SZ effects in the Magneticum Pathfinder Simulation: Comparison with the Planck, SPT, and ACT results
We calculate the one-point probability density distribution functions (PDF) and the power spectra of the thermal and kinetic Sunyaev-Zeldovich (tSZ and kSZ) effects and the mean Compton parameter using the Magneticum Pathfinder simulations, state-of-the-art cosmological hydrodynamical simulations of a large cosmological volume of . These simulations follow in detail the thermal and chemical evolution of the intracluster medium as well as the evolution of super-massive black holes and their associated feedback processes. We construct full-sky maps of tSZ and kSZ from the light-cones out to , and one realisation of deep light-cone out to . The local universe at is simulated by a constrained realisation. The tail of the one-point PDF of tSZ from the deep light-cone follows a power-law shape with an index of . Once convolved with the effective beam of Planck, it agrees with the PDF measured by Planck. The predicted tSZ power spectrum agrees with that of the Planck data at all multipoles up to , once the calculations are scaled to the Planck 2015 cosmological parameters with and . Consistent with the results in the literature, however, we continue to find the tSZ power spectrum at that is significantly larger than that estimated from the high-resolution ground-based data. The simulation predicts the mean fluctuating Compton value of for and . Nearly half () of the signal comes from halos below a virial mass of . Scaling this to the Planck 2015 parameters, we find .
keywords:hydrodynamics, method: numerical, galaxies: cluster: general, cosmic background radiation, cosmology: theory
One of the last unsolved problems in cosmology with the cosmic microwave background (CMB) is connected with unknown levels of the spectral distortion of the black-body spectrum of CMB. Two decades ago the Far Infrared Absolute Spectrophotometer (FIRAS) on board NASA’s Cosmic Background Explorer (COBE) provided the first limits (Fixsen et al., 1996) on the so-called - and -type spectral distortions, which originate from energy releases in the present and early universe (Sunyaev & Zeldovich, 1970b, 1980a). An enormous progress in technology of cryogenics and detectors of millimetre and sub-millimetre radiation enables us now to reach sensitivity which is 100 or even 1000 times better than that of FIRAS. A recently proposed PIXIE instrument is an example (Kogut et al., 2014).
While measuring the absolute monopole of the spectral distortions requires an absolute spectrometer such as FIRAS and PIXIE, the fluctuating component does not. Khatri & Sunyaev (2015) have recently demonstrated that the Planck data (Planck Collaboration et al., 2015c) permit to limit the mean of the fluctuating part of the -type distortion to be (where is the so-called Compton parameter), thereby reducing the observational upper bound by a factor of seven compared to the original COBE/FIRAS limit.111However, we should not forget that the COBE/FIRAS limit applies to the sum of the uniform and the fluctuating parts, whereas the Planck limit applies only to the latter.
The -type spectral distortions do not carry information about redshifts at which they were produced. Therefore, it will be a challenge to distinguish between the primordial -distortions originated from the pre-recombination era and those from a low-redshift universe induced by the large-scale structures. Nevertheless, in the standard thermal history of the universe, the latter component is expected to dominate by a couple of orders of magnitude222In principle, knowing the precise expectation for the low-redshift tSZ contribution would make it possible to estimate the detectable level of the primordial -type distortions from the pre-recombination era. In the absence of primordial non-Gaussianity, we do not expect any angular dependence of the -type signal before recombination. Therefore, characterizing the angular dependence of the fluctuating component of the -signal from the large-scale structures might open a possibility to separate that contribution from the primordial monopole (via, e.g., cross-correlation of tSZ with other tracers of the large-scale structure). If successful, we can retrieve information on the energy release at redshifts of .; namely, the dominant component of the mean is expected to come from the fluctuating component of the thermal Sunyaev-Zeldovich (tSZ) effect, i.e., inverse Compton scattering of low energy CMB photons on hot non-relativistic electrons in groups and clusters of galaxies (Sunyaev & Zeldovich, 1970a). Therefore, the new Planck bound provides a limit on the total thermal energy content of hot gas in the universe. Indeed, this bound is approaching the predicted value of the mean calculated by the previous generations of cosmological hydrodynamical simulations; for example, Refregier et al. (2000) find for a flat Cold Dark Matter (CDM) model with and (also see da Silva et al., 2000; Seljak et al., 2001; Zhang et al., 2004b). It is therefore timely to revisit this calculation with much improved, state-of-the-art simulations.
The Planck satellite has measured the power spectrum of fluctuations of tSZ on large angular scales (). The tSZ power spectrum is very sensitive to the amplitude of matter density fluctuations (Komatsu & Kitayama, 1999; Komatsu & Seljak, 2002). The power spectrum on large angular scales is particularly a powerful probe of the amplitude of fluctuations, as it is less sensitive to astrophysics of the core of galaxy clusters than that at (Komatsu & Kitayama, 1999; McCarthy et al., 2014).
Cosmological hydrodynamical simulations have proven essential in interpreting the observational data on the statistics of tSZ such as the power spectrum and the one-point probability density distribution function (PDF). The simulations have steadily improved over the past two decades, in terms of the numerical methods, the volume, the mass and spatial resolution, and implementation of baryonic physics such as cooling, heating, chemistry, and feedback processes (Persi et al., 1995; Refregier et al., 2000; da Silva et al., 2000; Seljak et al., 2001; Springel et al., 2001; da Silva et al., 2001b; Refregier & Teyssier, 2002; White et al., 2002; Zhang et al., 2002; Dolag et al., 2005a; Hallman et al., 2007; Roncarelli et al., 2007; Shaw et al., 2010; Battaglia et al., 2010, 2012; Munshi et al., 2013; McCarthy et al., 2014).
