A Covariance Matrix Derivation

Cosmological Information in the Intrinsic Alignments of Luminous Red Galaxies

Abstract

The intrinsic alignments of galaxies are usually regarded as a contaminant to weak gravitational lensing observables. The alignment of Luminous Red Galaxies, detected unambiguously in observations from the Sloan Digital Sky Survey, can be reproduced by the linear tidal alignment model of Catelan, Kamionkowski & Blandford (2001) on large scales. In this work, we explore the cosmological information encoded in the intrinsic alignments of red galaxies. We make forecasts for the ability of current and future spectroscopic surveys to constrain local primordial non-Gaussianity and Baryon Acoustic Oscillations (BAO) in the cross-correlation function of intrinsic alignments and the galaxy density field. For the Baryon Oscillation Spectroscopic Survey, we find that the BAO signal in the intrinsic alignments is marginally significant with a signal-to-noise ratio of and with the current LOWZ and CMASS samples of galaxies, respectively, and increasing to and once the survey is completed. For the Dark Energy Spectroscopic Instrument and for a spectroscopic survey following the EUCLID redshift selection function, we find signal-to-noise ratios of and , respectively. Local type primordial non-Gaussianity, parametrized by , is only marginally significant in the intrinsic alignments signal with signal-to-noise ratios for the three surveys considered.

I Introduction

The intrinsic alignments of galaxies are correlations between their positions, shapes and orientations that arise due to physical processes during their formation and evolution as tracers of the large-scale structure of the Universe. Examples of such processes are stretching by the tidal field of the large-scale structure Catelan et al. (2001); Hirata and Seljak (2004, 2010) or clusters of galaxies Thompson (1976); Ciotti and Dutta (1994), the interaction between the angular momentum of a galaxy and the tidal torque Peebles (1969), or dynamics along preferred directions (Zhang et al., 2009, i.e., the filamentary distribution of galaxies).

The observed alignments of Luminous Red Galaxies (LRGs) Eisenstein et al. (2001) have been shown to be adequately reproduced by the linear alignment (LA) model of Catelan et al. (2001); Hirata and Seljak (2004, 2010) on large scales (Mpc) in the redshift range Okumura and Jing (2009); Blazek et al. (2011). On smaller scales, Hirata et al. (2007); Bridle and King (2007) have suggested replacing the linear matter power spectrum by its non-linear analog. The non-linear alignment (NLA) model is widely used, despite the simplicity of the non-linear treatment, which does not evolve the non-linear scales consistently with the Poisson equation. A slight modification to the NLA model was proposed by Kirk et al. (2012), although once again without fully solving for the non-linear dynamics. Joachimi et al. (2011) showed that the NLA model reproduces observations of the alignments of LRGs up to and on scales of comoving projected separations Mpc resorting to the MegaZ-LRG sample Collister et al. (2007); Abdalla et al. (2011). On smaller scales, significant progress has been made in developing halo models that reproduce the currently available observational constraints and extend the forecasts of the intrinsic alignment contamination to weak lensing observables to higher redshifts and to satellite galaxies Schneider and Bridle (2010); Joachimi et al. (2013a, b). (Currently, there is no evidence of alignments for blue galaxies Mandelbaum et al. (2011) and we do not consider them in this work.)

The intrinsic shapes and alignments of galaxies have been explored as a contaminant in weak gravitational lensing observables (Hirata and Seljak, 2004; Joachimi and Bridle, 2010) but little is known about the cosmological information they encode. In the upcoming decades, imaging surveys like the Dark Energy Survey1 (DES) The Dark Energy Survey Collaboration (2005), Hyper-Suprime Cam2 (HSC)Miyazaki et al. (2012), Pan-STARRS3, the Kilo-Degree Survey 4 (KiDS), EUCLID5 Laureijs et al. (2011), the Large Scale Synoptic Telescope6 (LSST) Ivezic et al. (2008) and WFIRST7 Spergel et al. (2013) will map the large-scale structure of the Universe and explore the nature of dark energy Weinberg et al. (2012) by measuring the clustering of galaxies and the effect of weak gravitational lensing Bartelmann and Schneider (2001) over unprecedented volumes.

In a complementary effort, ongoing spectroscopic surveys such as the Baryon Oscillation Spectroscopic Survey Dawson et al. (2013), and upcoming ones, such as SDSS-8, the Dark Energy Spectroscopic Instrument (DESI) Levi et al. (2013), EUCLID, WFIRST, and the Prime Focus Spectrograph (PFS) Takada et al. (2012), will gather spectra of millions of galaxies and quasars with the goal of measuring the distance scale probed by Baryon Acoustic Oscillations (BAO) and the growth of structure through redshift space distortions (RSD).

In the context of these rich datasets, no probe of large-scale structure should be left unexplored. Intrinsic alignments are one such probe: an effect that was once just thought to be a contaminant to weak gravitational lensing measurements, could become a complementary source of cosmological information. In the LA model, the tidal field determines the strength and evolution of the alignments, which depend on the growth function of the matter density perturbations and their power spectrum. In particular, the LA prediction is thus sensitive to RSD, primordial non-Gaussianity and BAO. Another example of the potential of alignments to constrain cosmology was proposed by Schmidt and Jeong (2012), who suggested that gravitational waves from inflation could be detected in the intrinsic alignments of galaxies.

