yan-chuan cai et al- full-sky map of the isw and rees-sciama effect from gpc simulations

8/3/2019 Yan-Chuan Cai et al- Full-sky map of the ISW and Rees-Sciama effect from Gpc simulations

1/26


2/26

2 Cai et al.

CMB to the ISW and RS effects (e.g. Martinez-Gonzalez& Sanz 1990; Martinez-Gonzalez et al. 1990; Rudnick et al.2007; Inoue & Silk 2006, 2007; Tomita & Inoue 2008; Masina& Notari 2009a,b). However, on theoretical grounds such anexplanation seems unlikely because the estimated sizes of the non-linear structures responsible for the cold spot aretypically > 100 Mpc, which seems too large to occur in aCDM universe which assumes Gaussian initial conditions(see also Cruz et al. 2005; McEwen et al. 2005; Cruz et al.2006; McEwen et al. 2006; Cruz et al. 2007; McEwen et al.2008, for discussion of non-Gaussianity.). However, it is stillunclear whether or not the combined ISW and RS effectscan generate cold spots of a few degrees with the right am-plitudes and whether such large-scale non-Gaussianity canarise from large-scale structure. A full understanding of theISW and Rees-Sciama effect could be crucial in explainingthe oddities of these observations. Meanwhile, the increasein sensitivity of forthcoming CMB experiments opens pos-sibilities of exploiting CMB temperature uctuations downto arcmin scales (Planck 1 , ACT 2 , SPT 3 and APEX-SZ 4 ),at which the ISW and RS effect may also entangle withother large-scale astrophysics of interest, i.e. lensing (e.g.Verde & Spergel 2002; Nishizawa et al. 2008; Mangilli &Verde 2009) and the Sunyaev-Zeldovich (SZ) effect (Sun-yaev & Zeldovich 1972) (e.g. Cooray 2002; Fosalba et al.2003; Bielby et al. 2009). To disentangle all these effects isthe key to thoroughly exploiting the information encoded inthese upcoming CMB measurements.

N-body simulations are the ideal tool for investigatingthe phenomena discussed above since they treat the non-linear regime accurately and permit the construction of afull sky map of the ISW and RS effects and the full under-lying 3-dimensional light-cone. Maps of the ISW effect havebeen constructed from both simulations and observations(Barreiro et al. 2008; Granett et al. 2009). Most of themsimply adopt the linear approximation, using only the den-sity eld to estimate the time derivative of the potential. Wewill show in this paper that these linear maps are far fromaccurate. There are also maps constructed from ray-tracingthrough simulations (Tuluie & Laguna 1995; Puchades et al.2006), but these are limited by small simulation box sizesand are not adequate to explore very large-scale structures.A full sky map of the RS effect has been constructed usinga constrained high-resolution hydrodynamical simulation tomodel the RS effect in the very local universe (Maturi et al.2007). This map is useful for understanding the RS effectfrom within the radial distance of 110 Mpc, which is a verysmall volume. Maps from the ray-tracing of large cosmolog-ical volumes are still missing.

In this paper, we develop a new method of constructinga full sky light-cone of the time derivative of the potential,, using a large N-bodysimulation. Our method of comput-ing is fully non-linear and so should model the completeRS effect as well as the ISW component. Our Gpc box sizesimulation provides a sufficient number of independent largescale modes to investigate the ISW effect fully. We ray-trace

1 www.sciops.esa.int/PLANCK/2 http://www.physics.princeton.edu/act/3 http://pole.uchicago.edu/4 http://bolo.berkeley.edu/apexsz

through the light-cone to produce maps of temperature uc-tuations induced by the ISW and RS effects. Our maps covera large range of scales and cosmic time with high accuracyand allow us to investigate the ISW and RS effects. Themaps will also be a valuable source for understanding CMBsecondary non-Gaussianity arising from large-scale struc-ture. They may also prove useful for disentangling lensingand SZ effects from the ISW and RS effects.

The paper is organized as follows. In

2, we present the

basic physics and the mathematical description of the ISWand RS effects. In 3, we describe our method of computingCMB temperature perturbations from our N-body simula-tion and ray-tracing to produce full sky maps from light-cone data. In 4, we identify and discuss three characteristicnon-linear features of the temperature perturbations. Fullsky maps are presented in 5. Finally, in 6, we discuss ourresults and draw conclusions.

2 THE ISW AND REES-SCIAMA EFFECT

In a CDM universe, dominance of the cosmological con-stant, , causes the expansion factor of the universe, a, togrow at a faster rate than the linear growth of density per-turbations, . Consequently, the cosmological constant hasthe direct dynamical effect of causing gravitational potentialperturbations, /a , to decay. The ISW effect is causedby the change in energy of CMB photons as they traversethese linearly evolving potentials. A CMB photon passingthrough an overdense region, or cluster, will gain more en-ergy falling into the potential well than it later looses climb-ing out of the evolved shallower potential well. Therefore,overdense regions correspond to hot regions in a linear ISWmap. The converse is true for a photon passing through anunderdense region. Here, the potential uctuation is posi-tive and the CMB photon looses more energy climbing thepotential hill than it subsequently regains from its descent.Therefore, underdense regions appear cold in a linear ISWmap. Non-linear growth of the density perturbations mod-ies this picture, producing additional temperature pertur-bations the RS effect. In overdense regions, the acceleratednon-linear growth of structure acts to increase the depth of the potential wells resulting in a reduction in the CMB tem-perature, partially cancelling the ISW effect. In contrast, inunderdense regions the RS effect enhances the ISW effect assaturation of the density contrast in voids further suppressesgrowth of the gravitational perturbation. We will discussother situations in which the RS effect makes a signicantcontribution and analyze the morphology of the resultingfeatures in the sky maps in 5.The net induced ISW plus RS, hereafter ISWRS, tem-perature uctuation along a direction n can be written as anintegral of the time derivative of the gravitational potential,, from the last scattering surface to the present (i.e. Sachs& Wolfe 1967; Martinez-Gonzalez et al. 1990),

T (n) = 2c2

T 0 Z t 0

t L(t, n ) dt, (1)

where t is cosmic time, tL the age of the universe at the lastscattering surface, t 0 the present age, the time derivativeof the gravitational potential, T 0 the mean CMB temper-ature and c the speed of light. This is equivalent to the

c 0000 RAS, MNRAS 000 , 000000


3/26

ISW Ray-Tracing 3

integral over radial comoving distance, r ,

T (n ) = 2c3

T 0 Z r L

0(r, n ) adr, (2)

where r L is the comoving distance to the last scatteringsurface and a the expansion factor.

