Probing redshift-space distortions with phase correlations
Redshift-space distortions are a sensitive probe of the growth of large-scale structure. In the linear regime, redshift-space distortions are fully described by the multipoles of the two-point correlation function. In the non-linear regime, however, higher-order statistics are needed to capture the full information of the galaxy density field. In this paper, we show that the redshift-space line correlation function – which is a measure of Fourier phase correlations – is sensitive to the non-linear growth of the density and velocity fields. We expand the line correlation function in multipoles and we show that almost all of the information is encoded in the monopole, quadrupole and hexadecapole. We argue that these multipoles are highly complementary to the multipoles of the two-point correlation function, first because they are directly sensitive to the difference between the density and the velocity coupling kernels, which is a purely non-linear quantity; and second, because the multipoles are proportional to different combinations of and . Measured in conjunction with the two-point correlation function and the bispectrum, the multipoles of the line correlation function could therefore allow us to disentangle efficiently these two quantities and to test modified theories of gravity.
Cosmological galaxy redshift surveys, like the 6dF Galaxy Redshift Survey Jones et al. (2009), Sloan Digital Sky Survey Abazajian et al. (2009), WiggleZ survey Parkinson et al. (2012), VIPERS survey Scodeggio et al. (2018) or BOSS survey Alam et al. (2015), map the distribution of galaxies in redshift-space. Since the redshift of galaxies is affected by their peculiar velocity, the observed galaxy distribution is slightly distorted with respect to the real-space galaxy distribution. In the linear regime, these redshift-space distortions modify the two-point correlation function and the power spectrum, by adding a quadrupole and an hexadecapole modulation in the signal Kaiser (1987); Hamilton (1997). Measuring these multipoles has been one of the main goal of recent redshift galaxy surveys, see e.g. Alam et al. (2017). These measurements have been very successful and have provided constraints on modified theories of gravity Sanchez et al. (2017). Redshift-space distortions are indeed highly sensitive to the growth rate of perturbation , which is generically modified in alternative theories of gravity.
In the non-linear regime, the multipoles of the correlation function are however not fully tracing the information present in galaxy surveys. The non-linear gravitational evolution of the density and peculiar velocity generates indeed a flow of information into higher-order statistics. An obvious choice to capture this flow of information is to look at the three-point correlation function (or Fourier-space bispectrum), see e.g. Scoccimarro et al. (1999); Gaztanaga and Scoccimarro (2005); Gil-Marín et al. (2014) and refs. therein. However this estimator is a three-dimensional function with significant redundancies with itself and the two-point statistics, making its computation and information analysis a complex task.
Various alternative observables have been constructed in order to access information in the non-linear regime, see e.g. Hikage et al. (2002); Codis et al. (2013); White (2016); Scrimgeour et al. (2012). The goal of such observables is two-folds: first, part of the information present in the bispectrum has already been measured in the power spectrum. One can then wonder if it is possible to construct an observable which is less redundant with the power spectrum. And second, since the bispectrum is complicated in redshift-space, it would be interesting to construct an estimator which encodes the same type of information, but which is simpler to model.
In this paper, we study one possible alternative: the line correlation function. The line correlation function has been introduced in Obreschkow et al. (2013) and analytically modelled in real-space in Wolstenhulme et al. (2015). This observable is constructed from correlations between the phases of the density field. Since the two-point function is only sensitive to the amplitude of the density field, it seems promising to use in conjunction an observable which is targeted to measure the phases (see Scherrer et al. (1991); Ryden and Gramann (1991); Soda and Suto (1992); Jain and Bertschinger (1998); Chiang and Coles (2000); Coles and Chiang (2000); Chiang (2001); Watts et al. (2003); Chiang et al. (2004, 2002); Coles et al. (2004); Matsubara (2003); Hikage et al. (2004, 2005); Szepietowski et al. (2013) for other observables based on phase correlations). Fisher forecasts in real-space have shown that combining the line correlation function with the two-point correlation function does indeed improve parameter constraints on CDM cosmology by up to a factor of 2 Eggemeier and Smith (2017); Byun et al. (2017). The gain obtained from the line correlation function in the case of a Warm Dark Matter model or alternative theories of gravity, like the Symmetron and model is even stronger Ali et al. ().
