The first detection of the imprint of filaments on CMB lensing
Galaxy redshift surveys, such as 2dF 2dF (), SDSS SDSS (), 6df 6df (), GAMA GAMA (), and VIPERS VIPERS (), have shown that the spatial distribution of matter forms a hierarchical structure consisting of clusters, filaments, sheets and voids. This hierarchical structure is known as the cosmic web bond_theo (). The majority of galaxy survey analyses measure the 2-point correlation, but ignoring the information beyond a small number of summary statistics. Since the matter density field becomes highly non-Gaussian as structures evolve, we expect other statistical descriptions of the field to provide us with additional information. One way to study the non-Gaussianity is to study filaments, which evolve non-linearly from the initial density fluctuation. Several previous works have studied the gravitational lensing of filaments to detect filaments and learn their mass profile fil_lensing2 (); fil_lensing_maturi (); fil_lensing_clampitt (). In our study, we provide the first detection of CMB (Cosmic Microwave Background) lensed by filaments and we measure how filaments trace the matter distribution on large scales. More specifically, we assume that, on large scales, filaments trace matter with a constant filament bias, defined as the ratio between the filament overdensity and the mass overdensity. We propose a phenomenological model for the cross power spectrum between filaments and the CMB lensing convergence field. By fitting the model to the data, we measure filament bias.
The cross-correlations of CMB lensing with tracers of large-scale structure have been widely studiedSmith:2007rg (); Hirata:2008cb (); Bleem:2012gm (); Sherwin:2012mr (); Ferraro:2014msa (); Allison:2015fac (); Allison:2015fac (); Giannantonio:2015ahz (); Eg (); Doux:2016xhg (); Singh:2016xey () to test General Relativity, probe the galaxy-halo connection, etc. In our study, we detect the imprint of filaments on CMB lensing by cross-correlating filaments with the CMB lensing convergence () map. We use a filament intensity map, which is derived from the Cosmic Web Reconstruction filament catalogue fil_cat () 111Public in https://sites.google.com/site/yenchicr/catalogue from the Sloan Digital Sky Survey (SDSS) SDSS () Baryon Oscillations Spectroscopic Survey (BOSS) BOSS () Data Release 12 (DR 12) DR (). The filament finder (See Filament Finder in Method section) partitions the universe from 0.450 to 0.700 into slices with 0.005; filaments are found in each redshift bin as the density ridge of the smoothed galaxy density field yenchiSCMS (). At redshift , the filament intensity is defined as
where is the angular position, is the projected (angular) position of onto the nearest filament and is the uncertainty of filament at the projected position . The intensity function can be viewed as applying a Gaussian smoothing to the filament field; the smoothing scale here is chosen to be equal to the local uncertainty of filaments. Using the intensity map at each redshift bin, we construct the filament intensity overdensity map via
Namely, the filament intensity overdensity map is obtained by stacking all the intensity maps together from redshift 0.45 to 0.7 and then scaling it by the average of the intensity .
In this work we use the CMB lensing convergence map 222missing from the Planck Planck_overview () satellite experiment. The Planck mission has reconstructed the lensing potential of the CMB from a foreground-cleaned map synthesized from the raw Planck 2015 full-mission frequency maps using the SMICA code Planck (). The lensing convergence is defined in terms of the lensing potential as
We measure the cross angular power spectrum of CMB lensing convergence and filaments using standard techniques (See Estimator in Method section). We compute the error for each power spectrum by jackknife resampling 77 equally weighted regions that comprise the CMASS galaxy survey from where the filaments are detected.
We construct a phenomenological model to describe the cross-correlation of filaments and the CMB lensing convergence field. Instead of modeling the filament profile on small scalesfil_lensing_clampitt (); 2017MNRAS.468.2605E (); Higuchi:2014eia (), our model studies how filaments trace matter distribution on large scales through the use of the filament bias. We assume a CDM cosmology with Planck parameters from the 2015 release Planck_parameter (), where . In a spatially flat Friedmann-Robertson-Walker universe described by general relativity, the convergence field is
where is the comoving radial distance, is the redshift observed at radial distance, is Newton’s gravitational constant, is the present-day mean density of the universe, and is the comoving distance to the CMB. On linear scale, we assume that filaments trace the matter as , where is defined as the large-scale filament bias.