In this work we assume that red galaxies (the ancestors of low redshift LRGs, and which we will henceforth also refer to as LRGs) follow the NLA model and we study the cosmological information contained in the galaxy density-intrinsic shear cross-correlation function. In Section II, we summarize the tidal alignment model, we construct the galaxy density-intrinsic shear cross-correlation and we discuss possible contaminants and uncertainties of the model. In Section II.4 we study the typical scales at which the tidal field has an impact on the alignment of a galaxy. In Section II.5 we explore an alternative to the NLA model in which the alignment between a galaxy and the tidal field of the large-scale structure occurs instantaneously. In Section III, we study different cosmological observables that can be constrained using the intrinsic ellipticity-galaxy density cross-correlation: primordial non-Gaussianity (Section III.1) and the BAO (Section III.2). In Section III.1, we also analyse the effect of RSD on the cross-correlation. In Section III.3 we compare our results to the constraints the come from galaxy-galaxy lensing. In the Appendix we give a derivation of the Gaussian covariance matrix of the galaxy density-intrinsic shear cross-correlation in real space.

Unless otherwise noted, throughout this paper we work with the following Planck Ade et al. (2013) fiducial cosmology: , , , , , , Mpc and we define .

Ii The tidal alignment model

We define the ellipticity of a galaxy as , where is the ratio of the minor axis to the major axis of the best-fit ellipse to the galaxy image. The ellipticity can be decomposed in two components, and , with the position angle of the galaxy. The component indicates radial (if negative) or tangential (if positive) alignment of a galaxy with respect to another galaxy. The component measures the rotation with respect to Bernstein and Jarvis (2002). The distortion, , acting on a galaxy is the change in its ellipticity; it can also be decomposed into and . The relation between distortion and ellipticity is , where is the responsivity factor, the response of the ellipticity of a galaxy to an applied distortion Bernstein and Jarvis (2002).

When we measure the shapes of galaxies, we are measuring a combination of the effect of the tidal field , the effect of weak gravitational lensing on those shapes , and the intrinsic random shapes, :

(1)

The topic of this work is the correlation of with the galaxy field. We will consider as the fiducial model of intrinsic alignments the NLA model proposed by Bridle and King (2007). In this model, an intrinsic shear due to the tidal field of the primordial potential, ,

(2)

acts on a galaxy, where is some undetermined constant measuring the strength of the alignment, is Newton’s gravitational constant and the quantity in parentheses is the tidal tensor operator on the plane of the sky.

The primordial potential is evaluated at a redshift , during matter domination, when the galaxy was formed. This is an , since there is no first principle model from which the LA model is derived. The redshift at which intrinsic alignments are set is unconstrained, but Joachimi et al. (2011) have shown that the current measurements of the intrinsic alignments of LRGs are consistent with the primordial alignment model.

On large scales, the density field behaves linearly, , and can be related to the primordial potential through the Poisson equation,

(3)

in Fourier space, where is the mean density of the Universe at redshift , is the growth function and is the scale factor.

In practice, we can only measure shapes of galaxies at the positions of galaxies, hence, we observe the density-weighted intrinsic shear field, , with the galaxy overdensity field and , a scale-independent bias. 9 This weighting is of particular importance for intrinsic alignments, since galaxies that enter the correlation are physically associated. The details of the weighting might also depend on selection effects of the galaxy sample, such as flux or apparent size, which we do not model in this work. In the linear regime, this implies an effective rescaling of the bias, while in the non-linear regime, it can change the shape of the smoothing filter. A similar source-lens clustering is negligible in the context of galaxy-galaxy lensing, even at the typical current precision of photometric redshifts Schmidt et al. (2009).

The intrinsic shear in Fourier space is given by

(4)

where we have defined a smoothing filter for the primordial potential, , that removes the effect of the tidal field on scales smaller than the typical halo inhabited by LRGs. The purpose of applying a smoothing filter is to smooth the tidal field within the scale of the halo inhabited by the galaxy and to suppress the correlation due to non-linear effects on small scales that perturb the alignment. Numerically, the smoothing filter also avoids spurious features in the correlation function due to a sharp cut-off in space. We discuss the effect of the smoothing filter in Section II.4.

The density-weighted intrinsic shear field is given by a convolution in Fourier space,

(5)

where is the three dimensional Dirac delta.

The galaxy density field and the component of the weighted intrinsic alignment tensor are correlated with a power spectrum given by

(6)

In the NLA model, is replaced by the non-linear matter power spectrum in Eq. (6). Kirk et al. (2012) suggest a slightly modified version of the NLA model, where is constructed assuming that the tidal field does not undergo non-linear evolution, while does. This is also an approximation to the problem of correlating the non-linear density field with the primordial tidal field, but does not improve on the physical treatment of the non-linear dynamics. In this work we use the code CAMBLewis and Bridle (2002) to obtain the non-linear matter power spectrum.

ii.1 Modeling of the correlations

For modeling the correlations between galaxies and intrinsic shapes, which is the subject of this work, we will consider a redshift-space correlation function between two observables and ,

(7)

where is the comoving projected separation vector, in the plane of the sky, is the comoving distance along the line of sight, and is the perpendicular component of the separation vector projected along the line of sight, and is the redshift.