Here we derive an expression for the line correlation function in redshift-space. We show that the line correlation function can be expanded in Legendre polynomials and that almost all of the information is encoded in the first even three multipoles, i.e. the monopole, quadrupole and hexadecapole, similarly to the two-point correlation function. These multipoles are sensitive to the non-linear coupling kernels of the density and of the peculiar velocity. As such, the line correlation function provides a simple way to probe the non-linear evolution in redshift-space and consequently to constrain alternative theories of gravity in the non-linear regime, for example at the scales where screening mechanisms start to act. Note that our approach differs and complements the work of Eggemeier et al. (2015), which studies a modified version of the line correlation function (using an anisotropic window function) and is targeted to measure an anisotropic signal in two-dimensional Zel’dovich mock density fields.
The remainder of the paper is organised as follow. In Sec. II we derive an expression for the line correlation function in redshift-space, at second-order in perturbation theory. In Sec. III we expand the line correlation function in Legendre polynomials. We derive a general expression valid for any multipole . In Sec. IV we calculate numerically the first multipoles in a CDM universe and we show that the multipoles larger than are negligible. We conclude in Sec. V.
Ii The line correlation function of the observed number counts
Galaxy surveys measure the over-density of galaxies in redshift-space
where denotes the number of galaxies detected in a pixel situated at redshift and in direction , and is the average number of galaxies per pixel at a given redshift. The Fourier transform of the galaxy over-density111We use the Fourier convention and , , is characterised by an amplitude and a phase
The line correlation function of is then defined as
where is the inverse Fourier transform of . As discussed in Obreschkow et al. (2013), the cutoff at high has been introduced to avoid the divergence of the line correlation function due to an infinite number of phase factors at arbitrarily small scales, which do not carry any information.
We start by calculating the three-point correlation function of the phase of . At linear order in perturbation theory, is given by
where is the linear bias, is the Hubble parameter in conformal time , is the dark matter density field, is the peculiar velocity of galaxy and denotes radial derivative. The second term in Eq. (4) represents the contribution from redshift-space distortions. We assume here for simplicity that the galaxy distribution is related to the dark matter distribution through a linear bias . This assumption will break down at small scales and introduce a correction to the line correlation function, as shown in Eggemeier and Smith (2017). Note that contains various other contributions, namely relativistic effects and lensing effects Yoo et al. (2009); Bonvin and Durrer (2011); Challinor and Lewis (2011), but we neglect these terms here since we are mainly interested in small scales, where they are expected to be subdominant. We also neglect higher-order projection effects Nielsen and Durrer (2017), that may be relevant in some intermediate regime.
In Fourier space, takes the form
where is related to the Fourier transform of by
The three-point correlation function of the phase of , which enters in Eq. (3), can be expressed as
where is the probability distribution function of the field defined through . Note that here we have dropped the dependence in redshift in the argument of to ease the notation. Following Matsubara (2003); Wolstenhulme et al. (2015), we start by expressing the probability distribution function for using the Edgeworth expansion Scherrer and Bertschinger (1991); Juszkiewicz et al. (1995); Bernardeau and Kofman (1995), which is valid for mildly non-gaussian fields
where is a normalisation factor. Here and are the power spectrum and bispectrum of defined through
Note that since redshift-space distortions break statistical isotropy, depends not only on the modulus of but also on its orientation with respect to the direction of observation . Similarly the bispectrum depends not only on the shape of the triangle but also on its orientation.
Following the derivation in Wolstenhulme et al. (2015), we first discretise the field for a finite survey volume and then we integrate over the amplitude to obtain the probability distribution function of the phase
where we have defined
To calculate (13) explicitly we need an expression for . We work at second order in perturbation theory, where the density field and the velocity field take the form
Here denotes the linear density field,
with and . The kernels depend therefore very mildly on through and .
Inserting this into (5) we obtain for at second order
Combining this with the first order expression for
where denotes the linear power spectrum of at redshift and
We see that the phase correlation of the observed number count is sensitive to the non-linear coupling kernel of the density field , to the non-linear coupling kernel of the velocity field , and to the growth rate . Since redshift-space distortions are not isotropic, the phase correlations depend on the direction of observation . Note that here we work in the distant-observer approximation, where is the same for all galaxies.