The filament overdensity can be rewritten as
where is the mean filament intensity redshift distribution defined as
where is the filament intensity defined in eq. (1) and is the width of redshift slice, which is set to be 0.005. In cross-correlation, on scales smaller than the typical filament length, using filaments introduces additional smoothing compared to the true matter density. We model the smoothing as follows: the filaments have typical length and we lose all small-scale information about fluctuations along the filament; therefore, we take the corresponding filament power spectrum to be exponentially suppressed below the filament scale 1/(filament length) in Fourier space. Similarly, any matter in between filaments is either assigned to a filament or eliminated from the catalog (in underdense regions) . For this reason we also introduce a suppression in power in the direction perpendicular to the filaments, with suppression scale . We use two ways to model . The detailed models are shown later in the paper. Using the Limber approximation and the smoothing scale for small scales, the filament-convergence cross-correlation can be written as
where is the CMB lensing kernel, and is modeled as
where is the matter power spectrum. We use CAMB (Code for Anisotropies in the Microwave Background) 333http://www.camb.info/ to evaluate the theoretical matter power spectrum . The measurement of filament length is shown in Fig. 1. As shown in Fig. 1, we obtain very long filaments, and the mean (median) length of the filaments increases as a function of redshift due to the combination of two factors. Firstly, there is growth of the structure. While the number of galaxy clusters increases as the structure evolves, the length of filaments, acting as the mass bridges between galaxy clusters, will decrease. Secondly, the number of filaments detected also depends on the number density of galaxies, which, in the CMASS sample, is low and decreases as a function of redshift (See Supplement Fig.2 for galaxy redshift distribution and filament intensity redshift distribution). The large difference in the mean and median values of filament length indicates that the distribution of the filament length in each redshift bin is not Gaussian. We plot in the background the 2d histogram of filament length distribution as a function of redshift and filament length.
To check the validity of our model, we also compare the results to simulations. The excellent agreement that we find provides an important consistency check. The theoretical prediction for is shown in eq. (7). The matter-filament correlation is defined as
By taking the parameters that are slowly varying compared to , we can get
where . And for the filament catalogue, the effective redshift, which is defined as the weighted mean redshift of filament intensity, is 0.56. This approximation is not perfect, leading to a systematic bias in the prediction for . We propose an estimator for this systematic bias in Supplement Section 1 Eg (). As shown in Supplement Fig. 1, the systematic bias is less than 5%. Thus, the approximation only causes a negligible bias. In our analysis, we measure using 10 realizations of sky mocks of dark matter and corresponding filaments constructed from 10 N-body simulations (See sky mocks for dark matter and filament in Method section).
Fig. 2a shows the cross angular power spectrum of filaments and the CMB lensing convergence field. We bin our sample into 16 bins. The results from observation and simulation are shown as the blue crosses and green circles. Comparing simulation with data, we get with all 16 data points and without the first data point. The deviation of the first data point from the prediction is likely due to cosmic variance given the small sky area ( = 0.065) covered by the simulations. Thus, the first data can be very noisy. The result shows that the simulation is consistent with the data, indicating that the systematics in the data would not significantly affect our result.
We fit the model of eq. 7 to the data with filament bias () as the fitting parameter. We use two different smoothing methods to find . The first method is to define the perpendicular smoothing scale as the filament spacing, since any scale smaller than the filament spacing is smoothed out. The filament spacing is approximately the filament length. So, in this case, filament length is the overall smoothing scale for the effective power spectrum in eq. (8). The result is shown by the red line in Fig. 2. The best fit gives = 1.68 0.334. Since filaments also have width, filament spacing may be an overestimate of the smoothing perpendicular to filaments. In the second model, we fit for smoothing scale in the perpendicular direction as a free parameter, where we assume and we fit both for and constant using least fit. We get = 0.65 and correspondingly, . The result is shown as the orange line in Fig. 2.
We measure the significance of the cross-correlation detection by measuring the signal-to-noise ratio (SNR). Our SNR is defined as follows
where is the cross angular signal in bin , is the best-fit theoretical prediction for the cross signal in bin , and is the covariance matrix estimated from jackknife resampling. The final result is shown in Table 1. The SNR values for both models show a significant detection of the cross-correlation. On large scales, we find that the filaments trace the matter with the filament bias around 1.5, which is smaller than galaxy bias, as expected, since the filaments also trace underdense regions where there are fewer or even no galaxies.
