter and thus these poor constraints on w0 do contain all the lim. 2 information to compute (σsys ) . We consider that stars have an average Signal-to-Noise Ratio of 500 in space and 300 from the ground. We also consider a ?xed dilution factor (Rgal /RPSF )min of 1.5 for every survey. The bold line corresponds to the set-up of a DUNE-like survey (i.e. full-sky from space with medium depth and galaxy density) as taken as the reference in the scaling relation. Note that the surface of 20,000 deg2 (i.e. half of the sky) is the maximum surface of the sky relevant for cosmic shear. That is why it is often called ‘full-sky’. In the same spirit, 15,000 deg2 is the maximum surface of the sky (relevant for cosmic shear) that can be covered from Mauna Kea with a reasonable airmass.

parameter estimation. On scales smaller than the area containing N? stars there is not enough information coming from the stars to calibrate the PSF. Therefore, the systematic errors in the cosmological parameters may dominate over the statistical errors. This means that these small scales should not be used to constrain cosmology unless the variability is known to be extremely small. Our results show that for current cosmic shear surveys this scale is about an arcminute, which may explain residual systematics found on these scales in current analyses. In future all sky cosmic shear surveys, the data set will be increased by several orders of magnitude and a tight control of the PSF behaviour will be required to prevent this scale from being increased and to reach smaller scales. This places strong requirements on the hardware and observing strategy. For instance, for ground observations where the atmosphere prevents the PSF from being constant over several stars in a single exposure, Jarvis & Jain (2004); Jain et al. (2006) suggest the shear correlation functions should be calculated by cross-correlating galaxies in di?erent exposures. The PSF calibration error contribution would then be uncorrelated and averaged down. We also demonstrate how N? can depend on the complexity of the PSF. We de?ne a ‘complexity factor’ for two di?erent shapelet parametrisations and an elliptical gaussian parametrisation. This complexity factor is a number lower than the number of parameters used to describe the PSF and increases with the PSF complexity. There is a different complexity factor associated with each PSF shape parameter. For accurate galaxy shear measurements, we ?nd that the complexity factor for the PSF ellipticity is more important than that for the size. We summarise the dependence of N? as a function of complexity, star Signal-

to-Noise Ratio, galaxy size and cosmological requirements in a convenient scaling relation (see equation 50). Note that we work under the setup that the PSF can be perfectly described by the model, thus we do not include in our error budget the systematics due to a poor choice of the PSF model. We ?nd that pixelation degrades the PSF calibration accuracy. Our calculation of N? holds for the optimistic case of in?nitely small pixels (i.e. in?nitely high resolution). A ?nite pixel size increases N? . We have performed simulations to predict the increase in the case of an elliptical gaussian PSF. We show that for 2 pixels (or more) per FWHM, the rms scatter agrees with the standard deviation predicted for in?nitely small pixels to within 10%, which in turn can be translated into a 20% increase in the required number of calibration stars. For large pixels we see that the pixelation e?ect can be dramatic, for instance having 0.9 pixels per PSF FWHM would lead to a factor of 100 increase in the number of stars needed. This effect may be mitigated to some extent by using dithering, which is beyond the scope of this paper. The main results of this paper are for the high resolution case that provides a strict lower limit on N? .

Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear

11

Acknowledgments
We thank S? ebastien Boulade and collaborators at EADS/Astrium for stimulating discussions which helped initiate this study. We acknowledge Jo¨ el Berg? e for his very convenient on-line manual (http://www.astro.caltech.edu/?jberge/shapelets/manual) about the shapelets IDL pipeline, which Richard Massey makes available on http://www.astro.caltech.edu/?rjm/shapelets. A. Amara would like to thank Prof. K. S. Cheng and Dr. T. Harko of Hong Kong University for their hospitality. S. Bridle acknowledges support from the Royal Society in the form of a University Research Fellowship. L. Voigt acknowledges support from the UK Science and Technology Facilities Council.