Here we have used the Dirac Delta function to rewrite the three permutations in (21) with the same kernel multiplied by three different exponentials. In this way, the kernel depends on the direction of observation only through its scalar product with . We will see that this property is useful to solve analytically some of the integrals in (23).
Since redshift-space distortions break isotropy, the line correlation function depends not only on the modulus of the separation , but also on the orientation of the vector with respect to the line-of-sight: , as depicted in Fig. 1. In the rest of this paper, we will study the dependence of the line correlation function on . Note that in the case where , Eq. (23) is equivalent to the expression derived in Wolstenhulme et al. (2015).
Iii Multipole expansion of the line correlation function
In redshift-space, the two-point correlation function of can be written as the sum of a monopole, quadrupole and hexadecapole in the angle . At linear order in perturbation theory and using the distant-observer approximation, one can show that these three multipoles encode all the information present in the two-point correlation function Hamilton (1997).
Contrary to the two-point correlation function, the line correlation function cannot be simply expressed as a sum of the first three even Legendre polynomials only. However, we will see that the contribution from the multipoles larger than is actually negligible so that most of the information about redshift-space distortions is indeed encoded in the monopole, quadrupole and hexadecapole of .
Since the Legendre polynomials form a basis, we can expand the line correlation function as
where and denotes the Legendre polynomial of order . The multipole of order can be measured by weighting the line correlation function by the appropriate Legendre polynomial
Since in the distant-observer approximation, the direction of observation is fixed for all galaxies, we can choose on the axis without loss of generality. The integral over in Eq. (28) becomes then an integral over the direction of , which can be performed and gives rise to . Combining the remaining spherical harmonics into Legendre polynomials we obtain
Equation (32) contains a 6-dimensional integral. We now show how to reduce it to a 3-dimensional integral that we can compute numerically.
Let us denote by and the angular coordinates of and . Since we have fixed the direction of observation on the axis, we have . We first do a change of variables from , where and is the azimutal angle of around , see Fig. 2. The Jacobian of this transformation is 1, since it is a rotation. In Eq. (32), the only quantities that depend on and are the Legendre polynomials. We have
For any value of the integral over and can be done analytically, since the Legendre polynomials can always be expressed as a series of cosines. For odd ’s we find that the integrals vanish, as expected due to the symmetry of the line correlation function. We present here the derivation and explicit expression for the monopole , the quadrupole and the hexadecapole . In Appendix A we derive a general expression valid for any .
iii.1 The monopole of the line correlation function
For the monopole, the integral over and in Eq. (32) trivially gives since the Legendre polynomials are constant. The integral over can then be performed analytically
The monopole then simply becomes
enforces the condition . Here the kernel and , and the defined in Eqs. (29) to (31) can be expressed as functions of , and only. Equation (34) contains three integrals that can be computed numerically.
iii.2 The quadrupole of the line correlation function
To calculate the quadrupole, we first need to integrate the terms in the square bracket in Eq. (32) over and . As an example, let us look at the first term. We have
As for the monopole, the integral over can then be performed analytically
iii.3 The hexadecapole of the line correlation function
The hexadecapole can be calculated in a very similar way as the quadrupole. The only difference is that the integral over and in Eq. (35) contains Legendre polynomial of degree four instead of two. The resulting integral over can again been done analytically and we find
iii.4 General expression for the multipole
Following the same steps as for the monopole, quadrupole and hexadecapole, one can derive a general expression for the multipole of order . The detail of the derivation is presented in Appendix A. Here we only give the final expression
where denotes the Gauss hypergeometric function and
with the defined in Eq. (37).
We now calculate explicitly the multipoles of the line correlation function in a CDM universe with parameters Ade et al. (2016): and . In Fig. 3 we show the monopole, quadrupole, hexadecapole and tetrahexadecapole () at different redshifts. The monopole and quadrupole decrease with redshift, whereas the hexadecapole and tetrahexadecapole have a more complicated behaviour. The redshift dependence is governed by the coupling kernels and , the linear power spectrum, and the growth rate , which enters in a different way in the different multipoles.