We can relate the correlation function in redshift space to the power spectrum of the two observables and through:

(8)

Analogously to the choice of cylindrical coordinates in Eq. (7), we choose cylindrical coordinates in Fourier space (along the line of sight, and perpendicular to it, ). We will henceforth refer to as the angular average over the directions of (i.e., on the plane of the sky) of Eq. (7). The projected correlation function under the Limber approximation is defined as

(9)

where the weight function depends on the observables. The advantage of the estimator in Eq. (9) is that it only computes the correlation between galaxies in a box of length along the line of sight. This procedure reduces the contamination from other sources of correlation, as we will discuss in Section II.6. We apply here the weighting derived in Mandelbaum et al. (2011):

(10)

where is the redshift distribution of the galaxies in the sample, normalized to unity. This is the correct weighting when counting pairs of galaxies in the cylinder defined by coordinates . The comoving distance factors account for the change of comoving volume with redshift.

ii.2 Intrinsic shape correlations

In the tidal alignment model, depends on the primordial gravitational potential through Eq. (2). As a consequence, and the density field are correlated. This correlation can be directly measured, by taking galaxies as tracers of the density field and building the correlation function with their measured ellipticities. We will refer to the correlations between LRG positions and LRG shapes as . We will study the cosmological information imprinted on the correlations between the intrinsic ellipticities and the density field, as traced by galaxies. Under these assumptions, the cross-correlation between the observed density field and the observed shapes, given by Eq. (5), at a given redshift is

(11)

We have neglected here the contributions of galaxy-galaxy lensing and magnification bias to the correlation. We will discuss those contributions in Section II.6.

In the tidal alignment model, the and correlation functions were computed by Hirata and Seljak (2004, 2010). We incorporate the effect of RSD on large scales by transforming the non-linear matter power spectrum, to the anisotropic redshift-space distorted power spectrum, , through the transformation derived by Kaiser (1987) and Hamilton (1992) and applied to intrinsic alignments already in Blazek et al. (2011). Moreover, because of the smoothing filter acting on the primordial potential in Eq. (4), the tidal field power spectrum is further multiplied by the smoothing filter,

(12)

where Linder (2005), and where is the angle between and the line of sight. This approximation to the effect of RSD has been shown to be valid over an arbitrary range of scales Yoo and Seljak (2013).

The redshift space correlation functions, projected along the line of sight on the cylinder defined by , are given by:

(13)

where is the second order spherical Bessel function, and . Integrating over the redshift range of the LRG sample using the Limber approximation, we obtain:

(14)

We have chosen Mpc in agreement with Blazek et al. (2011).

For computing the auto-correlation functions between the different components of the ellipticity, and , we find it convenient to define the following functions,

(15)

With this notation, the auto-power spectrum of is given by

(16)

For obtaining , it suffices to replace by in the above equation

The smoothing filter has a larger impact on the auto-power spectra of the intrinsic ellipticities, and , since the smoothing filter appears squared in Eq. (16). From now on, we neglect quadratic terms in the power spectrum in Eq. (16), since they do not have a significant contribution to Hirata and Seljak (2004).

The auto-correlation functions of intrinsic ellipticities are given by

(17)
(18)

ii.3 Strength of alignment

We obtain from fitting the density-intrinsic shear cross-correlation. As illustrated by Eq. (1), the shape of a galaxy has contributions from the shear, the tidal field and a random noise component. For low redshift galaxies, the effect of gravitational lensing is negligible. We use low redshift observations of the intrinsic shapes of LRGs by Okumura and Jing (2009) to constrain the value of , the amplitude of the intrinsic alignment effect. We assume , the bias derived for this sample of LRGs from their clustering Blazek et al. (2011).

Okumura and Jing (2009) measured for LRGs in SDSS DR6 in the redshift range of and with a median redshift of . In that work, LRGs are selected by applying the technique of Reid and Spergel (2009), identifying central LRGs and removing satellites for each halo. The volume associated with each halo is a cylinder with dimensions Mpc and Mpc. This cut effectively removes the one-halo term contribution to .

We perform a least-squares fit to and obtain with a reduced , for Mpch. Figure 1 shows the results of our fit to the data from Okumura and Jing (2009) (their figure 3). If we only take into account points at Mpc for the fit, we obtain a consistent result of with a reduced . Blazek et al. (2011) also performed the fit on large scales, where the effect of the smoothing filter is not significant; our results are in agreement with theirs. Because we are interested in cosmological constraints on large scales, we adopt the latter value of as our fiducial amplitude for the remaining sections. Our numerical convergence tests for indicate a additional uncertainty in the value of .

We can compare the scatter in intrinsic ellipticities to the predicted value from the NLA model, which is obtained from the power spectrum of the gravitational potential as in Eq. (8) of Catelan et al. (2001):

(19)

Integrated over the range , the NLA prediction yields a much lower value for than the measured value for the LRGs of Okumura and Jing (2009), for which (). We are assuming here that is independent of redshift. In practice, this will most likely not be true, but it would require an unrealistic variation of at least an order of magnitude in to bring the NLA model in agreement with . The shapes used by Okumura and Jing (2009) are not corrected for the Point Spread Function (the response of an unresolved source to the combined effect of the telescope optics and the atmosphere). This results in an overestimation of but, once more, this effect, although highly significant Blazek et al. (2011), is not sufficient to account for the difference between the NLA prediction and the observed value. “Shape noise” due to a random component of ellipticity, , is the dominant contribution to the scatter and prevents us from estimating with this method.

