A&A 419, 439-447 (2004)
DOI: 10.1051/0004-6361:20035587
J. A. Rubiño-Martín - C. Hernández-Monteagudo - T. A. Enßlin
Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str.1, Postfach 1317, 85741 Garching, Germany
Received 28 October 2003 / Accepted 11 February 2004
Abstract
The Rees-Sciama effect produced in mergers of galaxy clusters
is discussed, and an analytical approximation
to compute this effect from numerical simulations is given.
Using this approximation and a novel model describing the physics
of the merger, we characterize the spatial properties and symmetries
of the Rees-Sciama signal.
Based on these properties, we propose a method
to extract the physical parameters of the merger, which
relies on the computation of the quadrupole moment
of the observed brightness distribution on the sky.
The relationships between the quadrupole coefficients and
the physical parameters of the merger (physical separation,
projection angle on the sky and angular momentum) are discussed.
Finally, we propose a method to co-add coherently the RS signals
from a sample of cluster mergers, to achieve a statistical
detection of the effect for those cases where individual signals
are masked by the kinetic SZ effect, the primordial
CMB components, and by observational noise.
Key words: galaxies: intergalactic medium - galaxies: clusters: general - galaxies: interactions - gravitational lensing - cosmology: cosmic microwave background
The integrated Sachs-Wolfe (ISW; Hu & Sugiyama 1994; Sachs & Wolfe 1967) effect produces CMB temperature fluctuations due to the accumulation of red- or blue-shift of photons travelling in time dependent gravitational potentials. Several recent studies (Fosalba & Gaztañaga 2003; Boughn & Crittenden 2004; Nolta et al. 2003) have claimed evidence for its detection in the WMAP data (Bennett et al. 2003). The non-linear contribution to the ISW effect, in which the density contrast producing the gravitational potential is in its non-linear regime, is usually called the Rees-Sciama (RS; Rees & Sciama 1968; Seljak 1996) effect.
Here we discuss the RS effect in the extremely non-linear regime of present day galaxy cluster mergers. The regime of moving single galaxy clusters was already examined by Birkinshaw & Gull (1983), Aghanim et al. (1998), Molnar & Birkinshaw (2003,2000), and Cooray (2002). However, galaxy clusters reach the largest velocities, and therefore the strongest RS signal, during mergers in which two or more clusters are invoked. Due to the slow centre-of-mass velocity of the merging system, the RS signals of the merging subclusters can partly cancel or increase each other, depending on the merger geometry and the location of the line-of-sight (LOS).
Furthermore, the spectral signature of the RS effect is a temperature change of the measured CMB photons, and therefore indistinguishable form the kinematic Sunyaev-Zeldovich (kSZ; Sunyaev & Zeldovich 1972,1980) effect spectrally. However, both effects have distinct morphologies, which in principle allow to discriminate them.
The focus of this work is to examine methods to extract the RS signature from merging pairs of galaxy clusters. The layout of the paper is as follows. In Sect. 2 we provide an approximation to compute the RS effect either from theoretical cluster models or numerical simulations. In addition, a brief description of the kSZ and the thermal Sunyaev-Zeldovich (tSZ) effects is also given, emphasizing their phenomenological differences with the RS effect. In Sect. 3 we present and use an analytic model of a merger of two galaxy clusters to characterize the typical morphologies of these effects (RS, kSZ and tSZ). Based on its spatial properties, we present in Sect. 4 a method to extract both the RS signal and the physical parameters describing the system from a given merger. A method to stack the signal from a sample of mergers is also proposed. Conclusions are presented in Sect. 5.
The CMB temperature change in a LOS direction
caused
by the RS effect can be written as (Cooray 2002)
In the following, we are dealing with well separated structures, galaxy clusters, which do not change significantly during the time of a photon passage. Thus, we explicitly neglect the intrinsic variation of the cluster potential while the photon is passing through it. This allows an approximative treatment.
