A&A 424, 389-408 (2004)
J. Chluba1 - R. A. Sunyaev1,2
1 - Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 86740 Garching bei München, Germany
2 - Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84/32 Moscow, Russia
Received 3 April 2004/ Accepted 11 May 2004
The CMB angular temperature fluctuations observed by COBE and WMAP enable us to place a lower limit on the spectral distortions of the CMB at any angular scale. These distortions are connected with the simple fact that the superposition of blackbodies with different temperatures in general is not a blackbody. We show that in the limit of small temperature fluctuations the superposition of blackbodies leads to a y-type spectral distortion. It is known that the CMB dipole induces a y-type spectral distortion with quadrupole and monopole angular distribution leading to a corresponding whole sky y-parameter of . We show here that taking the difference of the CMB signal in the direction of the maximum and minimum of the CMB dipole due to the superposition of two blackbodies leads to a spectral distortion with . The amplitude of this distortion can be calculated to the same precision as the CMB dipole, i.e. today. Therefore it may be used as a source with brightness of several or tens of K to cross calibrate and calibrate different frequency channels of CMB surveys with a precision of a few tens or hundreds of nK. We also discuss clusters of galaxies as possible sources for calibration purposes. Furthermore, we show in this work that primordial anisotropies for multipoles also lead to spectral distortions but with a much smaller y-parameter, i.e. .
Key words: cosmology: cosmic microwave background - cosmology: observations
The WMAP spacecraft measured the amplitude of the angular fluctuations of the cosmic microwave background (CMB) temperature with extremely high precision on a very broad range of angular scales, from 12' up to the whole sky. These temperature anisotropies and the existence of the acoustic peaks were predicted already long ago (Sunyaev & Zeldovich 1970a; Peebles & Yu 1970), but only now after BOOMERANG, MAXIMA, ARCHEOPS, WMAP and many ground based experiments like CBI, ACBAR, VSA, etc. do we know their precise characteristic angular scales and amplitudes. Only tiny fluctuations of the radiation temperature field have been observed, but nowadays with a precision to better than 1% down to degree angular scales.
It is commonly assumed that the spectrum in one direction of the sky is Planckian and that only the temperature changes from point to point. This follows from the nature of the main effects leading to the appearance of these fluctuations, i.e. the Sachs-Wolfe-effect (Sachs & Wolfe 1967) and the Doppler effect due to Thomson scattering off moving electrons (Sunyaev & Zeldovich 1970a) at redshift 1100. However, as will be demonstrated below, there are spectral distortions in the second order of . These distortions are inevitable when the CMB is observed with finite angular resolution or when regions on the sky containing blackbodies with different temperatures are averaged or compared with each other.
The CMB missions mentioned above have shown that there are fluctuations of the radiation temperature on the level of K-mK over a broad range of angular scales. One may distinguish two basic observational strategies: (i) absolute measurements, where the beam flux in some direction on the sky is compared to an internal calibrator ( COBE/FIRAS); and (ii) differential measurements, where the beam flux in one direction on the sky is compared to the beam flux in another direction ( COBE/DMR or WMAP). In the first strategy one observes a sum of blackbodies (SB) due to the average over the beam temperature distribution, whereas in the second two sums of blackbodies are compared with each other. Under these circumstances we will in general speak about the superposition of blackbodies, i.e. the sum and difference of blackbodies with different temperatures.
Any experiment trying to extend the great success of the COBE/FIRAS instrument, which placed strict upper limits (Fixsen & Mather 2002; Fixsen et al. 1996) on a possible - (Sunyaev & Zeldovich 1970b), , and y-type (Zeldovich & Sunyaev 1969), , CMB spectral distortion, will only have a finite angular resolution and would therefore observe a superposition of several Planck spectra with different temperatures corresponding to the maxima and minima on the CMB sky as measured with WMAP.
It is known (Zeldovich et al. 1972) that in the case of a Gaussian temperature distribution this will lead to a spectral distortion indistinguishable from a y-distortion, with a corresponding y-parameter which is proportional to the dispersion of the temperature distribution. Since the temperature fluctuations of the CMB indeed are Gaussian, this implies that the corresponding spectral distortions averaged over large parts of the sky should be of y-type. But here we are interested in the case of measurements with an angular resolution of a few arcminutes to degrees. In this situation, we deal with the limited statistics of finite regions with different mean temperatures and therefore it is not obvious what type of spectral distortion would be induced in each small patch of the sky. As mentioned above, a similar situation arises when we compare the signals from two regions on the sky, i.e. the difference of the intensities as is usually done in differential observations. Below it will be shown that for any observation of the CMB temperature fluctuations, there will be unavoidable spectral distortions due to the difference in the temperature of the radiation we measure and compare and that these distortions will be indistinguishable from a y-type-distortion. The biggest distortions arise due to the CMB dipole.