To compute accurately from our simulation we makeuse of the Poisson equation expressed in comoving coordi-nates, 2 (x, t ) = 4 G (t)a 2 (x, t ), which can be writtenin Fourier space as

( k, t ) = 32 H 0k

2

m ( k, t )a

. (3)

Here (t ) is the mean density of the universe, the densitycontrast ( )/ , H 0 and m the present values of the Hubble and matter density parameters and G is thegravitational constant. Taking the time derivative yields

( k, t ) = 32 H 0k

2

m "aa 2 ( k, t ) ( k, t )a #. (4)Combining this with the Fourier space form of the continuityequation, (k, t ) + i k

p( k, t ) = 0, we have

( k, t ) = 32 H 0k

2

m "aa 2 ( k, t ) + i k p( k, t )a #, (5)where p( k, t ) is the Fourier transform of the momentumdensity divided by the mean mass density, p(x, t ) = [1 +(x, t )]v (x, t ). Equation (5) enables us to compute (in-cluding the contributions of both the linear ISW effect andthe non-linear RS effects) to high accuracy using the densityand momentum elds of our simulation. We will refer to theresults obtained using equation (5) as the ISWRS.

We wish to contrast these ISWRS predictions withthe corresponding results from linear theory. In the linearregime, ( k, t ) = D (t)( k, z = 0), where D (t) is the lineargrowth factor. Substituting this into equation (4) yields

( k, t ) = 32 H 0k

2

m aa 2

( k, t )[1 (t)], (6)where (t) denotes the linear growth rate (t) d ln D (t)/d ln a. This equation represents the conventionalway of modelling the ISW effect and uses only the in-formation from the density eld. It is equivalent to as-suming that the velocity eld is related to the densityeld by the linear approximation p( k, t ) = i(k, t ) k/k 2 i (t)(k, t )( a/a ) k/k 2 . We will refer to results obtained us-ing this linear approximation for the velocity eld as LAV.Note that with this LAV approximation, simply scales

with time according to the ISW linear growth factor, G(t) =aD (t)[1 (t)]/a2 .

In the LAV approximation the density eld, , directlydetermines the potential eld, , and its derivative, .Hence the morphology of the eld is determined directlyby the form of the density eld, , with overdense regions( > 0) corresponding to positive regions of and under-dense regions corresponding to negative regions of . In con-trast, for the exact calculation, represented by equation (5),this correspondence is broken and the dynamics of the den-sity eld play an additional direct role in determining .This is fully discussed in 4.

3 CONSTRUCTING FULL SKY MAPS

In this section, we describe our method of constructing fullsky maps of the ISWRS effect using our large simulation.

3.1 The Gpc simulation

To investigate the ISWRS effect, we need a simulation of suf-ciently large volume to include the very large scale pertur-

bation modes (hundreds of Megaparsecs) necessary to ver-ify convergence with linear theory. At the same time, weneed sufficiently high resolution to investigate the effects of small-scale non-linearity on the structures resolved in thesky maps. Moreover, for high accuracy ray-tracing, we needthe redshift spacing of the simulation outputs to be smallenough that interpolation between neighbouring redshiftsdoes not introduce signicant systematic errors.

The dark matter only N-body simulation we employ fol-lows 22003 -particles in a 1 h 1 Gpc periodic box in a CDMcosmological model with = 0 .74, m = 0 .26, b = 0 .044,8 = 0 .8 and H 0 = 71 .5 km s 1 Mpc 1 , chosen for con-sistency with recent CMB and large scale structure data(Sanchez et al. 2009). The simulation, run on the COSMAsupercomputer at Durham, was designed in order to makemock galaxy catalogues for forthcoming surveys (e.g. Pan-STARRS1 and EUCLID) and, as such, resolves the darkmatter halos of luminous galaxies (Baugh et al. in prep).The simulation has a softening length (Plummer equivalent)of 0.023 h 1 Mpc which provides more than adequate reso-lution for our purposes. The initial conditions were set upat redshift z = 49 and the simulation was run using GAD-GET (Springel et al. 2005) with output at 50 snapshots, 48of which lay between z = 0 and z = 10, where we mainlyfocus our analysis. The redshift intervals between neighbour-ing simulation outputs correspond to about 100 h 1 Mpc inradial comoving distance. We have veried that this is ade-quate to ensure that the errors induced by interpolating between snapshots are less than 2% at Megaparsec scalesand even less at larger scales.

3.2 Map construction

Our method of constructing the ISWRS sky maps has twostages. First, at each output redshift, we construct an esti-mate of on a cubic grid. In the second stage, we propagatelight rays from the observer and, as we move along each ray,we interpolate from the grids and accumulate the T dened by the integral of equation (2) along the past light-cone of the observer. Finally, we use HEALPix (G orski et al.2005) to visualize the T map in spherical coordinates. Wenow describe these steps in more detail.

3.2.1 Cartesian Grids

We construct the density eld, (x ), by assigning the massof each dark matter particle to a 3D mesh of cubic gridcells using the cloud-in-cell assignment scheme (Hockney &Eastwood 1981). We apply the same scheme to accumulatethe momentum density eld, p(x ), by assigning the vectormomentum of each particle to a Cartesian grid. Then weperform four independent Fast Fourier Transforms to com-pute the Fourier space versions of the density eld, ( k), and

c 0000 RAS, MNRAS 000 , 000000


4/26

4 Cai et al.

the three components of the momentum eld, p( k). Theseare then combined using equation (5) to yield the eld inFourier space. Finally, we perform an inverse Fourier trans-form to obtain (x ) in real space on a cubic grid. We repeatthe nal two operations using equation (6) to obtain the al-ternative LAV eld. This whole process is then repeatedfor each of the 50 simulation outputs.

For our 1 h 1 Gpc simulation, we have used a grid of 10003 cells each of 1 h 1 Mpc on a side. The resolution of theN-body simulation is signicantly greater and would warrantusing a ner grid. However, this would be very demandingof both machine memory and hard disk space and is notnecessary to resolve structures over 10 h 1 Mpc accurately,which is the smallest scale of interest in this paper.

