Reconstruction of CMB Temperature Anisotropies with Primordial CMB Induced Polarization in Galaxy Clusters
Scattering of cosmic microwave background (CMB) radiation in galaxy clusters induces polarization signals determined by the quadrupole anisotropy in the photon distribution at the location of clusters. This ”remote quadrupole” derived from the measurements of the induced polarization in galaxy clusters provides an opportunity of reconstruction of local CMB temperature anisotropies. In this Letter we develop an algorithm of the reconstruction through the estimation of the underlying primordial gravitational potential, which is the origin of the CMB temperature and polarization fluctuations and CMB induced polarization in galaxy clusters. We found a nice reconstruction for the quadrupole and octopole components of the CMB temperature anisotropies with the assistance of the CMB induced polarization signals. The reconstruction can be an important consistency test on the puzzles of CMB anomaly, especially for the low quadrupole and axis of evil problems reported in WMAP and Planck data.
keywords:(cosmology:) cosmic microwave background – cosmology: theory
Large-scale anomalies have been reported in Cosmic Microwave Background (CMB) temperature map with several independent observations. A low quadrupole of CMB temperature anisotropies was first found in COBE data (Hinshaw et al., 1996) then confirmed by WMAP (WMAP Collaboration et al., 2003). The so-called axis of evil, which is an unusual alignment of the preferred axes of the quadrupole and octopole are found by several authors (de Oliveira-Costa et al., 2004; Land and Magueijo, 2005; Samal et al., 2008). Other anomalies include power asymmetry in north/south hemisphere (Eriksen et al., 2004; Hansen et al., 2009) and an anomalous cold spot (Cruaz et al., 2006). If the anomalies are not caused by foreground residuals or systematic effects, we are facing a challenge of understanding of fundamental physics and the nature of the cosmos.
The polarization of the CMB is expected to provide valuable information on the nature of CMB anomalies. It is generated through Thomson scattering of temperature anisotropies on the last scattering surface (Hu and White, 1997). If the anomalies in the CMB temperature are primordial, the polarization should thus exhibit similar peculiarities (see Vielva et al., 2011; Frommert and Ensslin, 2010, for examples related to the Cold Spot, axis of evil, respectively).
One of the question here is that: Is it possible to reconstruct CMB temperature map for the low multipole from other independent observations? If the answer is yes, then the reconstructed temperature map may be used to distinguish whether the anomalies are primordial or systematic. CMB polarization is one of the candidates. Due to the poor correlation coefficients (less than 0.5 for multipole and ), however, the reconstruction from only CMB polarization can not achieve our goal. The scattering of CMB photons in clusters of galaxies may shed light on this attempt.
The scattering of CMB photons in clusters of galaxies induces a polarization signal, which is determined by the quadrupole anisotropy in the photon distribution at the cluster location (Sazonov and Sunyaev, 1999). Therefore, the remote quadrupole in distant clusters, in principle, is investigable by the measurements of this induced polarization signal. The measurements of these signals were originally proposed to suppress the cosmic variance uncertainty (Kamionkowski and Loeb, 1997; Bunn, 2006; Portsmouth, 2004). Moreover, the magnitude of these signals gives some clues about the evolution of the CMB quadrupole to probe the dark energy. Of particular interest here is that it probes three-dimensional information of potential fluctuations around our last scattering surface (Seto and Sasaki, 2000). Even though this induced polarization in galaxy clusters has not beed observed, its detection contains rich information for study of cosmology.
In this Letter we propose an independent observation for the study of the CMB anomalies. We explore a practically useful cosmological probe from the measurements of remote quadrupole: the reconstruction of CMB temperature anisotropies for low multipole. Provided by the strong correlation of the remote quadrupole in low redshifts with the local CMB (Hall and Challinor, 2014), the reconstructed CMB temperature anisotropies are accurate enough for distinguishing the sources of low quadrupole and axis of evil problems.
2 Simulations of CMB Sky and CMB Induced Polarization in Distant Galaxy Clusters
The CMB temperature anisotropies and E-mode polarization from a single plane wave can be written as (Ma and Bertschinger, 1995)
where is Legendre polynomial, is the transfer function with for temperature anisotropies or -mode polarization, respectively. To compute the spherical harmonics coefficients of the temperature anisotropy and -mode polarization and some derivations later, we need the additional theorem of the spin-weighted spherical harmonics (Ng and Liu, 1999)
where , and are the Euler angles as being composed of a rotation around , followed by around the new and finally around . Then the spherical harmonics coefficients are
On the other hand, the polarization effect in the distant clusters of galaxies arises from the presence of the quadrupole component of the CMB in the rest-frame of a cluster. For a cluster located in the direction, the primordial CMB quadrupole induced polarization is (Ramos, da Silva and Liu, 2012)
where is the optical depth across the cluster, and are Stokes parameters in the unit of brightness temperature. We adopt here, by convection, () for a N-S (E-W) polarization component and () for a NE-SW (NW-SE) component. We have assumed that free electrons in a cluster see the same CMB quadrupole because the primordial CMB temperature quadrupole has variations on much larger scales than the extent of individual clusters (Ramos, da Silva and Liu, 2012). Using the addition theorem of spin-weighted spherical harmonics in Eq.(2), we rewrite Eq. (5) for a cluster located in any line of sight direction
All the CMB temperature anisotropy and its polarization and the polarization signal at distant clusters of galaxies can be computed directly from Eqs.(3) and (5), provided that the for each wave-mode is known. Since the evolution of is independent of the direction of , we may write , where is the primordial gravitational potential. This primordial gravitational potential that appears in Eq. (3) and (5) is the origin of CMB temperature and polarization fluctuations and primordial CMB induced polarization in galaxy clusters. Therefore, estimating the primordial gravitational potential in three-dimensional Fourier space is our first step for the reconstruction.