Figure 1: The observed galaxy-intrinsic ellipticity correlation function (diamonds), , from Okumura and Jing (2009) compared to our best fit intrinsic alignment model with a smoothing filter of typical scale /Mpc (solid line). The dashed and dotted-dashed lines correspond to the predictions of the intrinsic alignment model for the cases with Mpc and Mpc, respectively (using the best fit from the Mpc case). The dotted line corresponds to the linear alignment model (which uses the linear matter power spectrum).

ii.4 Smoothing filter

Hirata and Seljak (2004) suggest the use of a top-hat filter in space in Eq. (4) to remove the effect of the tidal field in galactic scales. Catelan et al. (2001) use a top-hat filter in real space. Blazek et al. (2011) do not use any particular smoothing because they focus on scales Mpc. These assumptions are all consistent with the assumption of linearity. In this work we use a well-behaved filter in -space in order to avoid non-physical behavior of real-space correlation functions. We thus consider the following smoothing filter as our fiducial filter:

(20)

The filter is a Gaussian function that decays by a factor at . This choice of smoothing filter removes spurious oscillations in the correlation function produced by a sharp cut-off in -space.

We can see in Figure 1 that the smoothing considerably changes the predicted correlation function. The smoothing filter that best reproduces the measured correlation function has Mpc. This is roughly consistent with the typical scale of an LRG halo found by Reid and Spergel (2009), of Mpch. The linear alignment model curve (with a filter of Mpc) lies between the non-linear alignment models with Mpc and Mpc. This is because, even though it does not capture the non-linear physics well enough, the linear alignment model still has non-zero power on small scales. Removing the smoothing filter for the LA model leads to changes of on scales Mpc, which are negligible compared to the effect of non-Gaussianity or the BAO in in Section III.

(a)  The auto-correlation function of predicted for the sample of Okumura and Jing (2009) for smoothing filters of different values.
(b)  Same as (a) for
Figure 2: The auto-correlation functions of the intrinsic ellipticity components: (left) and (right) predicted for the sample of Okumura and Jing (2009) for smoothing filters of different values. The LA curve is subject to smoothing with Mpc.

The smoothing filter also has a noticeable effect on the auto-correlation function of intrinsic shapes, . In Figure 2, we show the predicted correlation for the sample of Okumura and Jing (2009). For and , the effect of the smoothing filter is smaller on small scales in comparison to , but it is still present at scales of Mpc. As in Section II.3, we have neglected the quadratic contributions of the matter fluctuation power spectrum in this section. The auto-correlation functions of LRG shapes at were measured by Okumura et al. (2009) and shown to be in agreement with the LA model on large scales by Blazek et al. (2011). For the sample of Okumura et al. (2009), is consistent with at Mpc, where the effect of the smoothing filter is large.

ii.5 When were intrinsic alignments imprinted?

In the study by Hirata and Seljak (2004, 2010), intrinsic alignments are imprinted at the redshift of formation of the galaxy, . An alternative to the primordial alignment model is to assume that there is an instantaneous response of the galaxy shape to the tidal field. These two models result in different redshift dependence of the amplitude of the intrinsic alignment signal. In the case of instantaneous alignment, Eq. (4) is modified to be

(21)

where is the gravitational potential at redshift . Any model in which the galaxy shape and orientation is determined between the redshift of formation of the galaxy and the redshift of observation will have an amplitude that lies between the instantaneous and the primordial alignment case. The instantaneous and primordial alignment model differ from one another by a factor , the inverse of the growth function.

The constraints on the amplitude of the alignment signal obtained by Hirata et al. (2007) and Joachimi et al. (2011) are inconclusive regarding the redshift dependence of the alignment signal. Hirata et al. (2007) measured the correlation of LRGs in the SDSS DR4 sample () and in the 2SLAQ LRG sample () Cannon et al. (2006). Their results are consistent with the LA model, albeit with large error bars. Similarly, in Joachimi et al. (2011), the authors study the redshift dependence of by combining previous results for the LRGs of the SDSS DR4 by Hirata et al. (2007) with the MegaZ-LRG sample (at ) Collister et al. (2007); Abdalla et al. (2011), selected from SDSS DR6. They find no evidence for a redshift dependence other than the one proposed by the LA model. More specifically, in their Table , they show their constraints on the amplitude of the correlation for the MegaZ-LRG sample in two redshift bins with median redshifts of and . The amplitude is quoted relative to a fiducial value of and correcting for galaxy-galaxy lensing and magnification bias contaminations. At , it is not possible to distinguish between the instantaneous and the primordial models for those redshift bins. A halo model approach to intrinsic alignments, applied to the Millenium Simulation, displays an evolution consistent with the LA model as well Joachimi et al. (2013b).

In Figure 3, we explore the ability of future surveys to distinguish between the instantaneous and the primordial LA model. Having normalized the amplitude of the intrinsic alignment signal to unity today, we show the ratio between the instantaneous and the primordial alignment models for the correlation and the auto-correlation functions. This ratio is proportional to the growth function, , for , and to for . The instantaneous model has a steeper dependence with redshift. While this difference is really a function of scale, since the Kaiser factor of Eq. (12) for RSD depends on redshift, this affects very similarly for the surveys considered in this work, as will be seen in Section III.1 and Figure a, such that we only consider an overall variation in amplitude of the IA signal with redshift in this section. (Notice that in the non-linear regime, the details of the smoothing filter and the non-linear approximation adopted Bridle and King (2007); Kirk et al. (2012) have a significant impact on the evolution of alignment signal.)

The percentual difference in the amplitude of the alignment signal in the primordial and the instantaneous model is at and reaches at the median redshift of EUCLID. If we had assumed that the correlation length of galaxies is fixed, the bias would evolve proportionally to the growth function. An instantaneous LA model with constant bias has a similar redshift dependence to a primordial model with . This result stresses the necessity of measuring the bias of galaxies as a function of redshift, from the auto-correlation function of the positions of galaxies, complementarily to the intrinsic alignment correlation in order to distinguish between these models. While the dependence is steeper, this signal is also harder to measure due to shape measurement systematics.