3.2.2 Ray tracing through the light-cone

The next step is to choose a location for the observer andpropagate rays through the simulation. We choose to placethe observer at the corner of the simulation box at Carte-sian coordinate (0,0,0). We then used HEALPix (G orskiet al. 2005) to generate the directions of 3 145728 evenlydistributed rays, corresponding to an angular pixel scale of (6.87 )2 which is sufficient to resolve upto a spherical har-monic scale of l 1000. For each ray, we accumulate theintegral given by equation (2) by taking xed discrete stepsin comoving radial distance. At the location of each step,we nd the two output snapshots that bracket the lookbacktime at this distance. Using the Cartesian grids at each of these outputs we use the cloud-in-cell assignment scheme(Hockney & Eastwood 1981) to obtain the values of 1 and2 at the chosen position on the ray at these two lookbacktimes. Note that if the position along the ray lies outside thesimulation box, we use the periodic boundary conditions tomap the location back into the box. Finally, to estimate at the lookback time corresponding to the position on theray, we linearly interpolate (t)/G (t) with comoving radialdistance, r , using

(r 2 r 1 )G

= ( r 2 r )1G1

+ ( r r 1 )2G2

. (7)

Here G, G1 and G2 are the linear growth factors for atthe lookback time corresponding to the position on the lightray and the two neighbouring outputs that bracket it. Thevalues of r , r 1 and r 2 are the corresponding comoving radialdistances. This interpolation scheme guarantees that we re-cover the linear theory result exactly when is evolvingaccording to linear theory.

The maps that we present in Section 5 are constructed

and analysed using the HEALPix package (G orski et al.2005). We show maps corresponding to the contribution of the integral in equation (2) over different nite intervals of comoving radial distance. If we were to integrate over a co-moving distance larger than the 1 h 1 Gpc size of our simula-tion, the assumed periodic boundary conditions would createartifacts in the maps. For instance, in directions correspond-ing to the principal axes of the simulation volume the lightray would pass through the same location every 1 h 1 Gpc.The contributions to T (n ) from these replicas would addcoherently and lead to larger uctuations along these spe-cial directions than on average. In similar applications some

authors choose to rotate, ip and shift the replicated simula-tion cube. While this avoids the articial coherence betweenone replica and the next, it generates other problems includ-ing discontinuities of at cube boundaries. We have chosento evade this difficulty by mainly focusing on generating andanalyzing maps with a maximum radial depth equal to thesimulation cube size of 1 h 1 Gpc. It may still remain thecase that the same structure is seen in more than one di-rection, but this does not affect the mean power spectrumof the map and only produces non-Gaussian features on an-gular scales approaching the angular size subtended by thesimulation cube. Hence, the power spectra and small scalefeatures of the maps are not affected by the periodic natureof the simulation. 5

It is also important to establish that the resolution of the maps is not compromised by the spatial and temporalresolution of the simulation or our choices of size of grid celland integration step. The pixel size of our HEALpix mapsis 6.87 and matches the linear size of our Cartesian gridcells at an angular diameter distance of 500 h 1 Mpc. Thus,we would expect that beyond 500 h 1 Mpc our sky mapsare not lacking any small scale features due to the limitedresolution of our cubical grids. We have tested this using asmaller simulation of the same resolution but grided using0.5 rather than 1 h 1 Mpc cells. This test indicates that atl = 100, the larger, default, choice of cell size suppresses thepower in the maps made from the innermost 250 h 1 Mpcby just 10%. As we ray-trace further in the radial direc-tion, the accuracy improves, and is within 10% for l 1000for maps extending to 1000 h 1 Mpc. Additional tests haveshown that it is this cell size which is the limiting factor inthe resolution of our maps. For instance, we have checkedthat the maps are essentially unchanged if we make the in-tegration step size smaller than our default choice or if wehave more closely spaced simulation outputs. The full reso-lution of the N-body simulation could be exploited by usinga ner mesh but this is unnecessary for our purposes.

4 THE EFFECTS OF NON-LINEARITY

In this section, we compare our full non-linear estimates of the ISWRS effect with those of the LAV approximation. Weboth quantify and characterise the features that are gener-ated by non-linear gravitational evolution and elucidate thephysical processes by which they are generated.

Fig. 1 shows the temperature maps at three differentredshifts that result from computing the contribution to theISWRS integral equation (2) from a 100 h 1 Mpc thick sliceof the Gpc simulation. The top row shows the full ISWRS

calculation in which is computed using equation (5), whilethe middle row shows the result of using the LAV approxi-mation for given by equation (6). The LAV differs slightly

5 In fact, we nd that if we ignore this problem and producepower spectra directly from maps projected over a depth of 7000 h 1 Mpc or more, they do not differ signicantly from thosebuilt up from the power spectra of successive 1000 h 1 Mpc slices.Hence, at least for angular power spectra, our simulation size issufficiently large and the evolution of sufficiently rapid that thesuperposition of periodic replicas is not a concern.

c 0000 RAS, MNRAS 000 , 000000


5/26

ISW Ray-Tracing 5

Figure 1. Maps of T generated from a slab of thickness r = 100 h 1 Mpc taken from our Gpc simulation. From left to right, they aremaps at z = 0.0, 2.1, 6.2 respectively. From top to bottom, they are maps constructed using equation (5), which includes the ISW andRees-Sciama effects (ISWRS), maps constructed using the linear approximation for the velocity eld, equation (6), (LAV), and residualmaps of the top panels minus the middle panels, leaving essentially the Rees-Sciama (RS) contribution to the temperature uctuations.Note the individual temperature scales for each panel. At z = 0, we also show the momentum eld, averaged over the same slice of thesimulation, by the overplotted arrows. The three square boxes indicate the three regions we show in detail in Fig. 3 (solid-black box),Fig. 5 (dotted-black box) and Fig. 6 (dashed-yellow box).

from the standard ISW contribution to the temperature uc-tuations because to compute the ISW one should really usethe linear theory prediction for the density eld in equa-tion (6), whereas for the LAV we use the actual non-lineardensity eld, but assume the velocity eld is related to thedensity as in linear theory. However, the difference betweenthe LAV and the true ISW contribution is extremely small,

as we shall see when we compare their power spectra in Sec-tion 5. Consequently, the bottom row of panels, which arethe residual produced by subtracting the LAV maps fromthe corresponding ISWRS maps, are essentially the RS con-tribution to the temperature uctuations.