The recent progress in performing larger and more “complete” cosmological hydrodynamical simulations, which follow baryons in different phases such as in Warm-Hot Intergalactic Medium (WHIM,Cen & Ostriker (1999)), Intracluster Medium (ICM), stars and super massive black holes and their related feedback, allows now for a direct and detailed comparison of the predicted tSZ signals with observations. Such comparison may shed light on the recently reported tensions between cosmological parameters inferred from the primary CMB with the ones inferred from tSZ (Planck Collaboration et al., 2015c, b).
Scattering of CMB photons off electrons moving with a non-zero line-of-sight bulk velocity with respect to the frame of coordinates where the CMB is isotropic yields temperature fluctuations by the Doppler shift of light, and this is known as the kinetic Sunyaev-Zeldovich (kSZ) effect (Sunyaev & Zeldovich, 1970a, 1980b). There are two contributions to the kSZ signal: the reionisation contribution from , and the post-reionisation contribution. The calculation of the former requires detailed reionisation simulations including radiative transfer and is beyond the scope of this paper. The calculation of the latter is in principle simpler than the former. Precisely characterising the post-reionisation kSZ is important, as we must subtract this contribution from the total kSZ to extract the reionisation contribution which, in turn, can constrain the physics of reionisation (see Zahn et al., 2012; Park et al., 2013, and references therein). Roughly speaking, the post-reionisation kSZ power spectrum is twice as large as that from reionisation; thus, ten per cent uncertainty in the post-reionisation contribution results in twenty per cent uncertainty in the reionisation contribution (Park et al., 2015).
In a fully ionised universe, the kSZ effect is given by the line-of-sight integration of the radial momentum field, i.e., a large-scale velocity field modulated by a small-scale density fluctuations of electrons (see Park et al., 2015, and references therein). Therefore, the calculation of the post-reionisation kSZ requires a simulation whose box size is large enough to capture the long-wavelength velocity field, and the spatial resolution high enough to resolve non-linear structures of baryons responsible for scattering of CMB photons. The hydrodynamical simulations of the post-reionisation kSZ have also improved over the past decade (Springel et al., 2001; da Silva et al., 2001a, b; White et al., 2002; Zhang et al., 2002, 2004a; Roncarelli et al., 2007; Shaw et al., 2012; Dolag & Sunyaev, 2013). The measurements of the post-reionisation kSZ are improving rapidly as well (Hand et al., 2012; Li et al., 2014).
In this paper, we shall push the simulation frontier on the investigation of tSZ and kSZ further. The paper is organized as follows. Section 2 reviews the simulation method and the light-cone generation. Section 3 shows detailed comparisons of the simulation results with the observational data on the Coma cluster, and the one-point PDF and power spectrum of tSZ. We also show the simulation predictions for kSZ toward Coma, and the one-point PDF and power spectrum of kSZ. In section 4, we study how the mean Compton signal builds up over cosmic time. We summarise our findings in section 5.
We construct sky maps of tSZ and kSZ using two sets of simulations: the “local universe” simulation (Sec. 2.1) based on a constrained realisation of the local universe at , and the Magneticum Pathfinder simulation (Sec. 2.2). Combining these simulations, we construct full-sky maps of tSZ and kSZ out to , and one realisation of deep light-cone out to .
Both simulations are based on the parallel cosmological Tree Particle-Mesh (PM) Smoothed-particle Hydrodynamics (SPH) code P-GADGET3 (Springel, 2005). The code uses an entropy-conserving formulation of SPH (Springel & Hernquist, 2002) and follows the gas using a low-viscosity SPH scheme to properly track turbulence (Dolag et al., 2005b). Based on Dolag et al. (2004), it also follows thermal conduction at 1/20th of the classical Spitzer value (Spitzer, 1962). It also allows a treatment of radiative cooling, heating from a uniform time-dependent ultraviolet background, and star formation with the associated feedback processes.
Radiative cooling rates are computed by following the same procedure presented by Wiersma et al. (2009). We account for the presence of the CMB and the ultraviolet (UV)/X-ray background radiation from quasars and galaxies, as computed by Haardt & Madau (2001). The contributions to cooling from each one of 11 elements (H, He, C, N, O, Ne, Mg, Si, S, Ca, Fe) have been pre-computed using the publicly available CLOUDY photo-ionization code (Ferland et al., 1998) for an optically thin gas in (photo-)ionization equilibrium.
We model the interstellar medium (ISM) by using a sub-resolution model for the multiphase ISM of Springel & Hernquist (2003). In this model, the ISM is treated as a two-phase medium, in which clouds of cold gas form by cooling of hot gas, and are embedded in the hot gas phase assuming pressure equilibrium whenever gas particles are above a given threshold density. The hot gas within the multiphase model is heated by supernovae and can evaporate the cold clouds. A certain fraction of massive stars (10 per cent) is assumed to explode as supernovae type II (SNII). The released energy by SNII ( erg) triggers galactic winds with a mass loading rate proportional to the star formation rate (SFR) with a resulting wind velocity of km/s.
We include a detailed model of chemical evolution according to Tornatore et al. (2007). Metals are produced by SNII, by supernovae type Ia (SNIa) and by intermediate and low-mass stars in the asymptotic giant branch (AGB). Metals and energy are released by stars of different masses, by properly accounting for mass-dependent life-times (with a lifetime function according to Padovani & Matteucci, 1993), the metallicity-dependent stellar yields by Woosley & Weaver (1995) for SNII, the yields by van den Hoek & Groenewegen (1997) for AGB stars, and the yields by Thielemann et al. (2003) for SNIa. Stars of different masses are initially distributed according to a Chabrier initial mass function (IMF; Chabrier, 2003).
Most importantly, our simulations include prescriptions for the growth of black holes and the feedback from active galactic nuclei (AGN) based on the model of Springel et al. (2005) and Di Matteo et al. (2005) with the same modifications as in Fabjan et al. (2010) and some new, minor changes as described below.