Although the LA model reproduces the LRG alignments at , it does not make a prediction for the strength of alignment, i.e., the value of . The coupling of the baryons to the tidal field of the dark matter will ultimately determine the value of and the effective redshift of the tidal field. A possibility that we have not explored here is that there is a lag between the tidal field and the response of the galaxy. These variants to the LA model can only be fully studied with hydrodynamical numerical simulations of galaxy formation.

Figure 3: The ratio between the redshift evolution of the primordial alignment model and the instantaneous model. Vertical lines in gray indicate the median redshift of the surveys considered in this work, from low to high redshift: LOWZ and CMASS (BOSS), DESI and EUCLID.

ii.6 Contaminants to intrinsic alignments

In Eq. (11), we have assumed that the only contribution to the cross-correlation between the observed density field and the observed shapes comes from intrinsic alignments. There are additional contributions to the cross-correlation of the galaxy density field with galaxy shapes. The shear of a background galaxy by a lens gives a contribution that is referred to as galaxy-galaxy lensing (, the topic of Section III.3). At low redshift, there is a short baseline for matter fluctuations along the line of sight, hence the contamination can be neglected Blazek et al. (2011). At higher redshifts, the degree of contamination depends on the accuracy of galaxy redshifts. For intrinsic alignments, the correlation arises between galaxies that are physically associated and hence, separated by small physical distances. For galaxy-galaxy lensing, , the contribution to the correlation peaks for pairs of galaxies such that the background source is at twice the angular diameter distance between the lens and the observer. When redshifts are photometric, there is significant scatter of galaxies in redshift space such that a galaxy that in reality is behind a lens can be observed to be in the foreground, mixing the and signals.

In the most pessimistic scenario, where only photometric redshifts would be available for our fiducial targets, the increased uncertainty in the redshifts of the galaxies will induce a higher contamination of galaxy-galaxy lensing to the correlation measurement that can readily be modeled, as done by Joachimi et al. (2011) in the context of the MegaZ-LRG sample. In this context, a joint analysis of alignments, clustering and weak lensing cosmological constraints as in Joachimi and Bridle (2010) would be more adequate. Nevertheless, while we consider only spectroscopic redshifts in this work, typical photometric scatter for LRGs can be significantly smaller than for galaxies selected for cosmic shear Benítez et al. (2009). Moreover, since LRGs have been shown to trace group environments Zheng et al. (2009), it is also conceivable that a combination of photometric redshifts for these galaxies and the group members can result in redshift estimates for our fiducial targets that would be more accurate than using photometric redshifts alone.

The density field is also subject to lensing bias Schmidt et al. (2009): magnification increases the density of galaxies at a given redshift by increasing the brightness of galaxies in the background. Lensing bias contributes a term to the weighting of the tidal field by the observed galaxy population, where is the convergence field Bartelmann and Schneider (2001) and is the faint end slope of the luminosity function of LRGs. This introduces additional terms and in Eq. (11). The term arises when a galaxy is being magnified by the same matter overdensity that is shearing the second galaxy, and arises when a galaxy in the background is being magnified by the same matter overdensity that is producing the tidal alignment of the second galaxy.

We will only consider spectroscopic redshifts in this work. For the four samples of LRGs considered, we place an upper limit to the contamination of , and to the intrinsic alignment correlation. We apply the expressions derived by Joachimi et al. (2011) in their Section for the angular power spectra of each cross-correlation at the median redshift of each sample. We limit the integral to small line of sight separations (i.e., within ) and assume that there is no scatter in the redshifts of the tracers. We estimate the slope of the luminosity function required for computing the magnification correlations using the expression derived in the Appendix of Joachimi and Bridle (2010) and we assume limiting magnitudes of for the Baryon Oscillation Spectroscopic Survey10 (BOSS) Dawson et al. (2013), the Dark Energy Spectroscopic Instrument (DESI) Levi et al. (2013) and the EUCLID mission Laureijs et al. (2011), respectively. Under these assumptions, we find that the signal contributes less than to for all surveys. The contribution of is at least two orders of magnitude below the correlation and while is at the level of a few percent for LOWZ (LRGs at in BOSS), it is several orders of magnitude below for CMASS (described in more detail in Section III), DESI and EUCLID. The contaminations would decrease further if we decrease the line of sight correlation length, . While we have considered Mpc to ease the comparison of our results to those of Blazek et al. (2011), it would be possible to reduce this value to Mpc, since there is little contribution to the signal from above those scales Mandelbaum et al. (2006); Hirata et al. (2007).

Iii Cosmology from intrinsic alignments

In this section, we explore the cosmological information in the intrinsic alignment of LRGs. We consider three spectroscopic surveys: BOSS, DESI and the EUCLID mission. For each of these experiments, we constrain their ability to measure primordial non-Gaussianity and the BAO signature from the projected cross-correlation function of the galaxy density field and their intrinsic shears.

BOSS Dawson et al. (2013) is an undergoing survey of the SDSS-III collaboration. Among its targets, it has gathered spectra for low redshift LRGs, the LOWZ sample Parejko et al. (2013), and it has extended the LRG target selection of SDSS-II to Eisenstein et al. (2011); White et al. (2011), targeting bluer but massive galaxies in the CMASS galaxy sample, with a typical bias similar to that of the LRG population. The LOWZ sample has three times the typical comoving number density of SDSS DR6 LRGs in the redshift range of , with the same selection cuts but extended to fainter objects. While has not yet been measured for these samples, the linear alignment model has been shown to reproduce the alignments of the MegaZ-LRG sample Joachimi and Bridle (2010), with photometric redshifts up to . Hence, we consider both LOWZ and CMASS galaxies as tracers of the intrinsic alignment signal.