References
Albrecht, A., Bernstein, G., Cahn, R., et al. 2006, ArXiv Astrophysics e-prints Amara, A. & Refregier, A. 2007a, MNRAS, 381, 1018 —. 2007b, ArXiv e-prints, 710 Bahcall, J. N. & Soneira, R. M. 1980, ApJ, 238, L17 Bartelmann, M. & Schneider, P. 2001, Phys. Rep., 340, 291 Benjamin, J., Heymans, C., Semboloni, E., et al. 2007, MNRAS, 381, 702 Berg? e, J., Pacaud, F., R? efr? egier, A., et al. 2007, ArXiv e-prints, 712 Bernstein, G. M. & Jarvis, M. 2002, AJ, 123, 583 Fu, L., Semboloni, E., Hoekstra, H., et al. 2007, ArXiv e-prints, 712 Heavens, A. F., Kitching, T. D., & Taylor, A. N. 2006, MNRAS, 373, 105 Heymans, C., VanWaerbeke, L., Bacon, D., et al. 2006, MNRAS, 139, 313 High, F. W., Rhodes, J., Massey, R., & Ellis, R. 2007, ArXiv Astrophysics e-prints Hoekstra, H. 2003, ArXiv Astrophysics e-prints Hoekstra, H., Franx, M., Kuijken, K., & Squires, G. 1998, ApJ, 504, 636 Huterer, D., Takada, M., Bernstein, G., & Jain, B. 2006, MNRAS, 366, 101 Jain, B., Jarvis, M., & Bernstein, G. 2006, Journal of Cosmology and Astro-Particle Physics, 2, 1 Jarvis, M. & Jain, B. 2004, ArXiv Astrophysics e-prints Kahn, S. & LSST Collaboration. 2006, in Bulletin of the American Astronomical Society, Vol. 38, Bulletin of the American Astronomical Society, 1020–+ Kaiser, N., Squires, G., & Broadhurst, T. 1995, ApJ, 449, 460 Kuijken, K. 1999, A&A, 352, 355 Luppino, G. A. & Kaiser, N. 1997, ApJ, 475, 20 Massey, R., Heymans, C., Berg? e, J., et al. 2007, MNRAS, 376, 13 Massey, R. & Refregier, A. 2005, MNRAS, 363, 197

Massey, R., Rhodes, J., Refregier, A., et al. 2004, AJ, 127, 3089 Munshi, D., Valageas, P., Van Waerbeke, L., & Heavens, A. 2006, ArXiv Astrophysics e-prints Peacock, J. & Schneider, P. 2006, The Messenger, 125, 48 Refregier, A. 2003a, MNRAS, 338, 35 —. 2003b, ARA&A, 41, 645 Refregier, A. & Bacon, D. 2003, MNRAS, 338, 48 Refregier, A., Chang, T.-C., & Bacon, D. J. 2002, in The shapes of galaxies and their dark halos, Proceedings of the Yale Cosmology Workshop ”The Shapes of Galaxies and Their Dark Matter Halos”, New Haven, Connecticut, USA, 28-30 May 2001. Edited by Priyamvada Natarajan. Singapore: World Scienti?c, 2002, ISBN 9810248482, p.29, ed. P. Natarajan, 29–+ Refregier, A., Massey, R., Rhodes, J., et al. 2004, AJ, 127, 3102 Rhodes, J., , , et al. 2008, in prep. Stabenau, H. F., Jain, B., Bernstein, G., & Lampton, M. 2007, ArXiv e-prints, 710 Voigt, L. & Bridle, S. L. 2008, in prep

Appendix A: Propagation of the PSF error
In this appendix we detail how we derive equations 13 and 15 by propagating the error on each measurement of the PSF ellipticity and size to the error on the galaxy ellipticity and then to σsys . The 2 component galaxy ellipticity ?gal can be written 2 2 in terms of radii Robs and RPSF and ellipticities ?obs and ?PSF of the observed image and the PSF: ? =
PSF obs PSF obs PSF obs ) ? F12 ; 2(F12 + F22 ? F22 ? F11 F11 obs PSF obs PSF F11 ? F11 + F22 ? F22 2 2 ?obs Robs ? ?PSF RPSF = . (A.1) 2 2 Robs ? RPSF

2 Consider now that we have estimators of ?PSF and RPSF and the estimated values have small errors δ?PSF and 2 δ RPSF with respect to the true values. We can ?nd the deviation from the truth of a single measurement of one ellipticity component δ?gal,i by di?erentiation:

δ?gal,i ≈

??gal,i ??gal,i 2 δ?PSF,i δRPSF + 2 ?RPSF ??PSF,i

(A.2)

which can be expanded to give equation 13 and when combined with equation 12: ? ? 2 2 ? ? 2 gal PSF 2 2 ? δ RPSF = ? |δ?sys | + 2 2 Rgal Rgal + RPSF Rgal
4

|δ?PSF |

2

(A.3)

which leads to equation 15 when assuming that the ellipticity and size of the galaxy are uncorrelated (equation 14) and using equations 7 and 8.