The accretion onto black holes and the associated feedback adopts a sub-resolution model. Black holes are represented by collisionless “sink particles”, which can grow in mass by either accreting gas from their environments, or merging with other black holes. The gas accretion rate, , is estimated by the Bondi-Hoyle-Lyttleton approximation (Hoyle & Lyttleton, 1939; Bondi & Hoyle, 1944; Bondi, 1952):
where and are the density and the sound speed of the surrounding (ISM) gas, respectively, is a boost factor for the density which typically is set to and is the velocity of the black hole relative to the surrounding gas. The black hole accretion is always limited to the Eddington rate, i.e., the maximum possible accretion for balance between inwards-directed gravitational force and outwards-directed radiation pressure: . Since the detailed accretion flows onto the black holes are unresolved, we can only capture black hole growth due to the larger scale, resolved gas distribution.
Once the accretion rate is computed for each BH particle, the mass continuously grows. To model the loss of this gas from the gas particles, a stochastic criterion is used to select the surrounding gas particles that are accreted. Unlike in Springel et al. (2005), in which a selected gas particle contributes to accretion with all its mass, we include the possibility for a gas particle to accrete only with a fraction (1/4) of its original mass. In this way, each gas particle can contribute with up to four ‘generations’ of BH accretion events, thus providing a more continuous description of the accretion process.
As for the feedback, the radiated luminosity, , is related to the black hole accretion rate by
where is the radiative efficiency, for which we adopt a fixed value of 0.1. This value is assumed typically for a radiatively efficient accretion disk onto a non-rapidly spinning black hole (Shakura & Sunyaev, 1973; Springel, 2005; Di Matteo et al., 2005). We assume that a fraction of the radiated energy is thermally coupled to the surrounding gas; thus, is the rate of the energy feedback. is a free parameter and typically set to . The energy is distributed to the surrounding gas particles with weights similar to SPH. In addition, we incorporate the feedback prescription of Fabjan et al. (2010); namely, we account for a transition from a quasar- to a radio-mode feedback (see also Sijacki et al., 2007) whenever the accretion rate falls below an Eddington-ratio of . During the radio-mode feedback we assume a 4 times larger feedback efficiency than in the quasar mode. This way, we attempt to account for massive black holes that are radiatively inefficient (having low accretion rates) but are efficient in heating the ICM by inflating hot bubbles in correspondence of the termination of AGN jets. In contrast to Springel et al. (2005), we modify the mass growth of the BH by taking into account the feedback, e.g., . We introduced some more technical modifications of the original implementation, for which readers can find details in Hirschmann et al. (2014), where we also demonstrate that the bulk properties of the AGN population within the simulation are quite similar to the observed AGN properties.
2.1 Local Universe Simulation
The local universe simulation uses the final output of a cosmological hydrodynamical simulation of a constrained realisation of the local universe based on the nearly full-sky IRAS 1.2-Jy galaxy survey data (Fisher et al., 1994, 1995). The initial conditions are similar to those adopted by Mathis et al. (2002) in their study of structure formation in the local universe. The galaxy distribution in the IRAS 1.2-Jy galaxy survey is first smoothed by a Gaussian with a width of 7 Mpc and then linearly evolved back in time up to following the method of Kolatt et al. (1996). The resulting field is then used as a constraint on phases of random Gaussian fields (Hoffman & Ribak, 1991) for initialising the simulation.
The volume constrained by the observational data covers a sphere of radius centered on the Milky Way. This region is sampled with more than 50 million high-resolution dark matter particles and is embedded in a periodic box of on a side. The region outside the constrained volume is filled with nearly 7 million low-resolution dark matter particles, allowing a good coverage of long-range gravitational tidal forces. The gravitational force resolution (i.e., the softening length) of the simulation has been fixed to be (Plummer-equivalent), fixed in physical units from to and then stays constant in the corresponding co-moving units (e.g. ) at higher redshifts.
Unlike in the original simulation made by Mathis et al. (2002), where only the dark matter component is present, here we follow also the gas and stellar components. Therefore, we extend the initial conditions by splitting the original high-resolution dark matter particles into gas and dark matter particles having masses of and , respectively; this corresponds to a cosmological baryon fraction of 13 per cent. The total number of particles within the simulation is slightly more than 108 million, and the most massive cluster is resolved by almost one million particles.
In this simulation, we assume a flat CDM model with a present matter density parameter of , a Hubble constant of km/s/Mpc with , and an r.m.s. density fluctuation of .
2.2 Magneticum Pathfinder Simulation
The Magneticum Pathfinder333www.magneticum.org simulation follows a large box simulated using particles and adapting a WMAP7 (Komatsu et al., 2011) flat CDM cosmology with , , , , and an initial slope for the power spectrum of . Dark matter particles have a mass of , gas particles have depending on their enrichment history, and stellar particles have depending on the state of the underlying stellar population. For gas and dark matter, the gravitational softening length is set to (Plummer-equivalent), fixed in physical units from to and then stays constant in the corresponding co-moving units (e.g. ) at higher redshifts. For star particles it is accordingly half the values (e.g. at ). In this simulation, one gas particle can form up to four stellar particles.
2.3 Map Making
We construct maps from the simulations using SMAC (Dolag et al., 2005a), integrating both the tSZ and kSZ signals through our hydrodynamical simulations.
The tSZ signal in each pixel at a sky position is characterised by the Compton parameter defined by (Sunyaev & Zeldovich, 1972)
where and are the three-dimensional number density and temperature of thermal electrons, respectively, and , , , and are the Boltzmann constant, the Thomson scattering cross section, the electron mass, and the speed of light, respectively. The kSZ signal is obtained by (Sunyaev & Zeldovich, 1970a, 1980b)
where and is the radial component of the peculiar bulk velocity, defined such that for gas moving away from us.