It is usual to assume that the two-point correlation function of the has the form
and the power spectrum obeys a power-law
with being a normalization factor and a spectral index of the scalar perturbations.
To compute the simulation data, we first generate the three-dimensional primordial gravitational potential by drawing a random number from a gaussian distribution with variance for each wave-mode. The time evolution of the transfer function is calculated numerically by the CMBFast (Seljak and Zaldarriaga, 1996) Boltzmann code, assuming the cosmological parameters from the Planck 2013 results (Planck Collaboration et al., 2014). Different skies can be generated by changing the seed of our random number generator routine. Given the simulated primordial gravitational potential, we generate CMB data and using Eq.(3) and polarization data in clusters and using Eq. (5). The observed CMB induced polarization in clusters is linear proportional to the cluster’s optical depth, which may be extracted from X-ray surface brightness observations if the temperature profile is known. Here we simply assume . The position of the galaxy clusters is assumed to be randomly distributed in the universe from to . We perform the integration in Eqs.(3) and (5) by summing the contribution from each Fourier mode in spherical coordinates. In each radial direction of , we have sampled uniformly 240 modes in logarithm space from to /Mpc. The angular directions of are obtained by the Healpix scheme (Gorski et al., 2005) at resolution-5 map.
3 Reconstruction of CMB Temperature Anisotropies
We present in Fig. (1) a typical sky realization of the CMB temperature anisotropies () and its quadrupole and octopole components. In order to reconstruct this CMB temperature map with CMB induced polarization in galaxy clusters from the same realization, we first estimate the primordial gravitational potential using Bayes’ theorem
where is the prior probability, is the likelihood of obtaining the data given a realization of primordial gravitational potential . The estimation of is obtained by minimizing the function
where and are instrument noise of observation, is the number of Fourier modes in the fitting, is the number of observed clusters of galaxies, is the discretized , and are the reconstructed Stokes parameters in the th galaxy cluster with the estimated . The last term in the right side is the logarithm of the prior.
There must be many possible sets of corresponding to one observation within the desired uncertainty since the number of modes is much bigger than the number of data . Therefore, the reconstructed differ with the input, while the reconstructed Stokes parameters in the galaxy clusters do not change too much. However, reducing to avoid the overfitting produces another problem. It generates the spurious correlation between the induced polarization in clusters and the CMB temperature anisotropy and -mode polarization once we reduce the number of modes in radial direction. Moreover, the orthogonality property of the spherical harmonics may be broken once we reduce the number of modes in angular direction. When we perform the minimization, we use a smaller number of than the input. We reduce the number of sampling from 240 to 60 for the radial direction, and number of pixel from Healpix resolution-5 to resolution-3 map for the angular direction of . The total number of the parameters in the fitting is (complex variables). The relative errors of , and with in this configuration are less than . We also tested bigger but it does not significantly change our conclusion below.
In Fig. (2) we present the reconstructed temperature map () and its quadrupole and octopole components with the estimated , which is obtained by fitting to the simulated data from 300 galaxy clusters whose redshifts in our catalog. It is obvious that the reconstructed temperature map loses power on small angular scales. However, with the assistance of strong correlation of remote quadrupole with local CMB, the quadrupole component is almost completely reconstructed and the octopole component is also similar to the simulated one. It shows the feasibility of the reconstruction with our algorithm and the results should be useful for the study of anomalies in CMB maps. For the higher multipole, the reconstruction becomes worse due to the more deviation of the transfer functions from that of remote quadrupole with increasing .
We forecast the observation strategy for the induced polarization in galaxy clusters. To quantify the goodness of the reconstruction, we defined the dimensionless correlation coefficient and error of reconstruction with the input temperature power spectrum from simulation. We make 100 realizations and select the observed galaxy clusters with redshifts . We show the average of and as function of for and in Fig (3) with fixed . With tens of observed galaxy clusters the quadrupole component can be perfectly reconstructed and be useful for the investigation of low quadrupole problem. The reconstruction of higher multipole components improves with the increasing number of observed galaxy clusters. We also test the reconstruction by adding -mode polarization data (dashed curves). With the additional data for , the reconstruction of octopole and higher multipole anisotropies improves significantly through the correlation. The relation of the reconstruction and the depth of the observation is shown in Fig. (4). The remote quadrupole of observed in high redshift galaxy clusters probes the universe on scales smaller to the local CMB, so the reconstruction of octopole and higher multipole improves with increasing .
The procedure we have here is something of an idealization. We assume the optical depth through the galaxy clusters is precisely measured and the instrumental noise is significantly small (). We also ignore all the contaminations of the polarization signal, for example, the polarization induced by the transverse peculiar velocity of the galaxy clusters (Sazonov and Sunyaev, 1999; Ramos, da Silva and Liu, 2012) and the background polarization. The signal is probably detectable in the next-generation polarization experiments with broad frequency coverage such as PRISM (Prism Collaboration et al., 2013) but the separation of the quadrupole signal from the other contaminants would be an experimental challenge.
This work has been supported in part by the Ministry of Science and Technology, Taiwan, ROC under the Grants No. 104-2112-M-032 -007 - (G.-C. L.) and Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan under the Grants Nos. 15H05890 (N.S. and K.I.), 24340048 (K. I.) and 15K17646 (H.T.). H.T. also acknowledges the support by MEXT’s Program for Leading Graduate Schools PhD professional, ”Gateway to Success in Frontier Asia”.
- pagerange: Reconstruction of CMB Temperature Anisotropies with Primordial CMB Induced Polarization in Galaxy Clusters–LABEL:lastpage
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