In Newtonian approximation, the gravitational potential depends on the
total mass density
via
(2) |
By inserting Eq. (3) into Eq. (1) we approximate
all LOS to be parallel to the z-direction, we ignore the explicit
dependency of
and v on time, and integrate out the LOS integral, yielding
Nevertheless, its accuracy is sufficient to study the general difficulties in RS signal extraction from merging clusters of galaxies. Its simplicity allows easy inclusion into any code which calculates the kSZ effect from numerical simulations.
One can introduce the LOS projected momentum
(5) |
Equation (6) can also be reproduced using a different approach. The moving cluster of galaxies (MCG) effect (Birkinshaw & Gull 1983) was originally derived as a gravitational lensing effect. However, and as it was pointed out by Molnar & Birkinshaw (2000), this effect can be seen as a special type of RS effect, which is not caused by intrinsic variations of the gravitational field, but due to the movement of the cluster with respect to the rest frame of the CMB. Given that we have used exactly this assumption when deriving Eq. (4), then our expression is completely explained as a gravitational lensing effect.
Following Birkinshaw & Gull (1983) and Molnar & Birkinshaw (2000), the MCG effect for a cluster moving with velocity ,
for small values of the lensing deflection angle, can be written as
Since the kSZ effect has the same spectral signature as the RS effect,
it is crucial to include the latter into the considerations. Using similar
planar and time-independent approximations the kSZ effect reads
(9) |
Since the electron and dark matter velocity fields are not too
different, and the baryonic matter density follows roughly the dark
matter density (
,
with
the cosmic baryon mass
fraction), one can state that
Comparing Eq. (6) with Eq. (10) one recognises properties which hopefully allow to separate the RS signal from the kSZ effect:
Because of its different spectral characteristic the tSZ effect can
be separated from the the kSZ and RS effects. However, it has to be
considered simultaneously not only because of possible residuals from
incomplete removing of the tSZ effect from measurements, but also
since the tSZ effect helps to localise the gas distribution and
therefore to identify the regions which are expected to contain the
strongest kSZ contamination. The usual tSZ y-parameter is given by
(11) |
To be able to study the strength of the signal we construct a simplified model of a binary cluster merger. We grossly simplify the complicated gravitational and hydrodynamical process in order to have an analytic model. Nevertheless this model should provide us with maps of the RS, kSZ, and tSZ effects, which are sufficiently realistic for the purpose to study RS signatures by describing the typical strength and morphology of the effects.
We build the merger model out of two cluster models, labelled with the numbers 1 and 2, and each described by a NFW dark matter profile (Navarro et al. 1997) filled by isothermal hydrostatic gas. The two clusters approach each other. Each cluster velocity ( ) is calculated to be the velocity of a free-falling object in the gravitational potential of the other cluster. The total momentum of the merging system is assumed to be negligible.
Deformation of the clusters due to mutual tidal forces are ignored for simplicity, so that the dark matter and the gas of each cluster are streaming with the same velocity. Furthermore, hydrodynamical interactions of the gases of the two clusters are ignored, the gases are treated as non-interacting fluids. This coarse approximation is justified in so far that the kSZ and the RS effects both are only sensitive to momentum, and only in projection, but not to the thermodynamical state of the gas. The ignorance of shock waves exchanging the momentum of the gases therefore does not strongly change the signal strength, only its exact spatial morphology.