Let us consider rst the LAV maps. It is clear that thehot (red/yellow) and cold (blue/green) regions have a large

c 0000 RAS, MNRAS 000 , 000000


6/26

6 Cai et al.

Figure 2. Temperature perturbations along three light rays,shown in black, blue and red from z = 0 to z = 5 .8, correspond-ing to the comoving distance from r c = 0 to r c = 6000 h 1 Mpc.The top three panels show the temperature perturbations perunit comoving distance along each radial direction. Solid linesshow the results of the full ISW and Rees-Sciama effects (ISWRS)and dashed lines the results of the linear approximation for thevelocity eld (LAV). The fourth panel from the top shows the ac-cumulated temperature perturbations integrated along the radialdirection starting from the observer for ISWRS (solid lines) andLAV (dash lines). The bottom panel shows differences betweenthese accumulated ISWRS and LAV temperature perturbations.

coherence scale, approaching the size of the simulation box.The coherence scale is much larger than that of the un-

derlying density eld simply because of the k 2

factor inequation (4), which arises from the Poisson equation relat-ing and . As explained at the beginning of Section 2,hot and cold spots in these maps are induced by the de-cay of the gravitational potential in overdense and under-dense regions respectively caused by the dynamical effect of the cosmological constant, . In the higher redshift slices,the amplitude of the LAV/ISW uctuations are greatly re-duced. This is easily understood: as redshift increases becomes smaller and linear theory predicts density pertur-bations grow as a and so the corresponding potentialperturbations, /a , are constant. For = 0 there

are no ISW uctuations as CMB photons traversing sucha potential well will loose just as much energy leaving theperturbation as they gain on entering it.

The RS contribution to the temperature uctuations,shown in the bottom row of Fig. 1, have a much shortercoherence scale than the ISW uctuations. Their evolutionwith redshift is much more gradual than for the ISW uctu-ations. In the redshift z = 0 slice they are a minor contribu-tion to the overall ISWRS map (top row), but they become asignicant contribution at redshift z = 2 .1, generating smallscale structure within the smooth large-scale ISW uctua-tions. At z = 6 .2, the RS uctuations are almost completelydominant and while they are on a smaller scale than the ISWuctuations, they still produce lamentary structures whichcan be seen to be one hundred to a few hundreds of Mega-parsecs across. These panels give a visual conrmation of the two-point statistics presented in Cai et al. (2009), whichdemonstrated that both the importance and physical scaleof the RS effect become greater with increasing redshift.

Fig. 2 compares the ISWRS and LAV temperature per-turbations along three randomly chosen light rays (avoidingthe principal axes of the simulation box). In each of the topthree panels the solid line shows the overall ISWRS pertur-bation and the smoother dashed line the LAV contributionfrom z = 0 to z = 5 .8, corresponding to the comoving dis-tance from r c = 0 to r c = 6000 h

1 Mpc. As in the mapsof Fig. 1, we see that the LAV contribution varies moresmoothly and rapidly damps in amplitude as one progressesalong the rays to higher redshift. The RS contribution pro-duces high frequency uctuations around the LAV predic-tions with the amplitude of these high frequency uctuationsdecreasing with redshift at a much more modest rate. Thelower two panels of Fig. 2 show that while a large portion of the high frequency RS signal cancels out when accumulatedalong the line-of-sight, persistent residuals of the order of 1 K are also accumulated with a large contribution comingfrom the redshift range around a radial distance of about2000 h 1 Mpc.

In order to identify the non-linear physical processesthat give rise to the persistent features in the ISWRS maps,we have studied the momentum eld of our simulation. Theoverlaid white arrows in the left hand panels of Fig. 1 indi-cate the projected momentum eld in this slice of the sim-ulation. At higher redshifts we nd that the orientation of the momentum vectors is similar to that at z = 0, but theiramplitude increases, roughly in accord with the expectedlinear growth rate factor, . In the LAV panel (middle row)we see that on large scales the momentum eld is well cor-related with the temperature perturbations and hence withthe eld. Dark matter particles are moving towards hot

lumps (overdense regions, potential wells) and owing awayfrom cold lumps (underdense regions, potential hills). Thisis the scenario expected in linear theory. However, looking atthe smaller scale features evident in the ISWRS (top panel)maps we nd that this correlation between temperature andmomentum does not always hold. In fact, at high redshift,violations of the correspondence expected in linear theoryare very strong. Hence, guided by the momentum eld wehave identied three interesting non-linear phenomena in theISWRS maps, which are related to dipoles, convergent owsand divergent ows. Three regions exhibiting these phenom-ena are highlighted by the square boxes in Fig. 1 (one of

c 0000 RAS, MNRAS 000 , 000000


7/26

ISW Ray-Tracing 7

Figure 3. Maps of T in a slice of thickness r =50 h 1 Mpc and an area of 50 50 [h 1 Mpc] 2 . The x and y labels give the coordinatesof this set of maps in the full-box maps shown in Fig. 1. From left to right, they are maps at z = 0.0, 2.1, 4.2 respectively. From the rstto the third rows, they are maps constructed using equation (5), which includes the ISW and Rees-Sciama effect (ISWRS), maps usingthe linear approximation for the velocity eld, equation (6), (LAV), and residual maps of the top panels minus the middle panels, whichare essentially the Rees-Sciama (RS) contribution. The overplotted arrows show the projected momentum eld of the slice. The bottomrow shows log 10 ( + 1), where = ( )/ is the fractional perturbation in the projected density.

c 0000 RAS, MNRAS 000 , 000000


8/26

8 Cai et al.

Figure 4. Segments of the three light rays shown in Fig. 2, with the distances shown on the x-axis. The top panels show the temperatureperturbations per unit comoving distance in these chosen directions. Solid lines show the results of the full ISW and Rees-Sciama effects(ISWRS) and dashed lines those from the linear approximation for the velocity eld (LAV). The middle panels show the accumulatedtemperature perturbations along the radial direction for ISWRS (solid lines) and LAV (dashed lines). The bottom panels shows theaccumulated difference between the ISWRS and LAV perturbation. The left panel is an example of a dipole, the middle panel of aconvergent ow and the right panel of a divergent ow around an empty void.

which is split across the periodic boundary of the simula-tion). We zoom in and study each in detail in the followingsubsections.