As the tSZ and kSZ have different dependence on the observed frequencies, we use the following weighted sum of the two when showing the combined signal:
derived by Zeldovich & Sunyaev (1969), where . The temperature anisotropy due to tSZ is given by , where we ignore relativistic corrections.
We produced full-sky maps of the Compton and kSZ in the HEALPix (Górski et al., 1998) format with Nside=2048 from the local universe () and from the Magneticum Pathfinder simulation covering . We also produced one realization of a deep, light-cone covering . When we combine the deep light-cone with the full-sky maps, we use only the relevant parts, or , of the deep light-cone.
In figure 1, we show the full-sky maps of the Compton (top panel) and kSZ (middle) where the local universe simulation is combined with the Magneticum Pathfinder simulation. We also show of the deep light-cone (as indicated by the arrow) to give the impression of the additional SZ signals covered by the deep light-cone. The lower panels show the Compton from the deep light-cone () smoothed with a 9.66 arcmin FWHM Gaussian beam as well as the native resolution and the same with the kSZ signal added at 150 GHz (Eq. 5). The deep light-cone contains about 8000 galaxy clusters and groups with virial masses above , contributing to the most prominent structures visible in the zoom onto a sub-part of the Compton map shown in the right panel of figure 2.
3 Comparison with the Planck data
The thermal electron pressure determines the (non-relativistic) tSZ effect. Azimuthally-averaged radial profiles of thermal pressure in galaxy clusters identified in our hydrodynamical simulations follow the so-called “universe pressure profile” (Arnaud et al., 2010a). The stacked pressure profiles of galaxy-cluster-size halos in our simulation are in good agreement with the stacked pressure profiles inferred from the tSZ data on galaxy clusters detected by Planck (Planck Collaboration et al., 2013a) and South Pole Telescope (SPT) (McDonald et al., 2014). However, such comparisons allow only a statistical comparison of the averaged profiles.
Local, well-resolved galaxy clusters enable a more detailed object-by-object comparison. Here we make use of the fact that the constrained simulation of the local universe allows to cross-identify objects in the simulations with the real-world counterparts. Despite the relatively low spacial resolution (e.g., Mpc) of the constraints used for initialising the simulation (see Mathis et al., 2002, for details), local galaxy clusters like the Coma cluster have a remarkably similar counterpart in the simulation.
In the left panel of figure 3, we compare the radial profiles of tSZ (the Compton parameter given in Eq. 3), as well as the significant contribution of kSZ at 150 GHz (as given in Eqs. 4 and 5), toward the Coma cluster in the simulation with the Planck tSZ data for Coma (Planck Collaboration et al., 2013c). The blue line shows the Compton toward the “Coma cluster” in the local universe simulation out to a co-moving distance of 110 Mpc, while the thick black solid line shows the “Coma cluster” embedded in the deep light-cone out to . The latter has more signal in the outskirt of the cluster because of the contributions from hot gas outside Coma.
We find an excellent agreement between the simulation and the Planck data (shown as the symbols) in the inner part, kpc. The full light-cone integration (thick black line) agrees reasonably well with the Planck data up to a few Mpc. Given the uncertainties in the constrained simulation, this level of agreement is remarkable. Also the mean Compton () obtained from the deep light-cone (shown by the horizontal dotted line) agrees with the background subtracted from the Planck data (shown by the horizontal dashed line) to within a factor of two.
Finally, to get a feel for the magnitude of kSZ toward the “Coma cluster”, we show the sum of tSZ and kSZ at 150 GHz (Eq. 5) in the magenta line. Both include the full light-cone integration out to . For this particular realisation, we find a non-negligible (10 to 20 per cent) contribution from kSZ.444However, we should not compare the magenta line with the Planck data in the left panel of figure 3, as the Planck data shown here are obtained by combining the multi-frequency Planck data specifically to extract tSZ, and most (if not all) of kSZ has been removed. In particular, we find that the local universe simulation predicts that the overall halo of Coma is moving away from us at with respect to the CMB rest frame, yielding a negative kSZ signal up to Mpc radius. Being a merging system, the core of Coma in our realisation moves with even higher velocity (up to 800 km/s) as commonly seen in simulations (ZuHone et al., 2010; Dolag & Sunyaev, 2013) near the center.
We also find a large, positive kSZ signal in the outskirt, which is due to a massive, gas-rich infalling sub-structure moving toward us. See the right panel of figure 3 for a map of the mass-weighted velocity field around Coma. While the overall halo velocity predicted for Coma should be accurate to the extent of the precision of the constraints used by the local universe simulation, details of sub-structures present in this realization of the local universe are far outside the predictive power of such a simulation. Therefore, the prediction for a positive kSZ in the outskirt and the phase of movement of the core should be interpreted with caution.
3.2 The Compton Map
3.2.1 Simulation results
In figure 4, we compare the one-point PDF of the Compton map from the simulations (lines) with that observed by Planck (symbols; Planck Collaboration et al., 2014b; Khatri & Sunyaev, 2015; Khatri, 2015). The asymmetry of the left and right tails (skewness) of is due to galaxy clusters and groups along the lines of sight, as predicted previously (Yoshida et al., 2001; Rubiño-Martín & Sunyaev, 2003). In the absence of noise, the pixel values in the simulations are always positive, whereas Planck’s values can have negative values due to noise (or unaccounted other sources).
In the right panel, we have smoothed the simulated Compton map with a Gaussian beam with 10 arcmin FWHM, and added Gaussian noise to the simulations with and to approximate the noise estimates from “LIL” and “MILCA/NILC” maps of the Compton , respectively (see figure 3 in Khatri & Sunyaev, 2015; Khatri, 2015). Since Planck cannot measure the absolute value of , there is ambiguity in the exact zero point of the PDF. While the simulation agrees well with the “LIL” map, there seems a slight offset in the zero point with respect to the “MILCA/NILC” map.