In contrast to that, the tSZ effect is very sensitive
to the thermodynamical state of the
gas and has therefore to be treated with a little bit more care. Again
we do not try to model the exact spatial structure but try to get a
good estimate of the signal strength. For that, we have to estimate the amount
of shocked gas produced by the merger. We do this by measuring for
each volume element the excess kinetic energy of the gas of the two
superposed clusters with respect to the volume's bulk kinetic energy
and assuming this to be dissipated. The total thermal energy density
of a volume element is therefore approximated to be
(12) |
= | |||
(13) |
In this subsection, we present the equations outlined in the previous
one for the simulation of a merger of two galaxy clusters,
with a null total linear momentum. We assume a NFW dark matter
profile for each cluster,
(16) |
Figure 1: RS ( left image), kSZ ( central image) and the sum of both effects ( right image), for a pair of merging clusters with masses (left cluster) and (right cluster), with zero angular momentum, when observed with projection angle , and a physical separation of 1 Mpc. Contour lines in all plots correspond to (-64, -16, -4, -1, 1, 4, 16, 64) times the maximum of the map (dotted-lines for negative regions and solid lines for positive ones). | |
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For the computation of the RS effect, we use the 2-D projection of the
NFW profile (Eq. (14)), which is given by
Velocities are estimated following the formalism described in
Sarazin (2002), although we modified it to use the
gravitational energy associated to the NFW matter distribution
described by Eq. (14), because it is
more realistic and better behaved at r=0. For this latter profile, the
gravitational potential is found to be
(18) |
(19) |
Figure 2: Same as Fig. 1, but considering angular momentum, and with a larger projection angle. Here, , and the velocity vector is defined by the spherical angles . Contours have the same meaning as in Fig. 1. | |
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In this subsection, we use our toy model for two merging clusters to present two examples illustrating the spatial properties and symmetries of the effect. For definiteness, we choose the coordinate system with the Z-plane as the sky plane, with positive values pointing towards the observer. If we define as the angle between the direction joining the two centres of mass of the clusters (from M_{1} to M_{2}) and the X-axis, then we choose (i.e. clusters on the X-axis, and M_{1} on the left of the image). This coordinate system will be used throughout the paper, unless otherwise stated.
We first show in Fig. 1 the result for a merger with = . This is a very optimistic case (in terms of the RS signal strength compared to the kSZ effect), where the merger is practically face-on.
As pointed out in Sect. 2.3, the spatial pattern of the RS effect for a merging of two clusters shows a peculiar quadrupole structure. This quadrupole structure carries information about the velocity field of the clusters, and can exhibit, for example, if both clusters are in a pre- or post- merger state.
This fact is used in the next section to extract the RS signal in a map where kSZ is also present, given that the kSZ has a dipole spatial structure. On the other hand, it can also be seen in the figure that the RS effect has a much more extended pattern than the kSZ, because of its gravitational nature.
As a second example, in Fig. 2 we consider a more pessimistic case in which the velocity forms an angle of with the line of sight. We incorporate also angular momentum. In this case, = . Here a non-zero angular momentum is implicitly defined by assigning a velocity direction to the cluster M_{1} different from the direction joining the two cluster centres. The direction of is specified by the spherical polar angles , where is the usual polar angle from the Z axis, and the usual azimuthal angle measured from the X-axis. In the case presented in Fig. 2, the RS effect is practically hidden by the kSZ, although it can be marginally seen at large distances.
Once we have illustrated the morphology of the RS effect, we will study how to extract the RS information from observations of merging clusters.
The peculiar structure of the RS effect in merging clusters can be quantified studying the dipole and quadrupole moments of the observed intensity map. It is worth remarking that the quadrupole moment of the surface brightness is also used as a tool to extract the shear signal when studying weak lensing-induced distortion of faint background galaxies (Kaiser et al. 1995).
We propose the following recipe to proceed with future experiments aiming to measure this effect:
(20) |
For a 2D image, the -matrix only has 2 independent elements because it is symmetric, and tr() = 0. Thus, the matrix is specified by quoting Q_{xx} and Q_{xy}. However, the quadrupole matrix can be rotated to its principal axis, so that a single number characterises the matrix (one of the eigenvalues), plus an angle specifying the orientation of the principal axis. It is easy to show that the two eigenvalues for are , which suggests to introduce the total, coordinate independent quadrupole moment as .