4.1 Dipoles

In the RS maps, at scales of tens of Megaparsecs, large lumpsof dark matter moving perpendicular to the line-of-sight giverise to dipole features, i.e. a cold spot on the leading part of the lump and a hot spot on the trailing part. Fig. 3 showszoomed-in maps of the ISWRS, LAV and RS temperatureperturbations, along with a map of the projected densityeld, in the region around one such dipole at redshifts z = 0,2.1 and 4.2 for a 50 h 1 Mpc. The overplotted momentumvectors clearly show the bulk motion of a large lump of darkmatter. The length of the arrows indicate that this velocityis being damped and so decreases with decreasing redshift.

The LAV maps are very smooth and show no sign of thedipole feature at all, while the strikingly large dipole withamplitude of 1 K is clearly visible in the ISWRS maps(top row), particularly at the higher redshifts.

The physical origin of these dipole features, which are just a special case of the RS effect, can be understood interms of the evolution of the gravitational potential, , oneither side of the moving mass. At a xed position aheadof the moving mass, decreases as the mass and its gravi-tational potential well approaches. This will create a CMBcold spot as CMB photons passing through this point willgain less energy falling into the potential well than they

will subsequently loose climbing out of the then deeper well.Conversely, behind the mass, is increasing (becoming lessdeep) and so CMB photons gain more energy than they looseand this creates a hot spot.

As the maps in Fig. 3 are just the contributions to T from a thin, 50 h 1 Mpc, slice of our simulation, onemight worry that this articial truncation gives rise to ar-ticial edge effects. We have checked that this is not thecase by making corresponding maps projected through adepth of 1 h 1 Gpc, the box size of the periodic simula-tion. We nd that the dipole features are still clearly vis-ible, but, of course, they are somewhat perturbed by su-perposition of other perturbations along the longer line-of-sight. This nding is consistent with the accumulated dif-ference between the ISWRS and LAV T shown earlier inFig. 2 (bottom panel), which remain roughly constant forr c > 3000 h 1 Mpc in spite of some small-scale variations.

The amplitude of the temperature uctuations gener-ated by the dipoles is not large, but their characteristicdipole signature might enable them to be detected (Rubi no-Martn et al. 2004; Maturi et al. 2006, 2007). Dipoles gen-erated by moving galaxy clusters, which can equally bethought of as moving lenses, were predicted by Birkinshaw& Gull (1983), and later discussed by Gurvits & Mitrofanov(1986). Their detectability in the CMB and possible uses,for example to measure transverse motions of dark matter,have been further explored by Tuluie & Laguna (1995); Tu-luie et al. (1996); Aghanim et al. (1998); Molnar & Birkin-shaw (2000); Aso et al. (2002); Molnar & Birkinshaw (2003);

c 0000 RAS, MNRAS 000 , 000000


9/26

ISW Ray-Tracing 9

Figure 5. Like Fig. 3, but for slice of thickness r =100 h 1 Mpc, with an area of 200 200 [h 1 Mpc] 2 centred on a convergent ow.From left to right the panels show maps at z = 0.0, 2.1, 6.2 respectively. Note that the region shown here lies at the boundary of thesimulation box (see Fig. 1). To show it properly, we shifted the simulation box along the y-axis by 100 h 1 Mpc using the periodicboundary conditions, so that the y-axis ranges shown in the maps correspond to the combination of 890 to 1000 h 1 Mpc and 0 to90h 1 Mpc in the original simulation cube. The bottom row show the projected overdensity of the same region on logarithmic scale.

c 0000 RAS, MNRAS 000 , 000000


10/26

10 Cai et al.

Figure 6. As Fig. 3, but for a slice of thickness r =1000 h 1 Mpc, with an area of 300 300 [h 1 Mpc] 2 centred on a divergent ow.From left to right the panels show maps at z = 0.0, 2.1, 6.2 respectively. The bottom row show the projected overdensity of the sameregion on logarithmic scale.

c 0000 RAS, MNRAS 000 , 000000


11/26

ISW Ray-Tracing 11

Rubi no-Martn et al. (2004); Cooray & Seto (2005); Maturiet al. (2007). These studies considered only dipoles on thescale of galaxy clusters while our simulations reveal dipolesof K amplitude on scales ranging from 10 h 1 Mpc to afew tens of h 1 Mpc, the larger of which are seeded by bulkmotions on scales far larger than galaxy clusters.

Moving dark matter lumps will perturb the energies of CMB photons even if they move along, rather than trans-verse to the line-of-sight. Examples of the perturbations suchbulk motions cause are evident at several points along therays shown in Fig. 2. They appear as a sharp peak followedvery closely by a deep dip. The left-hand panels of Fig. 4show a close-up view of one of these features. Following theray on the top panel from the right to the left (i.e. movingin the direction of a CMB photon), the line dips and thenpeaks, indicating that a lump of dark matter is moving inthe opposite direction to the CMB photon. The local po-tential on the lumps leading part is getting deeper (coolingdown photons) while on the trailing edge the potential isbecoming shallower (heating up photons). The accumulated T (middle panel) is boosted and then suppressed as the raypasses through and the net effect is extremely small. This isto be contrasted to the case discussed previously when thelump moves transverse to the line-of-sight for which there isno cancellation.

4.2 Convergent Flows

In the RS and the higher redshift ISWRS maps of Fig. 1 wecan see several examples of cold regions surrounded by hotrings or laments which are centred on converging velocityows. These features are larger in scale than the dipoles,ranging from tens to hundreds of Megaparsecs. Fig. 5 showszoomed-in maps of the ISWRS, LAV and RS temperatureperturbations and the corresponding projected density eldwith overlaid velocity vectors at redshifts z = 0, 2 .1 and 6.2

for a 200 h 1

Mpc box centred on one such feature selectedfrom Fig. 1. In the LAV maps (middle row), we see onlya smooth hot lump, centred on the convergent ow, whoseamplitude increases towards low redshift as becomes in-creasingly important. In contrast, the RS contribution to thetemperature perturbations (bottom row) is negative at thecentre of the convergent ow and is surrounded by a pos-itive shell. Its amplitude evolves with redshift much moreweakly than the amplitude of the LAV perturbation. It isstrongest in the central panel, decreasing at the lowest red-shift due to the damping of non-linear growth by the latetime acceleration of the universe. At all but the lowest red-shift, this RS perturbation is a signicant contribution tothe overall ISWRS maps (top row). At high redshift, the

RS completely dominates over the LAV contribution turn-ing the hot spot predicted by linear theory into a cold spot,almost 200 h 1 Mpc in extent, with amplitude of order a K,surrounded by a hot shell. Even in the redshift z = 2 .1 slice,the RS contribution drastically changes the morphology of the LAV hotspot, producing a cold region in the very centreof the ow that is surrounded a hot lamentary shell.