As our maps have significantly higher spatial resolution than Planck, we find a larger excess of high values (black line in the left panel) if not smoothed. This is driven by the central parts of halos along the lines of sight. This can be best seen in figure 2, especially in the zoom-in in the right panel. However, once smoothed with the beam size of Planck’s map (10 arcmin FWHM), the excess due to these structures unresolved by Planck is reduced (red dashed line in the left panel), and the simulation and the Planck data are in much better agreement up to . The effect of the beam smearing in the light-cone map can be seen visually in the left most panel in the bottom panels of figure 1. With the smoothing size as large as this, the excess PDF at larger values of above the Planck data is dominated by the nearby structures that subtend large angles in the sky. We find that the excess is dominated by the structures in the local universe simulation. Remarkably, we could identify the source of the excess PDF as the structures such as Perseus-Pisces that are masked when the measurement is done. Using the same mask that retains 51% of the sky used by Khatri & Sunyaev (2015), we find an excellent agreement between the PDFs from the simulation (solid lines in the right panel) and the Planck data at all values of . The “core” of the PDF from the simulation at small values of is dominated by the structures in the deep light-cone beyond the local universe simulation.
We also convolve our map with the SPT beam (1.1 arcmin FWHM; magenta line in the left panel). In this case, the excess at large Compton values is only mildly suppressed, demonstrating that SPT-like instruments would be able to resolve almost all the contributions of structures resolved by our simulation. See Hill et al. (2014) for the measurement and interpretation of the one-point PDF of the ACT data.
Coming back to the unsmoothed PDF, we find that the tail of the PDF follows a power law shape over at least two orders of magnitude in Compton values. The slope of this power law is approximately as shown by the dashed line in the left panel of figure 5. To quantify non-Gaussianity of the PDF, we calculate low-order cumulants of . We find , , , and for the mean, variance, skewness, and kurtosis, respectively.
The middle panel shows the PDF of kSZ (solid line) as well as the contribution of the slice. It shows a non-Gaussian tail in agreement with the previous work (da Silva et al., 2001a; Yoshida et al., 2001; Hallman et al., 2009). The low-order cumulants of are , , , and for the mean, variance, skewness, and kurtosis, respectively. The ensemble average of the mean should vanish; a non-zero value from the simulation is due to cosmic variance. The right panel shows the PDF of the sum of tSZ and kSZ ( defined in Eq. 5) at various Planck frequencies. The contribution to the PDF of of the kSZ signal is negligible for , while it significantly modifies the PDF at smaller values of . As the PDF of kSZ is flatter at small values of kSZ than that of the Compton at small values of , the PDF of the sum of the two shows an excess probability in (see the solid lines figure 5). The dashed lines show the PDF for negative values of . This unique shape of the PDF, which changes as a function of frequencies in a predictable way, may be used to detect the kSZ signals in the data.
3.2.2 Analytical model of the PDF of Compton
The tail of the PDF of Compton can be calculated analytically following Hill et al. (2014). Specifically, we compute
where , is the virial mass with , is the differential comoving volume per steradian, and is the halo mass function given by
with derived from simulations. Here, is the mass enclosed within , in which the mean overdensity is times the mean mass density of the universe. We convert to using an NFW profile (Navarro et al., 1996, 1997) with the concentration parameter of Duffy et al. (2008) (see, e.g., Eq. 14 of Komatsu & Seljak, 2001). For the mass range we consider, we find to good approximation. For , we use the mass function from the Magneticum Pathfinder simulation (Eq. 1 and 3 of Bocquet et al., 2015, with the parameters for “ Hydro” given in their Table 2).
To compute the linear r.m.s. mass density fluctuation necessary in the fitting formulae of the mass function, we use the CAMB code (Lewis et al., 2000) to generate the linear matter power spectrum with the cosmological parameters of the Magneticum Pathfinder simulation.
The “” in Eq. 7 is the inverse function of
where is the electron pressure profile and is the proper angular diameter distance. That is to say, for a given value of , , and , we find the corresponding value of using this equation. is the mass enclosed inside , within which the mean overdensity is 500 times the critical density of the universe. Again, we convert to using an NFW profile with the concentration parameter of Duffy et al. (2008).
with , , , and the function is defined by
with , , , and (Planck Collaboration et al., 2013b). However, this fitting function for the pressure profile was derived for assuming hydrostatic equilibrium, which is known to be biased low relative to the true due to non-thermal pressure (see, e.g., Shi & Komatsu, 2014, and references therein). We thus rescale as and as to account for the hydrostatic mass bias. This correction brings the pressure profiles in the Magneticum Pathfinder simulation into good agreement with the Planck data (Planck Collaboration et al., 2013b).
Eq. 7 is valid when halos included in the calculation do not overlap in the sky. The overlapping fraction, , as defined by Eq. 10 of Hill et al. (2014), is 0.32 for . We show the result of the analytical model in the left panel of figure 5. The shape of the PDF in the tail is reproduced reasonably well. We also show the result for a smaller mass limit, , showing contributions from lower mass halos. However, this case gives , i.e., haloes overlap significantly in the sky, and thus the calculation cannot be trusted for lower values of . Nonetheless this calculation gives an idea about the contributions from lower mass haloes. Since the PDF in the extreme tail is sensitive to the core of the pressure profiles, a difference in details of the pressure profile in the simulation and that used in the analytical model shows up there: the simulation gives more pressure in the cores of galaxy clusters than that of Eq. 10.