We will now illustrate how to extract information about the physical parameters describing the merger from the suggested recipe. For definiteness, we will assume here that we have two observational maps available, one of the Compton y-parameter ( ), and another map of the brightness distribution on the sky at ( ). We will denote as the quadrupole moment associated to the y-parameter map, and , the one associated to the thermal temperature fluctuation map ( ). We will start by considering the case of zero angular momentum, and neglecting the CMB contribution. These other cases will be discussed in the following two subsections.
Figure 3: Dependence of the measured quadrupole (Q_{xx}) on the projection angle . These curves have been derived for the case of (left cluster), M_{2} = 5 (right cluster), separation of 1 Mpc, and the clusters placed along the X-axis (so Q=|Q_{xx}|) and we use (i.e. zero angular momentum). We performed the computation of the quadrupole in the whole image (quoted as "all''), and masking the region that in the y-parameter map has a signal above the 10% of the peak intensity (quoted as "outer''). We can see that when doing this, the quadrupole signal is dominated by the RS component. The flux is given in mJy, at the cross-over frequency of the tSZ effect ( ). The merging system is assumed to be at . | |
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In Fig. 3 we show the dependence of the observed quadrupole with the projection angle. The coordinate system has been placed at the "center-of-light'' of the y-parameter map, and the maps have been rotated so the cluster centres in the tSZ map lie along the X-axis (so we only quote Q_{xx} because Q_{xy}=0). Given that the weighting-factor in Eq. (21) is dimensionless, then has units of flux, so in order to quote these values in mJy, we have assumed that the merging system is located at an angular distance of 300 Mpc. The map-size used for the computation of the quadrupole was 10.24 Mpc. This size has been used throughout the paper^{}. For illustration purposes, the signal coming from the RS and the kSZ components has been computed separately, although the observable is the sum of the two quantities. Given that the RS signal has a more extended pattern than the kSZ, we have used a mask based on the tSZ map in order to enhance the signal from the RS effect. In this example, this mask was obtained by tapering all pixels with a temperature in the y-parameter map larger than 10% of the peak temperature. We can see that when using this taper, the quadrupole becomes totally dominated by the RS contribution.
For illustration purposes, we also present in Fig. 4 the same plot but for the dipole modulus ( ). We can see that this dipole signal is dominated by the kSZ contribution, and even when we mask 90% of the tSZ emission (note that the dipole signal from kSZ is also reduced by approximately the same factor) the signals from kSZ and RS are comparable, but in this latter case only for high values of the projection angle ( ). For (close to) face-on mergers, the kSZ contribution goes down because of the geometrical projection of the velocity ( ) while the RS component is non zero ( ). Therefore, as expected, we conclude that the quadrupole moment is better suited for the extraction of the RS signal, while the dipole moment is more suitable for the kSZ (or tSZ) component (note that when considering only the inner part of the map, where 90% of the tSZ emission comes from, then the dipole is totally dominated by the kSZ component).
Figure 4: Dependence of the measured dipole modulus ( ) with the projection angle . We consider exactly the same case and notation as in Fig. 3, but here we also show the dipole signal from the region where the 90% of the tSZ emission comes from (quoted as "inner''), as well as the complementary "outer'' region. | |
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Figure 5: Dependence of the measured quadrupole (Q_{xx}) on the separation between clusters. We consider the case , M_{2} = 5 , (so Q_{yy}=0) and we use (zero angular momentum). Notation and units are the same as in Fig. 3. Separations are measured from the most massive to the less massive cluster, so in this picture a positive separation represents a pre-merger state, where the two clusters are approaching each other, while a negative separation represents a post-merger state. | |
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In Fig. 5 we show the dependence of Q_{xx} on the physical separation, keeping fixed the other parameters. Here the separations are measured from M_{1} (the most massive cluster, i.e. the one with larger tSZ signal) to M_{2}, and the velocity for M_{1} is taken to be pointing towards the positive direction of X axis. Within this picture, a positive separation (i.e. cluster M_{1} on the left) represents a pre-merger state, where the two clusters are approaching each other, while a negative separation (i.e. cluster M_{1} on the right) represents a post-merger state. Thus, we see that the sign of the quadrupole components carries information about the merger kinematics. We also stress here that tapering the central region of the map makes the quadrupole dominated by the RS signal. Finally, it is worth remarking that depending on the particular configuration of the merging system, the RS signal adds-up coherently or cancels in the computation of the quadrupole (or dipole), so we find that the outer region has a larger quadrupole-flux than the total map for some physical separations.