In order to validate the detection of the cross power spectrum between filaments and CMB lensing convergence field, we perform a null test as follows. We rotate the CMB lensing convergence map by , and , and then we cross correlate these rotated CMB convergence maps with the filament intensity map. Fig. 2b shows that the cross signal with the rotated maps fluctuates around 0. for the three cross angular power spectra is 0.79, 0.75 and 1.04. in all three cases is close to 1, which means the cross-correlation between rotated CMB maps and the filament intensity map is consistent with 0.
We define the cross-correlation coefficient between the filament and galaxy maps as , where is the cross angular power spectrum of filaments and galaxies, is the auto angular power spectrum of filaments and is the auto angular power spectrum of galaxies. The result is shown in the left panel of Fig. 3a. The signal is highly correlated on large scales, since both galaxies and filaments trace the large-scale structure of the matter. However, the correlation decreases on small scales. Fig. (3)b shows the cross-correlation of and , where is the angular cross power spectrum of the CMB lensing convergence map and the CMASS galaxy catalogue. These two figures show that the maps are not totally correlated with large deviations at small scales. Establishing the amount of extra cosmological information present in the filaments field would require a joint analysis with galaxy clustering and lensing measurements; this is left to future work.
Filament finder. We obtain filaments from the publicly available Cosmic Web Reconstruction filament catalogue fil_cat (). This filament catalogue is shown to have strong agreement with the redMaPPer Catalogue, since most clusters in the redMaPPer Catalogue lie at the intersection of the filaments in the Cosmic Web Reconstruction filament catalogue, which is predicted by theory. The filament catalogue also has good consistency with the Voronoi model yenchiSCMS (). Furthermore, filaments in this catalogue show desired effects on nearby galaxies gal_property_fil (), and filaments within this catalogue are featured with a local uncertainty measurement at the positions of filaments yenchiSCMS (); fil_cat ().
Here is a brief description of how filaments are obtained. The filament finder begins by partitioning the universe from 0.450 to 0.700 into slices with 0.005. Then for each slice, the galaxies are smoothed into a density field by the kernel density estimator (KDE)KDE ():
where is the total number of galaxies in the slice, is the Gaussian function, is the angular position of the i-th galaxy at the slice at redshift , and is the smoothing parameter that is chosen depending on the density of galaxies in each redshift bin
where is the minimal value for the standard deviation of galaxies’ location within this slice along each coordinate. The detailed explanation for the selection method is shown in APPENDIX A of yenchiSCMS ().
The KDE can be viewed as applying a Gaussian smoothing to galaxies to reconstruct the underlying probability density field.
We then remove galaxies in low-density regions (details can be found in yenchiSCMS ())
and apply the subspace constrained mean shift algorithm (SCMS) SCMS (); yenchiSCMS () to find filaments. The SCMS algorithm returns points representing the ridges of the probability density function . The filament’s uncertainty is obtained by the bootstrap approach.
Filament length measurement. We get the filament intersections from Chen et al444Public in https://sites.google.com/site/yenchicr/catalogue fil_cat (). For each redshift bin, we use the hierarchical clustering method yenchiSCMS () to determine the number of branches at each intersection. The parameters in the hierarchical clustering are chosen to be the same as yenchiSCMS ():
where is the smoothing
bandwidth as in eq. 14. At each intersection, we find the nearest point to the intersection point from each branch, and we group the nearest point as the filament point belonging to that filament (See Supplement Fig.3 for an example of filaments grouped.). Then we keep finding the nearest point to the newly grouped filament to find the next filament point belonging to that branch.
We stop the loop if the distance between filament points is less than and the distance between a filament point and the other intersection point is larger than .
Estimator. We construct the filament map using the HEALPix pixelization with Nside=512. The CMB lensing convergence map is given directly by PLANCK using the HEALPix pixelization with Nside= 2048. We downgrade the lensing convergence map resolution to Nside=512 to cross-correlate with the filament map. The choice of resolution considers both the noise of the filament map and the smallest scale induced by the resolution. The resolution Nside = 512 gives us a pretty clean filament map; the scale of the pixel at the relevant redshift is approximately 3.5 Mpc, which is much smaller than the other length scales considered in the paper.