For the LOWZ sample, we consider a constant comoving number density of Mpc in the range of . The median redshift of the LOWZ sample is . We construct the redshift distribution of CMASS galaxies from spectroscopic redshifts obtained in SDSS-III data release 10 (DR10). Figure 4 shows the comoving number density as a function of redshift for the galaxies in LOWZ and CMASS. The median redshift of CMASS galaxies is . The area of the survey footprint, in the DR10 release, is deg. We do not attempt to model in detail the selection effects necessary for constructing a large-scale structure catalogue based on the CMASS sample as done in Anderson et al. (2013), we only expect to reasonably reproduce the comoving number density of CMASS galaxies as a function of redshift. In the final data release (DR12), BOSS is expected to have covered deg. In the following sections, we make forecasts both for the currently available sample of BOSS galaxies in DR10 and for DR12. For the LOWZ and CMASS samples, we assume the galaxies have a constant bias White et al. (2011); Nuza et al. (2013); Parejko et al. (2013).

DESI is a spectroscopic survey scheduled for operation between the years and . It will cover deg and it will target spectroscopic LRGs in the redshift range . We consider the comoving number density of LRGs observed by DESI, shown in Figure 4, as quoted in Table 3 of Levi et al. (2013). The median redshift of the LRG sample observed by DESI is . While DESI is a purely spectroscopic survey, we assume imaging will be available from a combination of other surveys.

The EUCLID mission Laureijs et al. (2011), scheduled to launch in 2020, will perform a weak gravitational lensing survey of sr ( deg) combined with a spectroscopic survey for measuring BAO. We model the redshift distribution of LRGs in an EUCLID-type survey in the range with:

(22)

where , and Smail et al. (1994); Joachimi and Bridle (2010), with a resulting median redshift of . We have chosen to normalize the distribution of Eq. (22) to approximately match the comoving number density of LRGs at lower redshift Zehavi et al. (2005). To assess the impact of the normalization choice in our results, we consider two normalizations: Mpc and Mpc. We show the resulting comoving number density of EUCLID LRGs as a function of redshift in Figure 4. Since the proposed spectroscopic targets in EUCLID are H- emitters Laureijs et al. (2011) (i.e., blue galaxies), LRG redshifts will be better constrained from photometry at a typical precision that will be for the overall population of galaxies Laureijs et al. (2011), which we do not include in our current modelling. For Mpc and Mpc, the number of LRGs in the EUCLID sample will be million and million, respectively.

Both for DESI and EUCLID, we consider that the bias of these tracers changes with cosmological epoch, becoming larger at higher redshifts. As a toy model for this process, we fix the correlation length of galaxies by making the bias proportional to the growth function. To normalize the bias, we match the LRG bias of the sample of Okumura and Jing (2009) at . This implies an almost linear increase in the bias from to in the range .

Figure 4: The comoving number density of LOWZ (dashed-triple dotted line), CMASS galaxies as observed in DR10 (solid line), DESI LRGs (dashed) and EUCLID LRGs (dotted), as a function of redshift. See Section III for a discussion of the underlying assumptions.

The typical systematic errors on galaxy shapes are subdominant compared to shape noise, and we consider a dispersion in the distortion of per component for all scenarios.

iii.1 Primordial non-Gaussianity

We consider a simple model of local non-Gaussianity (NG) where the non-Gaussian primordial gravitational potential is given by

(23)

where is the primordial Gaussian potential and is a constant that parametrizes deviations from Gaussianity Komatsu and Spergel (2001). A non-zero detection of would rule out any model of single-field inflation Creminelli and Zaldarriaga (2004). Current best constraints on the local type have been set by the Planck Collaboration at ( C.L.) from observations of the Cosmic Microwave Background temperature field Planck Collaboration et al. (2013). In the near future, tighter constraints will come from large-scale structure surveys, with EUCLID being capable of obtaining constraints to from galaxy clustering Laureijs et al. (2011).

The non-Gaussian density field can be derived from the Poisson equation to first order in as

(24)

where is the Gaussian density field. We can see from Eqs. (19) and (24) that primordial non-Gaussianity will have an effect on the ellipticity of the galaxies. The observable behaves similarly to , the rms variation of mass within a Gaussian sphere of radius . In the presence of primordial non-Gaussianity, there will be additional terms in Eq. (19). We can separate the contribution to the gravitational potential from long, , and short wavelength modes, , following the peak-background split formalism Cole and Kaiser (1989); Slosar et al. (2008),

(25)

and in doing so, we find that the effect of primordial non-Gaussianity on the scatter of intrinsic ellipticities is

(26)

On the other hand, local non-Gaussianity gives rise to a scale dependent bias of halos on large scales Dalal et al. (2008); Slosar et al. (2008):

(27)

where is the speed of light, is the linear matter transfer function at , is the spherical collapse linear overdensity and we have assumed a merger history consistent with , implying that LRGs have not undergone recent mergers. This assumption is consistent with the assumption of passive evolution of LRGs in the LA model, where the halo shape is determined at .