The quadrupole pattern of the RS effect associated to merging clusters can also be used to learn about the angular momentum of the system, at least in the plane of the sky: it is clear from Figs. 1, 2 that the direction of cluster velocities conditions the nature of the quadrupole pattern imprinted by the RS effect.
This is the motivation of the following approach, which is entirely based in the computation of the moments of the kSZ, tSZ and RS components, and hence is well defined for any morphology of the system:
Figure 6: Recovery of from . Thick solid line excludes that area containing 90% of the tSZ emission, whereas thin solid line considers the entire map. The dotted line gives the expected scaling . A projection angle of was used in the calculation. | |
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This is illustrated in Fig. 6, where we use the cluster parameters { , M_{2}=5 , , }, and we vary the azimuthal angle from 0 to radians. The quadrupole induced by the kSZ component perturbs our estimate of from as it is shown in the figure. Note that the value of at which we have the maximum departure from the real value is not constant, but depends on the relative amplitude ratio of the kSZ and RS quadrupoles, i.e. depends on the other physical parameters of the merger. Finally, we again obtain better results for the case in which we mask out the area where 90% of the tSZ emission comes from, even for cases with low values of the projection angle.
With respect to the amplitude of the angular momentum, one can see
that it is closely related to the cross component of the
tensor, provided the X-axis of our coordinate system is located along
the axis joining the cluster centres, and its centre is the system
centre of mass. In this reference frame, a merger with shows no
component. Indeed, for the case of two
point-like clusters, it can be shown that
,
with x_{1},x_{2} the coordinates of the cluster centres,
and J_{z} the projected component of the angular momentum along the
Z-axis (see Appendix). Therefore, one can make use of the transformations
of the components of the quadrupole under a
rotation of the coordinate system by an angle ,
Figure 7: Dependence of |Q^{T}_{xy} | with the absolute value of the projected component of the angular momentum, |J_{z}|. We use the same physical parameters of the previous figure, and computed for each value of the projected Z-component of the angular momentum (note that, for and , ). Under the symmetry of our system (clusters along the X-axis), the kSZ residual affects mostly the Q^{T}_{xx} values, leaving the crossed term Q^{T}_{xy} practically unaltered. | |
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The RS effect produces only a small temperature fluctuation making it very hard to be measured at individual clusters. It is therefore worth to develop ideas on how to co-add the signals from a large sample of clusters leading to a statistical RS effect detection.
The simplest approach could be to co-add the quadrupole strength Q, which is always positive. However, there are two problems with this. First, the intrinsic CMB temperature fluctuations contribute also to Q, although their contribution can relatively accurately be estimated by measuring at non-cluster positions. Second, the kSZ effect imprints some quadrupole structure, which can not be completely masked away. Estimates of its strength are not possible from the observational data alone and therefore would rely on numerical simulations of galaxy cluster mergers in a cosmological setting.
We therefore propose a different approach to stack the signal from several clusters. First, the cluster should be rotated in a way that its major axis (of the tSZ effect or X-ray image) is aligned with the X-axis. Then, the quadrupole tensor of the temperature fluctuations is calculated in this coordinate system and its components are co-added for all the clusters in the sample. We expect that for a sufficiently large number of clusters all contributions to cancel out due to symmetry properties except the ones from the RS effect.
