

Robustness of the pairwise kinematic Sunyaev-Zeldovich power spectrum shape as a cosmological gravity probe

Yi Zheng School of Physics and Astronomy, Sun Yat-sen University, 2 Daxue Road, Tangjia, Zhuhai, 519082, China

We prove from the constrained modified gravity (MG) galaxy mock catalogs that the shape of the pairwise kinematic Sunyaev-Zeldovich (kSZ) power spectrum has a strong constraining power on discriminating different gravity theories on cosmological scales. By varying the effective optical depth as a free parameter, we verify that the - (the linear growth rate) degeneracy in the linear theory of is broken down by the non-linear structure growth and the scale-dependence of in some MG theories. Equivalently speaking, the shape of alone could be used to tightly constrain the MG theories on cosmological scales. With a good knowledge of galaxy density biases, we verify that, a combination of the next generation galaxy spectroscopic redshift and CMB surveys, e.g. DESI+CMB-S4, could potentially discriminate and nDGP models from the general relativity at level using the shape of the galaxy pairwise kSZ dipole alone, when and .

98.80.-k; 98.80.Es; 98.80.Bp; 95.36.+x

Introduction.— The cosmological test of gravity theories has gain much more attention since the discovery of the cosmic acceleration. Profound progresses have been made, though there is still room to explore from the next generation of cosmological surveys, which will collect at least one order of magnitude more data than all the existing ones in the current literature. Therefore many next-to-leading-order cosmological/astrophysical effects will be detected at a high confidential level and will significantly benefit the cosmological gravity test. The kinematic Sunyaev-Zeldovich (kSZ) effect Sunyaev and Zeldovich (1970, 1972, 1980); Ostriker and Vishniac (1986) is one of these.

The kSZ effect describes one of the secondary CMB temperature anisotropies, induced by the CMB photons Thomson scattering off a bunch of free electrons with bulk motion. It is a measure of both the cosmological velocity field and baryon distribution in the universe, and the induced CMB temperature change


Here is the averaged CMB temperature, is the Thomson scattering cross-section, is the speed of light, is the unit vector along the line-of-sight (LOS), is the physical free electron number density, is the peculiar velocity of free electrons, and the integration is along the LOS given by . Eq. (2) assumes that the CMB photons scatter off only one cloud of moving free electrons surrounding the th halo until they reach the observer, and is the optical depth of the th halo. We adopt this assumption through out the letter. Furthermore, we assume a constant for all halos for simplicity, thus could be singled out in all calculations.

The velocity field of free electrons is believed to be a good tracer of dark matter velocity field and hence a promising test bed of cosmological models. Several works have detected the kSZ effect in the literature, although the signal-to-noise (S/N) has not reached Hand et al. (2012); Soergel et al. (2016); Schaan et al. (2016); Planck Collaboration et al. (2016); Hill et al. (2016); De Bernardis et al. (2017); Sugiyama et al. (2018); Planck Collaboration et al. (2018). By combining the next generation of galaxy redshift and CMB data, the significance of kSZ detection could potentially reach , by varying scenarios Sugiyama et al. (2018); Flender et al. (2016). This high detection significance will heavily benefit the constraint on the DE properties and the MG theories Sugiyama et al. (2017).

Though promising, this field always suffers from our poor understanding of . The calculation of is based on various complicated astrophysical processes, and could be only possible in hydro-dynamical simulations, which still suffers from numerous uncertainties nowadays. As one could tell from Eq. (2), is degenerated with the amplitude of velocity field. Therefore the uncertainty of will heavily degrade the constraining power of kSZ effect on cosmological models. Additionally, several systematic errors in the kSZ detection, such as the miscentering bias and scatter-in-mass bias, will systematically decrease the overall amplitude of kSZ signal Flender et al. (2016) and induce systematic biases if we rely on the amplitude of kSZ signal to constrain cosmology. Consequently, the conservative choice is to merely use the shape of the kSZ signal for cosmological analysis. This work will prove the robustness of this idea using high resolution MG N-body simulations.

The pairwise kSZ power spectrum. — Halos containing free electrons tend to move toward each other, due to the mutual gravitational attractions. This peculiar kinematic pattern leaves distinct feature in the CMB map, which could be captured by the pairwise kSZ estimator in the Fourier space Ferreira et al. (1999); Sugiyama et al. (2018),


is the pairwise kSZ power spectrum in redshift space, is the galaxy LOS pairwise velocity power spectrum in redshift space, is the survey volume, is the number of galaxies, and is the galaxy separation vector in redshift space.

In a series of papers Sugiyama et al. (2016, 2018, 2017), Sugiyama et al. proved that, assuming the global plane-parallel approximation and the amplitude of velocity field is proportional to , we could drive from the redshift space galaxy density power spectrum in the following way,


where is the scale factor, is the Hubble parameter at redshift . This relation holds for any object such as dark matter particles, halos, galaxies, and galaxy clusters with anisotropic clustering property (in redshfit space).