The components of the intrinsic shear field in Fourier space in the non-Gaussian case are

(28)

where the non-Gaussian density field in Fourier space is given by

(29)

and the observed galaxy density field is

(30)

An additional effect of primordial non-Gaussianity is to modify the effect of RSD on large scales by changing the bias (i.e., replacing by in the expression for in Eq. (12)) and to add terms to the Kaiser factor Schmidt (2010). While the change in the bias produces a significant effect that we take into account in this work, additional terms become relevant only on non-linear scales, smaller than Mpc Schmidt (2010). On small scales, the non-linearity of the density field can also give rise to non-Gaussian effects on the intrinsic alignments of galaxies. Hui and Zhang (2008) have explored these effects for a different alignment model than the one considered in this work. However, in our case, because we restrict to large scales, this effect can be safely neglected.

Intrinsic ellipticity-density cross-correlation

In Section II.2, we introduced the correlation functions of the galaxy density field and the observed shears. The full non-Gaussian contribution to the observed density-intrinsic shear correlation in Fourier space is given by

(31)

To order , Eq. (31) can be simplified to

(32)

where and components on the plane of the sky.

The non-Gaussian correlation function, integrated over , is

(33)

There are two non-Gaussian terms to order contributing to . The second term inside the brackets in Eq. (33) is the usual non-Gaussian bias on large scales, derived by Dalal et al. (2008). The mixing of scales in the second term of Eq. (32) gives rise to the third term inside the brackets. Interestingly, this term gives a constant constribution with scale. In all terms in Eq. (33), the bias that appears in the Kaiser factor of Eq. (12) is replaced by the non-Gaussian large-scale bias, .

The relative contribution from the third term to is at the level of . Moreover, because this term adds a constant to the bias, its effect is to rescale the correlation function. Given that the strength of the alignment cannot be derived from first principles, in practice it is not feasible to distinguish between this constant increase in the bias and intrinsic evolution of as a function of redshift at the level of .

(a)  LOWZ. DR12 points have been artificially displaced to higher for visualization purposes.
(b)  CMASS. DR12 points have been artificially displaced to higher for visualization purposes.
(c)  DESI.
(d)  EUCLID.
Figure 5: The correlation function of galaxies and ellipticities in a Gaussian cosmology (solid line) and a cosmology with primordial non-Gaussianity of (dashed line), with a smoothing scale of hMpc for the different surveys considered (LOWZ and CMASS in DR10 and DR12, DESI and EUCLID).

In Figure 5 we show the cross-correlation function for a cosmology with compared to a Gaussian correlation function for the four samples of LRGs described in Section III. The error bars have been computed by constructing the covariance matrix of the cross-correlation function, as described in the Appendix of this work. The correlation function has been tested for convergence due to numerical integration and our tests indicate un upper limit in the uncertainty due to convergence of .

The likelihood difference between a non-Gaussian and a Gaussian is obtained by summing over projected radius bins

(34)

where is the covariance matrix. Table 1 shows the estimated signal-to-noise ratios for detecting non-Gaussianity of for each sample up to Mpc. For DESI and EUCLID, extending the constraints to Mpc does not alter our predictions significantly. To show the impact of cosmic variance on these results, we include in Table 1 the constraints on the detection of non-Gaussianity in the case where cosmic variance is neglected. While shape noise typically dominates the covariance matrix on small scales, cosmic variance increases with scale, as does the effect of non-Gaussianity. For all samples considered, the for detecting is when cosmic variance is taken into account. As a consequence, for ongoing and upcoming surveys, primordial non-Gaussianity needs not be considered when removing the intrinsic alignments signal from gravitational lensing correlations.

Survey LOWZ CMASS DESI EUCLID
DR10 DR12 DR10 DR12
With cosmic variance 0.1 0.11 0.14 0.17 0.95 1.7 1.8
Without cosmic variance 0.26 0.33 0.20 0.24 1.3 2.1 2.5
Table 1: Signal-to-noise ratio for the detection of primordial non-Gaussianity of in the cross-correlation of the galaxy density field with intrinsic ellipticity for the surveys considered in this work. We have defined Mpc.

In Eq. (12), we introduced the Kaiser factor, the correction factor to the matter power spectrum when RSD are taken into account. In Figure a, we show the impact on the correlation function when the Kaiser factor is included compared to when its effect is ignored. In the right panel of Figure a, we also show the change in the , with RSD, when we apply a non-Gaussianity of . On large scales, the effects of RSD and primordial non-Gaussianity are similar: they both produce an enhancement in the correlation function. This is due to the rapid oscillation of the integrand in Eq. (14) along the length of the cylinder of . When projected, RSD and primordial non-Gaussianity have a similar scale-dependence, they both increase at large separations. Neglecting to model the effect of RSD can lead to a false detection of primordial non-Gaussianity of the local type.

(a)  Relative difference between the Gaussian with and without the effect of RSD.
(b)  Relative difference between the Gaussian and the non-Gaussian for .
Figure 6: Comparison of the effect of RSD on the cross-correlation function with the effect of primordial non-Gaussianity of the local type parametrized by for LOWZ, CMASS, DESI and EUCLID. In projection, the two effects have a similar scale-dependence, increasing at large separations.

iii.2 Baryon Acoustic Oscillations

We study the level of detectability of Baryon Acoustic Oscillations (BAO) in the intrinsic ellipticity-density field cross-correlation. We compute the likelihood difference between the case in which the cross-correlation has a matter power spectrum with baryon damping but no oscillations, Eisenstein and Hu (1998), and the case in which the effect of the baryons is present, :

(35)

In Figures a-d, we show the two correlation functions with and without the BAO feature for BOSS, DESI and EUCLID. In these figures, we see clearly that the BAO appears in the correlation successively as a trough, a node and a bump around the Mpc scale. The usual bump observed in the matter correlation function appears at Mpc, roughly coinciding with the location of the trough in . This feature has also been called a “shoulder” in the galaxy-galaxy lensing correlation function by Jeong et al. (2009). The correlation functions without wiggles have numerically converged to a tolerance of .