The explanation of this counter-intuitive conclusionthat overdense regions can become cold spots surrounded byhot-rings, rather than the hotspots predicted by the ISW,involves principally the same physics as is responsible forthe dipoles. In the CDM model, overdense regions grow as

the result of the inow of lumpy material, often along la-ments. Each of these inowing lumps will give rise to a dipolefeature. On the leading edge of the lump, the potential is de-creasing and CMB photons loose energy, while on the trail-ing edge the potential is increasing and CMB photons areheated. The only difference is that dipoles seen on the skyare indicators of lumps of dark matter with large transversemomentum, while these larger scale cold regions surroundedby hot rings are produced by larger scale coherent conver-gent ows. One can imagine splitting the convergence owinto many small lumps of dark matter moving towards thesame centre: the cold region in the centre is just the resultof stacking many leading parts of those lumps, while the hotring consists of their trailing parts. Our ndings regardingthe morphology of the RS effect in converging ows are ingeneral agreement with analytic models of forming clustersthat have been discussed by other authors (e.g. Martinez-Gonzalez & Sanz 1990; Lasenby et al. 1999; Dabrowski et al.1999; Inoue & Silk 2006, 2007).

The maps shown in Fig. 5 are the contribution to the T perturbation of just a 100 h 1 Mpc slice of our simu-lation. Again, one should consider whether this truncationgives rise to articial edge effects that would qualitativelyaffect the appearance of the cold spots and hot rings. Inparticular, if the slice articially removes a foreground orbackground section of the hot shell this will enhance thevisibility of the cold central region. We have directly testedwhether this is a strong effect by making untruncated mapswhose depth is the full 1000 h 1 Mpc size of the periodicsimulation. We nd that while there are some cases wherethe heating and cooling cancel each other out, normally thevisibility and contrast of the cold spot and hot ring featuresis not strongly affected. This is perhaps to be expected asthe hot shell is diluted because it is spread over a muchlarger area than is subtended by the cold spot. Moreover,the dominance of the cold spots can be seen directly in theplots of the T contributions along particular lines of sight.An example of a ray passing through a cold spot is shownby the middle column of Fig. 4. In the top panel the broadpeak predicted by the LAV (dashed line) corresponds to anoverdense region in the simulation. There is a convergentow around this overdense region that gives rise to boththe sharp central dip and surrounding upward uctuationsseen in the ISWRS result (solid line). The lower panels showthat the difference between ISWRS and LAV perturbationsis strongly dominated by the central cold spot.

4.3 Divergent Flows

Divergent ows surrounding voids or underdense regions also

produce characteristic features in the ISWRS and RS maps.However their effect is not simply the reverse of that of the convergent ows. Instead, the non-linear behaviour inthese void regions always acts to enhance the LAV perturba-tion producing stronger cold spots. Fig. 6 shows zoomed-inISWRS, LAV and RS temperature maps and correspondingprojected density eld of a 300 h 1 Mpc region centred onsuch a cold spot. This region is underdense and the LAVmaps (middle row) show the expected linear ISW behaviourof a cold spot which grows in amplitude with decreasingredshift as increases and the potential hill associatedwith the underdensity decays. The RS contribution (bottom

c 0000 RAS, MNRAS 000 , 000000


12/26

12 Cai et al.

row) also produces a cold spot at the centre of the region,but surrounded by a hot lamentary shell. Thus, perhapscounter-intuitively, the pattern of the RS contribution forthese divergent ows is the same, and has the same sign, asfor the convergent ows discussed above.

The reason why the effect of non-linearity in an under-dense region is not simply the reverse of its effect in an over-dense region is because the two situations are not symmet-rical. In an overdense region the overdensity is unboundedwhile in a underdense region the density cannot drop belowzero, so > 1. It is this saturation of the density contrastin voids that prevents the perturbation from growing in thenon-linear regime even as fast as linear theory would predictand so enhances the rate of decay of .

Again, the amplitude of the RS contribution evolveswith redshift much more weakly than the LAV contribu-tion. At z = 0, the LAV contribution to the ISWRS is over-whelmingly dominant, but at higher redshift, the RS effectcontributes by reinforcing the linear effect in the centre of the cold spot and suppressing it in the outer regions.

4.4 Overview of non-linearity

A useful summary of the three physical effects we have dis-cussed above is given by the schematic diagrams in Fig. 7.They depict the gravitational potential, /a , a t abaseline reference epoch, say when the CMB photon entersthe region, and both the LAV and fully non-linear predic-tions for the evolved potential at a later time, say whenthe CMB photon exits the region. These schematics yieldstraightforward interpretations of each of the phenomenawe have identied in the ISWRS and RS maps.

In void regions, the ISW effect is easily understood asthe result of the linear decay of the gravitational potential.CMB photons are cooled as they loose more energy climbingthe initial potential hill of the void region than they subse-

quently regain on departing the now shallower potential hill.This cooling is depicted by the heavy arrows in the left-handpanel of Fig. 7. One would expect non-linearity to acceler-ate the growth of and hence increase the rate of growthof the potential hill. This is what we observe to occur atthe edge of the voids and leads to a component of heatingof the CMB photons in this outer shell, as depicted by thelight upward arrows. However, once the centre of a void re-gion becomes almost empty, i.e. 1, the local densitycontrast stops growing, as it cannot become emptier. In thiscase, the expansion of the universe will reduce the height of the potential hill, just as it does in the linear regime in voids.Thus, in the centre of voids the RS effect has the same signas the linear ISW effect and the two reinforce each other to

produce cold spots at the centre of voids.In overdense regions such as galaxy clusters, the lin-ear decay of the gravitational potential well results in theheating of CMB photons as depicted by the heavy arrowsin the central panel of Fig. 7. The effect of non-linear in-fall and growth of the density perturbation is to shrink thescale of the potential well while deepening its central value.In the centre of the cluster this non-linear growth acts inopposition to the linear decay of the potential and cools theCMB photons, depicted by the downward light arrow. Wehave seen that at high redshift this effect can be strongerthan the linear ISW effect, resulting in cold spots centred

on overdense regions. At larger scales, the inowing regionshrinks the scale of the potential well and results in a furtherreduction in the depth of the potential well, reinforcing thelinear effect. Hence, in an outer shell around the cluster, thenon-linear effect heats CMB photons. This latter effect canalso be considered as the superposition of a set of dipolesarising from lumps in a surrounding inowing region.