The unprecedented high sensitivity of future or proposed space missions like PLANCK and CMBPOL or ground based instruments under construction like APEX, the South Pole Telescope ( SPT), the Atacama Cosmology Telescope ( ACT) and QUEST at DASI ( QUAD) will offer ways to investigate tiny secondary CMB angular and spectral fluctuations and should therefore add a lot to the success of previous missions. One target will be the measurement of the SZ effect from clusters, proto clusters or groups of galaxies and superclusters (Sunyaev & Zeldovich 1972) or signatures from the first stars in the universe (Oh et al. 2003). In the future CMB experiments will be so sensitive that it will be possible to investigate in detail the imprints of reionization and the traces of energy release in the early universe.
Basu et al. (2004) proposed a method to constrain the ionization history of the universe and the history of heavy element production using the properties of resonant scattering of CMB photons in the fine structure lines of oxygen, carbon and nitrogen atoms and ions produced by the first generation of stars. The strong frequency dependence of this effect permits one to extract the undisturbed angular dependence of the frequency-independent primary temperature fluctuations and thereby avoid cosmic variance. By comparing the signals in different frequency channels it is possible to investigate the contributions of the lines of different ions at different redshifts and therefore to examine different scenarios of element production and ionization histories in the low density regions of the universe, with overdensities less than 104. The sensitivities of PLANCK and ACT should be sufficient to detect the signals imprinted by the effects of resonant scattering, but the crucial point for the successful measurement of any small frequency-dependent signal is the cross calibration of the different frequency channels down to the limits set by the sensitivity of the experiments. Full sky missions like COBE/DMR or WMAP normally use the CMB dipole and its annual modulation to check the calibration of their instruments down to a level of K, whereas experiments with partial sky coverage like BOOMERANG directly use the CMB dipole for calibration issues (de Bernardis et al. 2000). But both methods permit one to cross calibrate different frequency channels only to a first approximation assuming that the dipole has a Planckian spectrum and the same amplitude at all frequencies. Unfortunately, this precision of the cross calibration will not be sufficient to detect the signals from the dark ages as discussed by Basu et al. (2004).
It is known that the motion system relative to the CMB restframe in addition to the dipole generates (in the second order of v/c) a small monopole and quadrupole contribution to the CMB brightness of the sky in the restframe of the observer (Sunyaev & Zeldovich 1980; de Bernardis et al. 1990; Bottani et al. 1992). Sunyaev & Zeldovich (1980), when they were discussing the radiation field inside a cluster of galaxies moving relative to the CMB restframe, have shown that the corresponding dipole-induced quadrupole has a non Planckian spectrum, which was then later derived by Sazonov & Sunyaev (1999). Kamionkowski & Knox (2003) later applied this solution to the case of our motion relative to the CMB restframe and proposed to use the dipole-induced quadrupole for calibration purposes.
The solution of Sazonov & Sunyaev (1999) is valid in the case of narrow beam observations. In this paper we choose an independent approach, which is based on the superposition of blackbody spectra with different temperatures, to look for the maximal and minimal spectral distortion obtainable from CMB maps. Our method allows us to calculate the value of the y-parameter for the dipole-induced monopole and quadrupole for a beam with finite width or equivalently for any average of the signal over extended regions on the sky. Most importantly we show that the difference of the sky brightness in the direction of the maximum and minimum of the CMB dipole, corresponding to the maximal difference of the radiation temperature on the CMB sky, leads to a y-type spectral distortion with an associated y-parameter of . We propose here to use this spectral distortion arising due to the CMB dipole to cross calibrate the frequency channels of a CMB experiment in principle down to the level of a few tens of nK. We discuss different observing strategies in order to maximize the inferred spectral distortion (Sect. 7).
In this paper, we first give a short summary of the basic equations necessary in the following derivations and define some of the terminology used (Sect. 2). We then discuss the underlying theory for small spectral distortions (Sect. 3) and show that in this limit even the distortions arising due to the superposition of two blackbodies (Sect. 4) with close temperatures are well described by a y-type solution. Furthermore, we discuss in detail the spectral distortion due to the superposition of Planck spectra with different temperatures arising from to the CMB dipole (Sect. 5) and from the higher multipoles (Sect. 6) using generated CMB sky maps for the WMAP best fit model. We discuss the spectral distortions arising in differential measurements of the CMB temperature fluctuation (Sect. 7) and how to use the spectral distortions induced by the CMB dipole to cross calibrate the frequency channels of CMB experiments (Sect. 8). We end this work with a discussion of the consequences of the obtained results for some of the highly demanding tasks which may be addressed by future CMB projects (Sect. 9) and finally conclude in Sect. 10.