In Figure 8, we show the estimated cumulative signal-to-noise ratio of the detection of the BAO in the cross-correlation when we take a fixed upper limit to the interval of Mpc and we vary the lower bound. The displays a significant increase at Mpc. We do not extend our computation of the below the scale of Mpc due to differences in the prediction of in the non-linear scales by CAMB/halofit and the analytical predictions of the case without BAO by Eisenstein and Hu (1998). For the BOSS samples, the at Mpc reaches and for LOWZ and CMASS in DR10, respectively, and and for those samples in DR12. For the LOWZ sample only, cosmic variance has a significant impact on the estimate due to the smaller volume probed at low redshift. For DESI, the cumulative predicted is at Mpc and for EUCLID, it is and for with Mpc and with Mpc, respectively. Thus, the BAO signature could be detected at high significance in DESI and EUCLID. In these surveys, large-scale structure probes will be combined to achieve tighter cosmological constraints. The addition of the correlation to the ensemble of observables probed by these surveys is worth considering in the context of measuring the evolution of the distance scale of the BAO.

(a)  LOWZ. DR12 points have been artificially displaced to higher for visualization purposes.
(b)  CMASS.DR12 points have been artificially displaced to higher for visualization purposes.
(c)  DESI.
(d)  EUCLID (Mpc).
Figure 7: The correlation function of galaxies and ellipticities in the cases: with baryons but no wiggles Eisenstein and Hu (1998) (solid line) and for our fiducial case (dashed line). The data points correspond to the radially binned correlation function with errors predicted from the covariance matrix. At Mpc, the BAO appears as a decrement, a node and an increment in with respect to the “no wiggles” case, rather than the usual bump present in .
(a)  LOWZ
(b)  CMASS
(c)  DESI
(d)  EUCLID
Figure 8: The signal-to-noise ratio of the BAO feature in the cross-correlation as a function of the lower bound of the interval of projected radius for which it is calculated, keeping the upper bound fixed at Mpc. We see a clear increase in the when the interval comprises the scale of the BAO, Mpc. The converges to for LOWZ DR12(a), CMASS DR12 (b) and DESI (c), respectively, at Mpc. For EUCLID (d), the predicted depends on the assumed comoving number density normalization, yielding and for Mpc and Mpc, respectively.

iii.3 Comparison to galaxy-galaxy lensing

The detectability of the BAO and primordial non-Gaussianity in the galaxy-galaxy lensing signal () were studied by Jeong et al. (2009). In the case of primordial non-Gaussianity, Jeong et al. (2009) obtain the following expression for the projected surface mass density profile of a lens at :

(36)

where MMpc is the mean comoving mass density. This expression takes into account the scale dependent bias of the lenses in a similar spirit as our Eq. (33). The main difference between Eq. (33) and Eq. (36) is the effect of the scale-dependent bias on RSD in the case of IA. We compare in Figure 9 the fractional difference in and in when non-Gaussianity of is considered (similar results for are presented in Figure 5 of Jeong et al. (2009)). In the galaxy-galaxy lensing case, we choose and the bias values at the median redshift of the surveys considered. For LOWZ and CMASS, the bias is fixed at , while for DESI and EUCLID, we scale the bias keeping the correlation length fixed and normalizing it to match the bias derived by Blazek et al. (2011) for the sample of Okumura and Jing (2009) at . The fractional change in the inferred surface mass density profile and are similar because they result from the scale-dependent bias. The effect of non-Gaussianity also enters through the RSD factor.

Jeong et al. (2009) find that, for lens samples of LRGs and clusters at , they cannot distinguish between and but they can achieve a significant detection of the BAO feature in the projected surface mass density from galaxy-galaxy lensing. As suggested in the previous section, and could be combined to yield a more robust detection of the BAO.

Figure 9: The fractional effect of primordial non-Gaussianity () and the BAO feature in the projected surface mass density profile obtained from galaxy-galaxy lensing for lenses at the median redshifts of LOWZ, CMASS, DESI and EUCLID LRGs.

Iv Conclusions

The intrinsic alignments of LRGs may become a cosmological tool in the near future. We have presented forecasts for the detection of the BAO signature and of primordial non-Gaussianity in the cross-correlation function of LRG positions and shapes for ongoing and upcoming surveys. Our predictions are based on the LA model of Catelan et al. (2001) and further development by Hirata and Seljak (2004, 2010) and the NLA model of Bridle and King (2007), which yield similar results on scales above Mpc but differ on non-linear scales. Our results are expressed in terms of the projected correlation function of galaxy positions and intrinsic shears, .

At low redshift, we constrain the strength of alignment, parametrized by , by means of the observational results of Okumura and Jing (2009) for . Our result is consistent with that of Blazek et al. (2011). We explore the effect of smoothing the tidal field on different scales and we find that it has a significant impact on non-linear scales in the cross-correlation function of positions and shapes. The observational constraints are consistent with a smoothing over the scales of a typical LRG halo. The impact of the smoothing filter is smaller on small scales and larger on large scales in the auto-correlation functions of intrinsic shears compared to