3.3 The angular power spectra of tSZ and kSZ
3.3.1 Simulation results
To calculate the angular power spectrum of tSZ, we separately use the full-sky map obtained from the local universe simulation in and the light-cones over in . As these simulations are performed with different cosmological parameters, we rescale the amplitudes of the tSZ power spectra by to Planck 2015’s best fitting CMB cosmological values, and (“TT+lowP+lensing” of Planck Collaboration et al., 2015a). We have checked that this scaling agrees with the scaling predicted by the analytical calculation presented in section 3.3.2
Figure 6 shows the tSZ power spectrum measured by Planck at 150 GHz (red symbols with error bars; Planck Collaboration et al., 2015c). Two dashed lines show the tSZ power spectrum from the local universe at low multipoles and that from the light-cone at high multipoles, while the black solid line shows the sum of the two. This division in multipoles is consistent with the previous work showing that the nearby structures dominate at low multipoles simply because they appear larger in the sky (Refregier et al., 2000; Komatsu & Seljak, 2002; Dolag et al., 2005a; Hansen et al., 2005).
The black solid line agrees with the Planck data well at all multipoles measured by Planck, i.e., . Our conclusion that the predicted tSZ power spectrum with Planck 2015’s TT+lowP+lensing parameters agrees with the measured power spectrum is consistent with the finding of the Planck team (Planck Collaboration et al., 2015c). McCarthy et al. (2014) show that the Planck 2013 parameters with and (Planck Collaboration et al., 2014a) over-predict the tSZ power spectrum, and show that another set of parameters with and (Spergel et al., 2015) gives the tSZ power spectrum in agreement with the measurement. The latter set is close to the Planck 2015 parameters with lensing that we use in this paper; thus, our conclusion is consistent with their results. Using the scaling, for example, the former set gives 32% larger power than the 2015 parameters, while the latter gives 4% smaller power.
However, at , our prediction is significantly higher than the measurements reported by SPT and Atacama Cosmology Telescope (ACT) collaborations (Reichardt et al., 2012; Sievers et al., 2013). This finding is not new: the previous calculations of the tSZ power spectrum also over-predict the power at compared to the SPT and ACT data, although the degree of overestimation varies depending on the details of baryonic physics implemented in the models (e.g., McCarthy et al., 2014; George et al., 2015; Ramos-Ceja et al., 2014).
Whether this discrepancy at poses a serious challenge to theory is unclear, given that the SPT and ACT do not have as many frequency channels as Planck. Distinguishing the primary CMB, tSZ, and the other extra-galactic sources by their frequency dependence is more challenging for SPT and ACT. (On the other hand, Planck does not have angular resolution to resolve the power at .) The tSZ signal is a sub-dominant contribution to the power spectrum at compared to the extra-galactic sources. The magenta symbols with error bars in figure 6 show the SPT power spectrum with the best-fitting primary CMB power spectrum subtracted. The difference between the magenta symbols and a magenta vertical line at indicates the amount of the extra-galactic power that needs to be subtracted. At least, our tSZ power spectrum does not overshoot the magenta data points that provide a firm upper limit on the tSZ power at these angular scales.
Next, we show the kSZ power spectrum from the light-cones by the blue solid line in figure 6. Our result at high multipoles () agrees with that of Shaw et al. (2012). The upturn at lower multipoles can be understood by the contribution from the longitudinal velocity contributions that did not fully cancel by the line-of-sight integration (Hernández-Monteagudo & Ho, 2009). At 150 GHz, the kSZ amplitude becomes comparable to tSZ at , and becomes even dominant at lower multipoles. (Note that the Planck data shown in this figure remove most of the kSZ signals by construction, and thus should not be compared with the blue line.) However, as the kSZ signal is dominated by the largest modes present in the simulation (see figure 1), even larger cosmological volumes will be needed to obtain a fully converged kSZ results from such light-cones.
3.3.2 Analytical model of the tSZ power spectrum
To check accuracy of scaling the tSZ power spectrum to other cosmological parameters, we compute the tSZ power spectrum using an analytical model. Ignoring a small contribution from the correlation between two distinct dark matter halos (Komatsu & Kitayama, 1999), we model the SZ power spectrum as (Komatsu & Seljak, 2002)
where . The 2D Fourier transform of the Compton parameter, , is given by
where and . The electron pressure profile, , is given by Eq. 10.
For , we use three different sets of fit parameters obtained from numerical simulations in the literature. First, we use the mass function from the Magneticum Pathfinder simulation (Eq. 1 and 3 of Bocquet et al., 2015, with the parameters for “ Hydro” given in their Table 2. Using “ DMonly” gives a similar result: the difference in the power spectrum is less than four percent at all multipoles). This mass function should give the result that is most consistent with the black solid line shown in figure 6. The other mass function fits are taken from Tinker et al. (2008) and Tinker et al. (2010).
To compute the linear r.m.s. mass density fluctuation necessary in the fitting formulae of the mass function, we use the CAMB code (Lewis et al., 2000) to generate the linear matter power spectrum with the Planck 2015 “TT+lowP+lensing” parameters: , , , , , and (Planck Collaboration et al., 2015a).
The light blue solid line shows in Fig. 6 the analytical model calculation with the mass function of Bocquet et al. (2015), which is in good agreement with the results obtained directly from the simulation at , while it under-predicts the power at higher multipoles. This can be understood partially as due to inhomogeneity in the distribution of pressure within halos, as the analytical model assumes a smooth distribution of pressure. Battaglia et al. (2012) show that inhomogeneity increases the power at these high multipoles by 20%. The remaining difference may be due to the pressure profile of low-mass halos (less than ) in the Magneticum Simulation being slightly different from the Planck pressure profile given in Eq. 10, as well as to the normalisation of the mass function for low-mass halos (see the next paragraph). In any case, the overall agreement between the simulation and the analytical model is satisfactory.