We see that the monopole dominates over the other multipoles by at least one order of magnitude. Note that the quadrupole, hexadecapole and tetrahexadecapole become negative at large separation, whereas the monopole is always positive. The -dependence of the multipole is governed by the sum of the spherical Bessel functions , weighted by different and -dependent prefactors. It is therefore not surprising that the multipoles can change sign. Note that this is not specific to the line correlation function: the monopole of the two-point correlation function in redshift-space does indeed also change sign at large separation, see e.g. Samushia et al. (2014).
In Fig. 4 we show the relative contribution due to redshift-space distortions
for and 8. Note that is equivalent to Eq. (30) for in Wolstenhulme et al. (2015). We see that redshift-space distortions generate a correction of 7 percent in the monopole, at small separation and high redshift. The quadrupole and hexadecapole are a few percent of the monopole. The tetrahexadecapole is always less than a percent of the monopole, and the multipole is less than 0.1 percent. Most of the information about redshift-space distortions is therefore captured by the first three even multipoles.
In Fig. 5 we show the relative contributions at different redshifts. We see that in all cases the contributions due to redshift-space distortions increase as the redshift increases.
The reason for which the contribution due to redshift-space distortions is suppressed with respect to the density contribution can be understood by looking at the expression for the multipoles, Eqs. (34), (36) and (38). These expressions are all proportional to the difference between the density kernel and the velocity kernel . This follows from the fact that the correlation between phases in Eq. (13) is proportional to the weighted bispectrum . This weighted bispectrum probes the difference between the linear relation between and and the non-linear relation. If these relations are the same, then and the function defined in Eq. (22) reduces simply to . We recover then the expression for the line correlation function in real-space. Hence by measuring the line correlation function in redshift-space we probe the fact that the relation between the density and the peculiar velocity is different at linear and at second order in perturbation theory. In other words, we probe the difference between the continuity and Euler equation at linear and second-order in perturbation theory.
As such the line correlation function is complementary to the two-point correlation function in redshift-space. The two-point correlation function probes indeed the linear relation between density and velocity by measuring the growth rate . The line correlation function adds information since it probes the non-linear relation between the density and velocity by measuring the difference . This clearly shows that phase correlations encode a different type of information than the two-point correlation function. Modified theories of gravity generically modify both the growth rate Gleyzes et al. (2016); Alonso et al. (2017); Leung and Huang (2017) and the coupling kernels and Bernardeau and Brax (2011). Hence the line correlation function in redshift-space is expected to be useful to constrain modifications of gravity.
In Fig. 6, we compare the contribution to the monopole (34) generated by the kernel only, by the kernel and by the difference . We see that the difference is significantly smaller than the individual contributions from and . This explains the suppression of the redshift-space distortion signal, with respect to the signal in real-space. Note that here we are using second-order perturbation theory, which does not account for the effect of Fingers of God at small scales. As shown in Gaztanaga and Scoccimarro (2005); Gil-Marín et al. (2014), those have a strong impact on the bispectrum in the non-linear regime. In a future work, we will study the line correlation function beyond perturbation theory, accounting for the Fingers of God, to see if they enhance the multipoles.
In Fig. 7, we plot the prefactors for the monopole, quadrupole and hexadecapole, which depend on the growth rate
We see that these prefactors evolve slowly with redshift, showing that the line correlation function is less sensitive than the two-point correlation function to variations in the growth rate. We also see that these prefactors are smaller than 1 at all redshift, which also explains why the redshift-space correction is significantly smaller than the density contribution.
Note that the different dependence of the multipoles in the growth rate is very interesting, since it provides a way of disentangling it from the parameter . The two-point correlation function measures indeed the combination (see e.g. Alam et al. (2017)). The monopole of the bispectrum has been shown to measure a different combination, , which in combination with the two-point function allows to disentangle and Gil-Marín et al. (2015). Here we see, from Eqs. (34), (36) and (38), that the multipoles of the line correlation function are sensitive to yet three other combinations of and . Combining these measurements has therefore the potential to tighten the individual constraints on and . In a future work, we will do a detail forecast on the constraints we expect from the line correlation function on , and the coupling kernels and .