We measure the cross angular power spectrum for the filament catalogue and the CMB lensing convergence field using a pseudo- estimator:
where is the sky fraction observed by both the filament catalogue and the CMB lensing convergence field, is the spherical harmonic transform of the lensing convergence field, and is the spherical harmonic transform of the filament intensity overdensity. The spherical harmonic transform and are computed using HEALPY.
Sky mock for filaments and dark matter. We use -body simulation runs using the TreePM method (Bagla2002, ; White2002, ; Reid14, ). We use 10 realizations of this simulation based on the CDM model with and . These simulations are in a periodic box of side length 1380Mpc and particles. A friend-of-friend halo catalogue is constructed at an effective redshift of . This is appropriate for our measurement since the galaxy sample used has effective redshift of 0.57. We use a Halo Occupation Distribution (HOD) (Peacock2000, ; Seljak2000, ; Benson2000, ; White2001, ; Berlind2002, ; Cooray2002, ) to relate the observed clustering of galaxies with halos measured in the -body simulation. We have used the HOD model proposed in Beutler13 () to populate the halo catalogue with galaxies.
where is the average number of central galaxies for a given halo mass and is the average number of satellite galaxies. We use the HOD parameter set () from Beutler13 (). We have populated central galaxies at the center of our halo. The satellite galaxies are populated with radius (distance from central galaxy) distributed out to as per the NFW profile; the direction is chosen randomly with a uniform distribution.
The sky mocks of dark matter and galaxy are obtained from the simulation box using the method described in 2014MNRAS.437.2594W (). We use publicly available “MAKE SURVEY” 555https://github.com/mockFactory/make_survey code to transform a periodic box into the pattern of survey. The first step of this transformation involves a volume remapping of the periodic box to sky coordinates preserving the structure in the simulation. This is achieved by using the publicly available package called “BoxRemap”666http://mwhite.berkeley.edu/BoxRemap 2010ApJS..190..311C (). The BoxRemap defines an efficient volume-preserving, structure-preserving and one-to-one map to transform a periodic cubic box to non-cubical geometry. The non-cubical box is then translated and rotated to cover certain parts of the sky. We then convert the cartesian coordinate to the observed coordinate, which is right ascension, declination and redshift. We down-sample the galaxies with redshift to match the mock redshift with the redshift distribution observed in data. We request the reader refer to 2014MNRAS.437.2594W () for more details. We then apply the filament detection algorithm to these simulated mocks using the method described in Filament Finder.
|model 1||model 2|
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Acknowledgements We thank Anthony Pullen and Elena Giusarma for helpful discussion, Martin White for providing us the N-body simulations, Alex Krolewski, Benjamin Horowitz for comments on the draft. S.Ho is supported by NASA and DOE for this work. S.He is supported by NSF-AST1517593 for this work. S.A. is supported by the European Research Council through the COSFORM Research Grant (#670193). S.F. thanks the Miller Institute for Basic Research in Science at the University of California, Berkeley for support. Some of the results in this paper have been derived using the HEALPix package 2005ApJ…622..759G ().
Author Contributions S.He led the project and most of the manuscript writing. S.A. provided the sky mocks for galaxies and dark matter particles as well as wrote the text relative to sky mock for Filaments and Dark Matter in the Method section. S.F. helped with the theoretical modeling and the interpretation of the results, as well as writing part of the manuscript. Y.C. provided the filament intensity maps for data and simulations. S.Ho conceived the idea of cross-correlating filaments with CMB lensing. All authors contributed to the interpretation of the data and commented on the manuscript.
Competing interests Authors declare no competing financial interests.
Appendix A Supplementary information
a.1 Error calculation for from simulation
We derive the relation between and in eq. (10). By removing the appropriate functions from the integrands, which are slowly varying compared to , the correct expression is
However, the approximations required to produce this expression are not perfect, causing the estimation of from simulation to slightly deviate from the true value of . We estimate the deviation by relating the theoretical prediction for (eq. (7)) and (eq. (9)) by the following equation
The result is shown in Supplement Fig. 1.
We see that the is less than 5% from unity, which is much smaller than , where is the error for . Thus, the approximation for converting to only causes a negligible bias.