The analytical models with the other mass functions yield similar, but different, results. The only difference between the mass functions of Tinker et al. (2008) and Tinker et al. (2010) is that the latter forces the normalisation of the mass function by , where is the mean mass density of the universe. This normalisation mainly changes the abundance of low-mass halos which are not well resolved by their N-body simulations. As a result, the latter mass function (shown as the dashed light blue line in figure 6) gives larger power at high multipoles where the contributions from low-mass halos dominate.
At , the tSZ power spectra with both Tinker et al. mass functions are slightly larger than that with the mass function of Bocquet et al. (2015) (20 and 15 per cent larger at and 100, respectively). Indeed, the fitting formula for the mass function of Bocquet et al. (2015) gives similarly smaller than the fits of Tinker et al. mass functions at the relevant mass scales, i.e., . Bocquet et al. (2015) explain this by the way the fits are performed; namely, the actual data of the N-body simulations are similar between Bocquet et al. and Tinker et al., but the fitting procedures give slightly different results. Bocquet et al. use the Bayesian likelihood approach taking into account properly the Poisson nature of the mass function measured from the simulation, whereas Tinker et al. use the statistics. The former results do not depend on the bin size of the mass in which the fits to are performed, whereas the latter results do. Our results highlight the importance of better understanding the high-mass end of the mass function for the study of the tSZ power spectrum.
4 Mean Compton
4.1 Simulation results
The deep light-cone yields the mean Compton of (shown as the dotted horizontal line in figure 3) for the cosmological parameters used in the Magneticum Pathfinder simulation: and . We now study how builds up over cosmic time and how much objects of different masses contribute to . To this end, we proceed similar to the construction of the different slabs contributing to the light-cone. However, to avoid that the volume (and therefore the statistics) decreases with decreasing redshift like the volume swept by a light-cone of a given area in the sky, we produce maps of the full simulation at each redshift. This allows us to compute at every snap shot with the same precision given by the comoving box size of our simulation, which is slightly more than 2 Gpc.
Figure 7 shows d/dln as a function of , i.e., the time evolution of the contribution per logarithmic redshift interval to the overall signal. We find that all redshifts below contribute almost equally to within 30%. To study which masses contribute, we also show d/dln with high-mass halos above a certain mass threshold removed from the simulation. At , half of the signal comes from , whereas at smaller halos dominate. We find that, even at , there is more than 10% of d/dln coming from outside of resolved objects, e.g., the diffuse baryon component. This fraction increases at higher redshifts and reaches almost 30% at .
We also show an order-of-magnitude comparison (gray solid line) by Khatri & Sunyaev (2012) of the contribution from the epoch of reionisation and from low redshift WHIM. The first one is based on the reionisation optical depth inferred from CMB observations by WMAP whereas the later one is based on the simulations of Cen & Ostriker (1999) and assumes that the temperature of IGM is K at and K at and does not include the contribution of massive clusters of galaxies. While these rough estimates cannot be compared quantitatively with the simulation results, it demonstrates that these contributions are much smaller than those from hot gas in galaxy clusters in the simulation.
The dashed gray line shows the analytical model of the contributions of halos computed with the mass function of Tinker et al. (2010) and the Planck pressure profile with the mass bias of 1.2. The analytical model does a reasonable job describing the simulation result, although it is systematically lower than the simulation by 20% at . See section 4.2 for more detailed discussion on the analytical model.
The dashed black lines show the scatter due to kSZ at 150 GHz. The kSZ is most prominent at high redshift and starts to contribute even earlier than tSZ does.
In figure 8, we show the evolution of the cumulative mean Compton . The solid lines show , while the dashed lines show . The latter clearly shows that the mean Compton receives significant contributions up to . We find that cluster-size halos with contribute only 20% of the total signal, and nearly half of the signal, , comes from (which can be detected by stacking; Planck Collaboration et al., 2013d; Gralla et al., 2014; Greco et al., 2015) and the diffuse baryons outside halos.
4.2 Analytical model
where is Eq. 13 with . However, as the mean Compton receives significant contributions from lower mass halos compared to the power spectrum, we use . As our goal in this section is to confirm the result of the previous section, we use the cosmological parameters of the Magneticum Pathfinder simulation. All the other details of the calculation are the same as in section 3.3.2.
We find for the mass functions of Bocquet et al. (2015), Tinker et al. (2008), and Tinker et al. (2010), respectively. The mean Compton receives significant contributions from low-mass halos for which these mass functions differ significantly. In particular, the former two fitting functions do not satisfy the normalisation constraint on the mass function, . A reasonable agreement between the results from the simulation and the analytical model with the mass function of Tinker et al. (2010), which does satisfy the normalisation constraint, is encouraging; however, more study on a low-mass end of the mass function is necessary. The remaining difference of order 20% relative to the simulation is due to the Planck pressure profile with the mass bias of 1.2 being slightly lower than the pressure profiles in the simulation in lower mass halos (). The same trend can be seen in the tSZ power spectrum at shown in figure 6. For example, we find with the mass biases of 1.15 and 1.1, respectively. However, the mass bias of 1.1 would yield too large a tSZ power spectrum at low multipoles, , to agree with the simulation.
Using the Planck 2015 parameters, the mass function of Tinker et al. (2010) and the mass bias of 1.2, we find . Using this to scale the simulation result, we find .
Recently, Hill et al. (2015) use the analytical model with the mass function of Tinker et al. (2010) and the pressure profile of Battaglia et al. (2012) to obtain for the WMAP9 parameters with and (Hinshaw et al., 2013). A few minor details on the calculation are different: while they integrate the pressure profile out to 2.5 times the virial radius, we integrate out to ; their redshift integration is from to 6; and the lower boundary of their mass integration is . Changing the upper integration boundary from to in Eq. 13 and adjusting the integration boundaries in Eq. 14, we find for the Planck pressure profile with the mass bias of 1.2 and the same WMAP9 parameters. This value is significantly lower than their value. We find that this is due to the difference in the pressure profiles; while the pressure profiles of the Magneticum Pathfinder simulation, the Planck pressure profile with the mass bias of 1.2, and the profile of Battaglia et al. (2012) agree in the high-mass end, (Planck Collaboration et al., 2013a), the latter profile gives significantly larger pressure than the Planck profile in lower masses that dominate in .