The right-hand panel of Fig. 7 depicts the situation thatgives rise to a characteristic dipole perturbation of the CMB.A moving cluster, or other overdense region, gives rise to amoving potential well. Linear theory does not model themovement of the potential well and would instead predict asimple static decaying potential well, resulting in a hot spotas indicated by the heavy upward arrows. In contrast, themovement of the potential well leads to a rapid deepeningof the potential well on the leading edge of the cluster and acorresponding rapid increase of the potential on the trailingedge. This gives rise to cooling of the CMB photons on theleading edge and heating on the trailing edge as indicatedby the light arrows.

5 SKY MAPSWe have used the methods described in 3 to construct fullsky maps of the ISWRS effect using both the density andvelocity elds, and using the LAV approximation. We havealso generated maps of the RS effect, by subtracting the LAVmaps from the corresponding ISWRS maps. In all of the skymaps we have removed both the monopole and dipole usingthe REMOVE DIPOLE subroutine of HEALPix (G orskiet al. 2005).

The top panel of Fig. 8 shows the ISWRS map madeby integrating along the line-of-sight of the observer overthe range 0 < z < 0.17, corresponding to distances from theobserver, r c , in the range 0 < r c < 500h

1 Mpc. The whole

sky is dominated by a few large hot and cold features withamplitudes of a few K to 10 K, as expected from lineartheory. The bottom panel shows the ISWRS map integratedfrom 0.17 < z < 0.57, corresponding to 500 < r c < 1500h 1 Mpc. The typical angular size of the features in thisshell is smaller than in the top panel. As we will discuss laterin this subsection, the smaller angular size of the featuresin the bottom map relative to the top map is due largelyto the fact that there is a cut-off in the power spectrum of the simulation used to build the sky maps. Unlike in thereal universe, there is no power contributing to our maps onscales larger than the fundamental modes of the simulation.This defect can easily be remedied: as the missing modes areessentially in the linear regime, it would be straightforward

simply to add additional longer wavelength modes by handbefore doing the line-of-sight projection to make the maps.This extra power, however, would be the same in maps of the ISWRS effect and the LAV approximation, and so wouldvanish identically from the maps of the RS effect. For thisreason we have chosen not to add these linear modes to theISWRS and LAV maps.

The corresponding maps, but with the LAV approxima-tion, are shown in Fig. 9. Comparing Fig. 8 with Fig. 9, wesee that the large-scale distribution is essentially identical,but differences are apparent on smaller scales. The differencemap made by subtracting the LAV map from the ISWRS

c 0000 RAS, MNRAS 000 , 000000


13/26

ISW Ray-Tracing 13

Figure 7. Schematic diagrams of the evolution of the gravitational potentials that produce the three characteristic features of thenon-linear RS effect discussed in Section 4. The solid lines depict for a void (left), cluster (middle) and moving cluster (right) thegravitational potential at a reference epoch, say when a CMB photon enters the potential. The dashed and dotted lines show predictionsfor the evolved potential at a later time, say when the CMB photon exits the region. The dashed lines are the linear prediction of theLAV approximation and the dotted line represents the fully non-linear result. The heavy up and down arrows indicate the heating orcooling of the CMB that is predicted by the LAV approximation (essentially the linear ISW effect). The lighter arrows indicate the RScontribution to the temperature perturbations, i.e. the difference between the fully non-linear and LAV contributions.

map to reveal the RS effect, is shown in Fig. 10. In the toppanel, covering the range 0 < z < 0.17, we see some strikinglarge-scale structures with amplitudes that are about 1 K,which is about 10% of the amplitude of the features seen inthe ISWRS and LAV maps themselves. A few large dipoles,ranging from a few degrees to over 10 degrees are clearlyvisible, indicating the large-scale bulk ow of matter in thelocal universe of our observer. The bottom panel shows theRS effect for a projection covering the range 0 .17 < z < 0.57.Many more dipoles are visible, with typical angular scales of a few degrees. In addition, the map has large-scale lamen-tary structure which is coherent over lengths of up to tensof degrees. The typical amplitude of these features is a fewK, again about 10% of the amplitude of the features in thecorresponding ISWRS and LAV maps.

In Fig. 11 and Fig. 12, we show a patch of sky of size40 40 degrees, for the redshift intervals of 0 .57 < z < 1.07(1500 < r c < 2500h 1 Mpc) and 1 .07 < z < 1.78 (2500 80 (a few degrees) and completelydominate for l > 200. At still smaller scales the kineticSZ effect is expected to dominate and at such scales a fulltreatment would have to incorporate additional modellingof this contribution. The RS contribution to the tempera-ture maps is strongly non-Gaussian with a skewed one-pointdistribution. In future work it will be interesting to inves-tigate the RS contribution to higher order statistics as itsnon-Gaussian characteristics might limit the ability to de-tect primordial non-Gaussianity in the underlying primaryCMB uctuations. Combining our full-sky maps with mockgalaxy catalogues built from the same N-body simulationswill be a powerful tool for developing cross-correlation tech-niques aimed at extracting the ISWRS signal from redshiftsurveys.

ACKNOWLEDGEMENT

YC was supported by the Marie Curie Early Stage TrainingHost Fellowship ICCIPPP, which is funded by the Euro-pean Commission. YC acknowledges the support of grantDE-FG02-95ER40893 from the US Department of Energy.We thank Carlton Baugh, Elise Jennings and Raul Angulofor providing the Gpc simulation, which was carried out onthe Cosmology Machine at Durham. We thank Istvan Sza-pudi and Ben Granett for useful discussion. We also thankRavi Sheth, Lam Hui and the anonymous referee for use-ful comments. YC thanks Andrew Cooper for useful dis-cussion on technical details and careful reading through of the paper, and Lydia Heck for computing support. CSF ac-knowledges a Royal-Society Wolfson Research Merit Award.This work was supported in part by the STFC rolling grantST/F002289/1.