The components of have both signs with equal probability, since the CMB fluctuations are not correlated with the foreground cluster and the signs of quadrupole components are not rotational invariant (they reverse after a rotation). For the same reason, we expect that the lensing-induced cluster signal in the CMB (Seljak & Zaldarriaga 2000) will also cancel out, because the quadrupole moment is a linear function of the temperature fluctuations, for which we expect both signs with equal probability.
Also the components of have both signs with equal probability: mirroring a cluster through the X-Y plane going through its centre of mass reverses the sign of the kSZ map, and due to linearity also the sign of all components. Since a cluster and its mirrored counterpart are equally likely in the cluster sample, their contributions to the co-added quadrupole cancel each other statistically.
The components of have both signs with different probabilities. Since galaxy clusters are growing, one should find more clusters with converging velocity fields than with diverging ones. Diverging velocity fields should be present shortly after core passage of the dark matter clumps of two merging clusters. However, dynamical effects due to gravitational interactions (violent relaxation) should convert rapidly the organised bulk motion of the dark matter particles into an undirectional motion. This reduces the statistical contribution of diverging matter flows to the co-added compare to the contribution of the converging matter flows of early stage mergers.
For this reason, only a contribution from the RS effect due to the growing large-scale structure should survive statistically. However, there is one possible contamination to the described measurement scheme: any residual tSZ contamination to the temperature fluctuations is expected to be co-aligned with the major axis, and therefore to add coherently to , unless they appear with both signs with equal probability. This point could be achieved by taking relativistic corrections to the tSZ effect into account.
Therefore, further theoretical investigations are required before this stacking method can be fully established. This issue will be addressed in a follow-up paper.
In this paper we have presented a formalism that can be easily incorporated to N-body simulation codes in order to predict the Rees-Sciama effect in merging systems of clusters of galaxies. For the typical range of velocities in those systems (), the obtained expression (Eq. (4)) can also be seen as a gravitational lensing effect produced by a moving lens.
Using simple modelling for the cluster merger, we have illustrated the morphology and symmetries of this effect, and we have developed a method to extract the signal, which should be applicable to realistic maps. This method is based on the computation of the (weighted) dipole and quadrupole moments of the brightness distribution. In particular, it has been shown how the quadrupole moment is related to the kinematic properties of the merger, so we can extract information about the dynamical state of the system: pre-merger or post-merger, and the magnitude of the angular momentum.
Since we expect that in the near future the observation of a single cluster merger will be extremely difficult, given the weak signal strength, we have proposed a simple method of stacking the signal from a large number of clusters in order to extract their RS signature statistically. The procedure is straight-forward: for a sample of clusters (e.g. a complete, or a merging cluster sample) a coordinate system is attached to the center of light (X-ray or tSZ map) so that the X-axis is aligned with the major elongation of the gas. Then the quadrupole moments of the CMB temperature fluctuations (outside the gas region, defined by the tSZ effect) are calculated for each cluster. Finally the individual quadrupole moment components are co-added for the sample. Intrinsic CMB fluctuations and kSZ effect residuals should cancel out statistically, leaving only a signature of the RS effect. A detailed investigation of this stacking method will be carried out in a follow-up paper.
Acknowledgements
We acknowledge useful comments by S. D. M. White and M. Bartelmann. J.A.R.M. and C.H.M. acknowledge the financial support provided through the European Community's Human Potential Programme under contract HPRN-CT-2002-00124, CMBNET. We thank an anonymous referee for useful comments.
In this section, we prove that for two point-like clusters, is directly proportional to the component of the angular momentum projected along the line of sight, provided the coordinate origin is located at the mass centre of the system and the X-axis is aligned to the axis joining the clusters. Thus, the z=0 plane is the image plane.
Let us assume that the coordinates of clusters are given by
,
for i=1,2. Their linear momenta will be labelled as
,
i=1,2 and we set the total
momentum of the system to zero, so
.
Also note that from the choice of the
coordinate system, then
and
are related.
The xy component of the RS-induced quadrupole is then given by:
with
the LOS projected component
of the angular momentum. Therefore,