These discrepancies need to be understood before we obtain an accurate estimate of the expected level of . Nevertheless, the original conclusion from the previous generation of cosmological hydrodynamical simulations (Refregier et al., 2000; da Silva et al., 2000; Seljak et al., 2001; Zhang et al., 2004b) seems robust: the expected from the large-scale structure is of order and is only one order of magnitude lower than the FIRAS bound. It is also encouraging that none of these estimates exceed the new Planck bound on the fluctuating part of the mean parameter, (Khatri & Sunyaev, 2015).
4.3 Buildup of the Compton Pdf
Finally, we study contributions from different redshifts to the build up of the PDF of the Compton signal. Figure 9 shows the PDF of d/dln at four different redshifts. Although these PDFs look at first glance similar to the one of the full light-cone (as shown in figure 5), the tail does not follow a simple power law. In general, with decreasing redshift, the PDF becomes broader and less sharply peaked, especially when compared to the PDF at .
In figure 10, we show PDFs with high-mass halos above a certain mass threshold removed from the simulation. At any given time, the tail is dominated by the most massive halos of that time.
The Magneticum Pathfinder simulations, state-of-the-art cosmological and hydrodynamical simulations, follow in detail the thermal and chemical evolution of the ICM as well as the evolution of super-massive black holes and their associated feedback processes. These simulations reproduce the average ICM pressure profiles measured by Planck (Planck Collaboration et al., 2013a) and SPT (McDonald et al., 2014). At the same time, the stellar mass functions of galaxies and the luminosity functions of the AGN population agree well with observations (Hirschmann et al., 2014). The improved numerical methods and increased computing power available today have enabled these simulations to follow a large enough cosmological volume to construct realistic light-cone maps, which can be compared with observations in detail.
In this paper, we have computed the tSZ and kSZ effects toward the counterpart of the Coma cluster in the local universe simulation, and statistics of the SZ effects from the light-cone maps, including the one-point PDF and power spectrum of tSZ and kSZ, and the mean Compton parameter. We have then compared these predictions on tSZ with the Planck, SPT, and ACT data. Our findings are summarised as follows:
The tSZ radial profile of Coma in the local universe simulation embedded in the background from the deep light-cone agrees well with that in the Planck data.
The local universe simulation predicts that the halo of Coma is moving away from us at km/s with respect to the CMB rest frame, thus yielding a negative kSZ within the central region of Coma. The magnitude of kSZ is ten percent of tSZ at 150 GHz. On the other hand, because Coma is a merging system, we find a significant relative motion of the core (even increasing the negative signal in the center) and a significant positive kSZ in the outskirt, which comes from a infalling sub-structure moving toward us. This makes a positive kSZ contribution to tSZ at 150 GHz at distances beyond 1 Mpc from the center of Coma.
The predicted one-point PDF of the Compton agrees with that measured by Planck, once the simulations are smoothed to the resolution of Planck’s map (10 arcmin). Given the much smaller beam size, we expect that ACT- and SPT-like instruments will see almost the full PDF that is resolved by our simulations. The tail of the full PDF follows a power-law with an index of .
The tSZ power spectrum measured from the simulation agrees with that of the Planck data at all multipoles up to , once the power spectrum is rescaled to Planck 2015’s “TT+lowP+lensing” cosmological parameters with and (Planck Collaboration et al., 2015a). We have confirmed and understood this result using the analytical model.
Consistent with the previous work, we continue to find the predicted tSZ power spectrum at that is significantly higher than that estimated by ACT and SPT. Whether this poses a challenge to theory is unclear, but our prediction is still well below the firm upper bound on tSZ given by the SPT data points with the primary CMB subtracted.
The simulation predicts the mean Compton value of for and . When the contributions from halos above a virial mass of are removed, we find ; thus, nearly half of the signal comes from such low-mass halos and diffuse gas outside halos. This remaining signal would pose a challenge to detecting the primordial -distortions.
Using the analytical model, we scale the Compton value from the simulation to the Planck 2015 parameters with and , finding . This is still lower than, but not far away from, the new Planck bound, (Khatri & Sunyaev, 2015).
The one-point PDF and the power spectrum of kSZ from our simulations agree broadly with the previous work. While our box size is large, the contribution to kSZ is still dominated by the largest modes within the box and originates mainly from high redshifts. Therefore, unlike for tSZ, we have not yet obtained a reliable, converged result on kSZ on large scales, . Simulations following even large cosmological volumes are needed.
In short, the main conclusion from our study is that all the properties of tSZ found in the Magneticum Pathfinder simulation and the local universe simulation agree well with the Planck data. This includes the tSZ power spectrum, which was previously found to be in tension with the Planck 2013 parameters (Planck Collaboration et al., 2014b; McCarthy et al., 2014). Now, the tSZ power spectrum calculated for the Planck 2015 parameters including CMB lensing information agrees with the measurement at all multipoles up to .
We thank N. Battaglia, J. C. Hill, and A. Saro for many helpful discussions, and R. Khatri for providing us with the PDFs of the Compton parameter estimated from the Planck data. E.K. thanks E. Jennings for her help in calculating the mass function of Tinker et al. (2008). K.D. acknowledges the support by the DFG Cluster of Excellence “Origin and Structure of the Universe”. We are especially grateful for the support by M. Petkova through the Computational Center for Particle and Astrophysics (CPAP). Computations have been performed at the “Leibniz-Rechenzentrum” with CPU time assigned to the Project “h0073”. Information on the Magneticum Pathfinder project is available at http://www.magneticum.org.
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