REFERENCES

Aghanim N., Prunet S., Forni O., Bouchet F. R., 1998,A&A, 334, 409

Arnau J. V., Fullana M. J., Saez D., 1994, MNRAS, 268,L17+

Aso O., Hattori M., Futamase T., 2002, ApJ, 576, L5Barreiro R. B., Vielva P., Hernandez-Monteagudo C.,

Martinez-Gonzalez E., 2008, IEEE Journal of SelectedTopics in Signal Processing, vol. 2, issue 5, pp. 747-754,2, 747

Bielby R., Shanks T., Sawangwit U., Croom S. M., RossN. P., Wake D. A., 2009, ArXiv e-prints

Birkinshaw M., Gull S. F., 1983, Nature, 302, 315

Boubekeur L., Creminelli P., DAmico G., Nore na J.,Vernizzi F., 2009, ArXiv e-printsCai Y.-C., Cole S., Jenkins A., Frenk C., 2009, MNRAS,

396, 772Cooray A., 2001, Phys. Rev. D, 64, 063514Cooray A., 2002, Phys. Rev. D, 65, 103510Cooray A., Seto N., 2005, Journal of Cosmology and Astro-

Particle Physics, 12, 4Cruz M., Cayon L., Martnez-Gonz alez E., Vielva P., Jin

J., 2007, ApJ, 655, 11Cruz M., Martnez-Gonz alez E., Vielva P., Cay on L., 2005,

MNRAS, 356, 29

Cruz M., Tucci M., Martnez-Gonz alez E., Vielva P., 2006,MNRAS, 369, 57

Dabrowski Y., Hobson M. P., Lasenby A. N., Doran C.,1999, MNRAS, 302, 757

Fosalba P., Gazta naga E., Castander F. J., 2003, ApJ, 597,L89

Francis C. L., Peacock J. A., 2009, ArXiv e-printsFullana M. J., Saez D., Arnau J. V., 1994, ApJS, 94, 1Giovi F., Baccigalupi C., Perrotta F., 2003, Phys. Rev. D,

68, 123002Gorski K. M., Hivon E., Banday A. J., Wandelt B. D.,

Hansen F. K., Reinecke M., Bartelmann M., 2005, ApJ,622, 759

Granett B. R., Neyrinck M. C., Szapudi I., 2009, ApJ, 701,414

Granett B. R., Szapudi I., Neyrinck M. C., 2009, ArXive-prints

Gurvits L. I., Mitrofanov I. G., 1986, Nature, 324, 349Hockney R. W., Eastwood J. W., 1981, Computer Simula-

tion Using ParticlesHu W., Dodelson S., 2002, ARA&A, 40, 171Inoue K. T., Silk J., 2006, ApJ, 648, 23

Inoue K. T., Silk J., 2007, ApJ, 664, 650Lasenby A. N., Doran C. J. L., Hobson M. P., DabrowskiY., Challinor A. D., 1999, MNRAS, 302, 748

Mangilli A., Verde L., 2009, ArXiv e-printsMartnez-Gonz alez E., Gallegos J. E., Arg ueso F., Cayon

L., Sanz J. L., 2002, MNRAS, 336, 22Martinez-Gonzalez E., Sanz J. L., 1990, MNRAS, 247, 473Martinez-Gonzalez E., Sanz J. L., Silk J., 1990, ApJ, 355,

L5Masina I., Notari A., 2009a, Journal of Cosmology and

Astro-Particle Physics, 7, 35Masina I., Notari A., 2009b, Journal of Cosmology and

Astro-Particle Physics, 2, 19Maturi M., Dolag K., Waelkens A., Springel V., Enlin T.,

2007, A&A, 476, 83Maturi M., Enlin T., Hern andez-Monteagudo C., Rubi no-

Martn J. A., 2007, A&A, 467, 411Maturi M., Ensslin T., Hernandez-Monteagudo C., Rubino-

Martin J. A., 2006, ArXiv Astrophysics e-printsMcEwen J. D., Hobson M. P., Lasenby A. N., Mortlock

D. J., 2005, MNRAS, 359, 1583McEwen J. D., Hobson M. P., Lasenby A. N., Mortlock

D. J., 2006, MNRAS, 371, L50McEwen J. D., Hobson M. P., Lasenby A. N., Mortlock

D. J., 2008, MNRAS, 388, 659Molnar S. M., Birkinshaw M., 2000, ApJ, 537, 542Molnar S. M., Birkinshaw M., 2003, ApJ, 586, 731Nishizawa A. J., Komatsu E., Yoshida N., Takahashi R.,

Sugiyama N., 2008, ApJ, 676, L93Page L., Barnes C., Hinshaw G., Spergel D. N., Weiland

J. L., Wollack E., Bennett C. L., Halpern M., Jarosik N.,Kogut A., Limon M., Meyer S. S., Tucker G. S., WrightE. L., 2003, ApJS, 148, 39

Panek M., 1992, ApJ, 388, 225Puchades N., Fullana M. J., Arnau J. V., S aez D., 2006,

MNRAS, 370, 1849Quilis V., Ibanez J. M., Saez D., 1995, MNRAS, 277, 445Quilis V., Saez D., 1998, MNRAS, 293, 306Rees M. J., Sciama D. W., 1968, Nature, 217, 511

c 0000 RAS, MNRAS 000 , 000000


26/26

26 Cai et al.

Rubi no-Martn J. A., Hern andez-Monteagudo C., EnlinT. A., 2004, A&A, 419, 439

Rudnick L., Brown S., Williams L. R., 2007, ApJ, 671, 40Sachs R. K., Wolfe A. M., 1967, ApJ, 147, 73Saez D., Arnau J. V., Fullana M. J., 1993, MNRAS, 263,

681Sanchez A. G., Crocce M., Cabre A., Baugh C. M., Gaz-

tanaga E., 2009, ArXiv e-printsSmith R. E., Hern andez-Monteagudo C., Seljak U., 2009,

Phys. Rev. D, 80, 063528Springel V., White S. D. M., Jenkins A., et al. 2005, Nature,

435, 629Sunyaev R. A., Zeldovich Y. B., 1972, Comments on As-

trophysics and Space Physics, 4, 173Tomita K., Inoue K. T., 2008, Phys. Rev. D, 77, 103522Tuluie R., Laguna P., 1995, ApJ, 445, L73Tuluie R., Laguna P., Anninos P., 1996, ApJ, 463, 15Verde L., Spergel D. N., 2002, Phys. Rev. D, 65, 043007Vielva P., Martnez-Gonz alez E., Barreiro R. B., Sanz J. L.,

Cay on L., 2004, ApJ, 609, 22Zhang R., Huterer D., 2009, ArXiv e-prints

yan-chuan cai et al- full-sky map of the isw and rees-sciama effect from gpc simulations

Documents