12

Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

Fig. 4. Non-gaussian PSFs used to study the e?ect of complexity in section 4.2 (10 pixels per PSF FWHM, logarithmic color scale, the normalisation is such as the total ?ux is 103 ). Each of them is described with a di?erent shapelet model, described in table 2.
label gauss a b c d e f g h i j k l m n o p nmax 4 5 6 7 8 9 10 11 12 4 6 8 10 12 14 16 diamond (yes or no) no no no no no no no no no yes yes yes yes yes yes yes Ncoef 15 21 28 36 45 55 66 78 91 9 16 25 36 49 64 81 Ncoef
m=2(0)

2(3) 2(3) 3(4) 3(4) 4(5) 4(5) 5(6) 5(6) 6(7) 1(3) 2(4) 3(5) 4(6) 5(7) 6(8) 7(9)

complexity factor Ψ? (ΨR2 ) √ √ 2( 2) 3.27(2.58) 3.27(2.58) 4.47(3.74) 4.47(3.74) 5.66(4.90) 5.66(4.90) 6.83(6.06) 6.83(6.06) 8.00(7.21) 1.63(2.58) 2.83(3.74) 4.00(4.90) 5.16(6.06) 6.32(7.21) 7.48(8.37) 8.64(9.52)

Table 2. Properties of the PSF models investigated in section 4.2. The ?rst one is the elliptical gaussian (labelled ‘gauss’). For each of the 16 other basis sets, we study one PSF (labelled a to p) shown in ?gure 4. From left to right, the columns are: nmax and diamond de?ned in section 3.3 and by Massey & Refregier (2005); the number Ncoef of m=2(0) coe?cients in the basis; the numbers Ncoef of coe?cients with m = 2 (and m = 0) in the basis and the complexity factors de?ned by equations 39 to 41.

Appendix B: Complexity factors for shapelet models
As stated in section 3.3, any object can be described by the sum of polar shapelet functions χn,m weighted by a set

of complex coe?cients fn,m (see equation 38). The scale parameter β and the center of the basis can be tuned to get a sparse description of the object. Most of the time, a good choice is to take β 2 = R2 /2 and the center of the basis corresponding to the centroid of the surface bright-

Paulin-Henriksson et al.: PSF calibration requirements for dark energy from cosmic shear

13

ness. Then, for simple objects like stars with reasonable substructures and tails, the required number of coe?cients is typically less than 20. The ‘diamond’ con?guration, de?ned for nmax even, consists in setting fn,m = 0 when |m| > nmax ? n. For given values of β and the center of the basis, equations 50, 54 and 55 of Massey & Refregier (2005) can be summarised by: √ F (0) = 4π β × S0 √ 16π β 3 2 R = S1 F (0) = 2β 2 × S1 /S0 (B.1) √ 3 16π β S2 ? = F (0) R2 = S2 /S1 (B.2) with:
nmax

1. We neglect the term proportinal to ?2 in equations B.9 and B.10. This is justi?ed by the ‘small ellipticity’ assumption adopted all along this paper. 2. We choose R2 = 2β 2 in equations B.6, B.7 and B.8.

S0 =
n even=0 nmax

fn0 (n + 1)fn0
n even=0 nmax

(B.3) (B.4) (B.5)

S1 = S2 =

n(n + 2)fn2 .
n even=2

From equations B.1, B.2 and 21, it follows that: σ [R2 ] = 1 × 2 β 2 × ψR2 S 2 β2 1 × 2 × ψ? σ [?] = S R
max 1 R2 (n + 1 ? 2 )2 N/2 + 1 n even=0 2β

(B.6) (B.7)

with:

n

ψR2 =

(B.8)

and, for the non-diamond con?guration:
2 ψ? =
max 1 n(n + 2) + ?2 (n + 1)2 N/2 + 1 n even=0

n

=

N (N + 4) + ?2 3

N (N + 4) + 1 3

(B.9)

or alternatively, if the diamond con?guration is used:
2 ψ? =
max max 1 (n + 1)2 [n(n + 2)] + ?2 N/2 + 1 n even=0 n even=0

n

?2

n

=

N (N ? 2) + ?2 3

N (N + 4) + 1 3

(B.10)

where N is the largest even integer lower than or equal to nmax . Note that σ [?] is the standard deviation of one component of ?, according to our de?nition given in equation 8. To obtain equations 39 to 41, we make two simpli?cations:


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