Instead of detailed modelling of Sugiyama et al. (2016); Okumura et al. (2014), we provide here a toy model to qualitatively understand the numerical results. We introduce a simple RSD model, with a linear Kaiser term and a Gaussian Finger-of-God (FoG) term,


Here is the linear galaxy bias, denotes the cosine of the angle between and the LOS, and is the linear dark matter power spectrum at redshift . In linear theory, the LOS velocity dispersion . By substituting Eq. (6) into Eq. (5), we derive the corresponding formula for ,


Eq. (7) characterizes two circumstances at which the degeneracy could be broken. One is when the MG theory predicts a scale-dependent , which will change the shape of in a characteristic way and break the degeneracy. The other is the non-linear structure growth, which generates higher order terms such as the shape kernel and the FoG term. The combination of the term and these higher order terms will also break the degeneracy.

In order to illustrate the robustness of above two mechanisms in breaking the degeneracy, and also to quantify the robustness of the shape as a cosmological gravity probe, we will measure and compare the dipoles from high-resolution general relativity (GR) and MG simulations in the following sections.

The MG simulations and mock catalogs.— We study two representative MG theories in this work, the gravity De Felice and Tsujikawa (2010) and the normal branch of DGP (nDGP) gravity Dvali et al. (2000). The gravity acquires a scale-dependent and we adopt the Hu Sawicki (HS) functional form of , where the deviation from GR is characterized by the free parameter . The nDGP model predicts a scale-independent , and this model has one parameter of length dimension, below which gravity becomes four dimensional.

simulations are run by the ECOSMOG code Li et al. (2012); Bose et al. (2017) and nDGP simulations are run by the ECOSMOG-V code Li et al. (2013); Barreira et al. (2015). The simulation box size is and the particle number is . Three values are chosen for f(R) gravity simulations, denoted as F4, F5, and F6. Two values are adopted for nDGP gravity simulations, named as N1 and N5. The levels of deviation from GR are in the sequence of F4F5F6, and N1 N5. All simulations have the same background expansion quantified by the WMAP9 cosmology Hinshaw et al. (2013),

and we run 5 realizations for each gravity model.

The snapshot is analyzed in this work, to represent the survey volume at of the future survey. The dark matter halo catalogs are geenrated using ROCKSTAR Behroozi et al. (2013), then by tuning the HOD parameters suggested in Zheng et al. (2007), the mock galaxy catalogs are generated. Different HOD parameters are applied to different MG simulations, and all “constrained” galaxy catalogs are guaranteed to have the identical galaxy number density and the projected correlation functions . This effectively fixes the uncertainties from the complicated galaxy density biases and we could therefore focus solely on the physical deviations induced by the MG theories at cosmological scales. The HOD parameters of the GR simulations are the best-fit HOD parameters from the CMASS data Manera et al. (2013). The detailed description of the simulations and catalogs could be found in Hernández-Aguayo et al. (2019).

 Assuming a fiducial
Figure 1: Assuming a fiducial , the error bars are calculated by . is the standard dispersion of LOS directions, and is the simulation box volume. All directions are assumed to be independent from each other. For conciseness, we only plot the error bars of GR simulation for illustration.

The pairwise kSZ dipole.— We study the dipole of in this section. The first step is to calculate . For the computational convenience, we adopt an estimator equivalent to Eq. (4), namely Sugiyama et al. (2016)


where and are the momentum field and the density fluctuation field in redshift space. and are the corresponding Fourier counterparts. We sample the and fields on regular grids using the nearest-grid-point (NGP) method and calculate and fields by the Fast Fourier transform (FFT) method.

We further adopt Sugiyama et al. (2017), and Eq. (4) gives . Using the Legendre polynomials , the multipole of is defined as


The measured dimensionless dipoles of the dark matter and mock galaxies are shown in the top panels of Fig. 1 111We also calculate and compare the octopoles of , but find negligible differences () of between different gravity simulations..

We see that all simulations give the of similar trend, with a turnover at around . However different gravity theories predict of different amplitudes and shapes, and the MG dipoles are more distorted than that of GR. The deviations of F4 and N1 simulations from GR case are larger than that of F5/F6 and N5 simulations, as expected.

Instead of making careful comparisons between Eq. (7) and Fig. 1, we generally discuss the impact of the shape kernel on the dipole shape here. is physically a consequence of the competition between Kaiser and FoG effects, which makes it a decreasing function ranging from unity to as increases. Together with the FoG term, induces the turnover of at around . Furthermore, as crosses a typical for a given , value will transfer from negative to positive, and continues to grow as Fig. 1 illustrates 222, thus gravity generating larger will predict smaller . Galaxies have , thus their will be smaller than that of dark matter. Both predictions are consistent with Fig. 1. Moreover, it might be possible to use to constrain , which is beyond the scope of this letter.. Finally, different gravity theories predict different , , and . These physical quantities interact together in Eq. (7) and generate diverse kSZ dipoles shown in Fig. 1.