Finally, in Fig. 8 we show the relative contribution from redshift-space distortion as a function of the orientation222Since the multipoles larger than are negligible, we can write the line correlation function as . of the line and the separation
We see that redshift-space distortions has the largest impact at small separation and when the three-points are aligned with respect to the direction of observation (). In this case, the contribution from redshift-space distortions can reach 17 percent. This reflects the fact that redshift-space distortions modify the apparent radial distance between galaxies, but not their apparent angular separation. As a consequence, it is the largest when the galaxies are at different radial distances but in the same direction.
One would then naively expect that in the other extreme, i.e. when , the redshift-space contribution would vanish. This corresponds indeed to the case where the three points are at the same redshift, but in different directions. From the cyan dashed line in the top panel of Fig. 8 we see however that . This can be understood in the following way: suppose that the three pixels, which are at the same redshift, are all situated in an over-dense region. As a consequence the galaxies inside each pixel are falling toward the center of the pixel. The pixels in redshift-space look therefore denser than they are in real space. Now since the three pixels are situated in the same over-dense region, this effect induces a correlation between the three pixels. This in turns generates an additional correlation between the phases of . This effect is independent of the orientation of with respect to . It simply comes from the fact that correlated density fields generate correlated velocity fields. Hence, even though at there is no change in the apparent distance between the pixels, there is still an effect due to the fact that the size of each pixel changes in a correlated way. Note that this effect is not specific to the line correlation function, but it also exists in the two-point correlation function of galaxies: the redshift-space two-point correlation function at is not the same as the real-space two-point correlation function .
In this paper, we have derived an expression for the line correlation function in redshift-space, which is valid at second-order in perturbation theory. We have expanded the line correlation function in Legendre polynomials and we have derived a generic expression for the multipoles . We have calculated explicitly the first multipoles in a CDM universe and we have found that the monopole, quadrupole and hexadecapole encode almost all of the information in redshift-space.
We have shown that the multipoles are sensitive to the difference , i.e. to the difference between the non-linear evolution of the density field and the non-linear evolution of the velocity field. As such the line correlation function is highly complementary to the two-point correlation function, which is sensitive to the linear growth rate of the density and velocity fields. This shows that correlations between phases encode different information than the two-point correlation function. Our expressions for the multipoles further show that each of them is sensitive to a different combination of the growth rate and of . Combining this with a measurement of the two-point correlation function, which is sensitive to the product can therefore break the degeneracy between these two parameters. In a future work, we will forecast how well this can be achieved with current and future surveys.
Our derivation relies on second-order perturbation theory. It is however well known that redshift-space distortions are not fully described by the second-order coupling kernel even on mildly non-linear scales Scoccimarro et al. (1999); Gaztanaga and Scoccimarro (2005); Gil-Marín et al. (2014). In a future work, we will investigate how the multipoles change if we introduce non-linear effects, like Fingers of God. We expect such effects to enhance the redshift-space distortion signal, since they will increase the difference .
Finally, let us note that the line correlation function targets a very particular choice of phase correlations, namely those that appear along a line, i.e. along filaments. It may be interesting to investigate other configurations, where the redshift-space distortions signal may be enhanced with respect to the real-space signal.
We thank Joyce Byun and Pierre Fleury for useful discussions. CB and FOF acknowledge support by the Swiss National Science Foundation. DO thanks for support the Research Collaboration Award PG12105206 of the University of Western Australia.
Appendix A Calculation of the multipoles
In Section III we have performed a multipole expansion for the line correlation function. All information of our statistical measure can be encoded in a (infinite) sum of multipoles given by (32). Here we show how three of the six integrals in this expression can be solved analytically for any order of multipole , namely the angular integrals , and .
Since only the kernel and the Legendre polynomial are functions of these angles, the challenge is to solve the following expression
with constraint .
In order to solve the integrals, we express the Legendre polynomials as
Due to the binomial coefficients, the only terms that contribute to the sum will be those with the same parity as . Using the binomial expansion, can be rewritten as
Therefore, the integrals over the axial angles and can be trivially solved and it yields
The integral over takes the form333To simplify the notation we drop the argument in and which are both functions of .