In the bottom panels of Fig. 1, we show the dipole differences between the MG and GR simulations . The nDGP model predicts a constant deviating from GR case at all scales, while the of model coincides with that of GR at large scales and then increases towards smaller scales Hernández-Aguayo et al. (2019). Therefore, at large scales (e.g., ), the deviations of nDGP dipoles are in general larger than those of models and in the other way around at small scales (e.g., ). Assuming (1) a fiducial survey volume of at , which is typical for the next generation of galaxy redshift surveys, e.g. DESI DESI Collaboration et al. (2016), and (2) the CMB resolution is well within the dark matter halos and instrument noise is negligible, which could be achieved by CMB-S4 Carlstrom et al. (2019) survey, we calculate the S/N of the differences between MG and GR dipoles, where


is the number of free parameters here. , and .

The estimated S/N are listed in the bottom panels of Fig. 1. We find that, the F5 and N1 gravity models could be discriminated from GR at a level using , denoting its great potential on cosmology.

Similar to Fig.
Figure 2: Similar to Fig. 1, but with a varying .

degeneracy breaking.— In the realistic data analysis, it is however difficult to have accurate knowledge of in advance and thus the amplitude of kSZ power spectrum could not be used to constrain cosmology. The key question to ask is that, “without knowing , can we still discriminate kSZ signals predicted by different gravity theories?” The answer is yes, that we could discriminate them using the shape of the galaxy pairwise kSZ dipole.

In order to illustrate this point, we implement the following test. We fix the GR as it is in the previous section. Then in calculating the MG dipoles, we replace by , and vary between to fit the GR dipole. Consequently we obtain the best fitted , and the corresponding S/N of the GR-MG dipole differences are calculated by eq. 11 and . We consider this S/N as the constraining power of the gravity model from the shape of the galaxy pairwise kSZ dipole, and equivalently, it illustrates how much the degeneracy is broken by the shape. The results are plotted in Fig. 2.

It is shown that, (1) in nDGP cases, the amplitudes of change by up to, and the S/N of the GR-MG dipole difference degrades by up to , but not by , while (2) in cases, the amplitudes change by up to and S/N degrades by up to 333The S/N of F4 and F5 even slightly increases due to .. This discrepancy illustrates the facts that (1) non-linear structure growth breaks the degeneracy in the kSZ signal, and (2) the scale-dependence of in models further breaks the degeneracy as discussed previously. More importantly, we prove that using the shape alone, we could discriminate F5 and N1 gravity from GR at level, by a DESI+CMB-S4 like combination. This conclusion overcomes the obstacle in the kSZ cosmology originating from the poor understanding of and will have a significant implication prospect in the next generation of galaxy and CMB surveys

Moreover, we choose a moderate in the analysis, which corresponds to a comoving scale around . Therefore our conclusions are immune from the systematics affecting the small scale kSZ signal, e.g., the complicated small scale astrophysical processes.

Discussions.— In conclusion, we verify that the shape of the galaxy pairwise kSZ dipole has great constraining power on gravity models at cosmological scales. This constraining ability does not depend on the small scale kSZ signal, thus immune from the complicated small scale astrophysical processes. This probe is in particular useful for a self-consistent test of GR from cosmological data, where our main target is to falsify GR rather than to determine the “true” gravity model. We find that, with a good knowledge of galaxy density biases, a DESI+CMB-S4 like survey combination could discriminate F5 and N1 gravity models from GR at level, illustrating the promising implication potential of shape in the future.

As a proof-of-concept paper, this work is simplified and could be improved in several aspects. (1) It would be beneficial to develop more accurate analytical models or numerical emulators for . (2) We assume that we have good knowledge of galaxy density bias in this work, while in reality the biases or HOD parameters fitted from will have uncertainties. (3) We assume a constant in this work, while in reality could be correlated with and . (4) The current error estimation assumes negligible CMB instrument noise. (5) To obtain better forecasts of the next generation galaxy and CMB surveys, the realistic kSZ catalogs matching their survey designs would be necessary. These issues will be addressed in our future works. Finally, we would like to use this work to motivate more theoretical and observational kSZ studies on the cosmological model constraints in the future.

We appreciate Baojiu Li and Wojciech Hellwing for providing the simulation and catalog data. We thank Pengjie Zhang, Baojiu Li, Naonori S. Sugiyama, Weiguang Cui and Ziyang Chen for useful discussions. This project has also benefited from numerical computations performed at the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM) University of Warsaw under grants no GA67-17 and GA65-30.