To obtain the relative difference in temperature (5)
corresponding to a y-distortion from the relative difference in
intensity (4) the relation
Now, using the Taylor expansion of Eq. (8) up to the second order in
|Figure 1: as a function of for different values of according to Eq. (11): for (solid), (dashed), (dotted) and (dashed-dotted). The right ordinate corresponds to , with T0=2.725 K.|
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The temperature difference should not be larger than a few percent of the temperature of the reference blackbody, otherwise corrections due to higher orders in will become important and lead to additional distortions (see Sect. 3). Fortunately, the temperature differences on the CMB sky are sufficiently small to neglect these corrections.
Equation (10) also shows that in general the inferred temperature difference is frequency-dependent. At frequencies below the inferred temperature difference is close to the true temperature difference and frequency-independent within the sensitivity of the experiment. In this case, we will speak about a temperature distortion or fluctuation, emphasizing that it is frequency-independent. It is possible to eliminate the temperature distortion using multifrequency measurements. For frequency-dependent terms become important, which we will henceforth call spectral distortions.
When the CMB sky is observed with finite angular resolution or equivalently if the brightness of parts of the sky (not necessarily connected) is averaged one deals with the sum and more generally with the superposition of blackbodies. Here we develop a general formalism to calculate the spectral distortions arising for arbitrary temperature distribution functions in the limit of small temperature fluctuations and derive criteria for the applicability of this approximation. We first discuss the basic equations necessary to describe the spectrum of the sum of blackbodies (SB) as compared to some arbitrary reference blackbody (Sect. 3.1) and then generalize these results to the superposition of blackbodies (Sect. 3.2).
|Figure 2: Frequency dependence of : Dotted line , which is equivalent to a temperature distortion, solid line , which is equivalent to a y-distortion as given by Eq. (5), dashed line and dashed-dotted line . The function becomes larger than unity for GHz or , larger than 5 for GHz or and larger than 10 for GHz or . For GHz or it follows gy= 0.5. Also shown as vertical lines are the PLANCK LFI (dashed) and HFI (dash-dotted) frequency channels.|
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Inserting Eqs. (15) into (12) the relative
between the SB spectrum and the reference blackbody
can be derived
In Eq. (17) the term proportional to the first
corresponds to a temperature
distortion which introduces a frequency-independent shift in the
measured temperature difference. It is possible to eliminate this
contribution by multifrequency measurements, since it does not change
with frequency. The term proportional to the second moment is
indistinguishable from a Compton y-distortion as given by Eq. (4) with y-parameter
In the Wien region of the CMB spectrum (
again starting from Eq. (12) and using relation (7) one can deduce
Using multifrequency observations it is possible to eliminate the
temperature distortion (
). In this case,
the leading term in the signal corresponds to a y-distortion. Here
we are interested in the question when higher order moments start to
contribute significantly to the total spectral distortions.
In the RJ region using Eq. (19b) one can find that
|Figure 3: as a function of for different values of according to Eq. (22) assuming that : For (solid), (dashed) and (dotted). The right ordinate corresponds to , with T0=2.725 K.|
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In the Wien region the situation is a bit more complicated. Given some
one can find the frequency
above which the
relative contribution of higher order moments will become
important. To find
one has to solve the non-linear equation
In the RJ limit the sum of blackbodies (SB) is again a blackbody with temperature . The RJ temperature can be directly measured at sufficiently low frequencies and hence it is convenient to compare the SB to a reference blackbody of temperature . This makes any distortion due to the SB vanish at low .
In the following, we will call
is defined as the relative
difference between the temperature T of a given blackbody inside the
beam and the RJ temperature of the SB, i.e. the beam RJ temperature
For this choice of reference blackbody the temperature
distortion vanishes and we obtain
It is important to note that for any given temperature distribution the minimal distortion arising due to the sum of Plancks corresponds to a y-distortion with y-parameter . Therefore by calculating the second beam moment it is possible to place a lower limit on the distortions arising from any average over not necessarily connected regions of the CMB sky.
For the sum of two Planck spectra with temperatures T1 and T2 of
relative weights w1 and w2=1-w1 the RJ temperature is given by
For the first five beam moments one may
Here we want to derive the equations describing the comparison of the
spectrum of the SB with an arbitrary reference blackbody. This example
can be applied for absolute measurements of the CMB.
Starting again from Eq. (17) and using the results
obtained for the beam spectral distortions, one can rewrite the moments
in terms of the beam
Equation (28) shows that the spectral distortion of the SB spectrum with respect to the some reference blackbody has the following three contributions:
where Ti are the
temperatures of the blackbodies inside the beam i, and applying
formula (17) one may obtain
can be rewritten using the beam moments
Here we want to clarify some aspects of the dependence of the inferred
y-parameter on the chosen reference temperature. If we assume that
we can find
the reference temperature at which the total y-distortion vanishes:
Now we are interested in the change of
change from one reference temperature
one can again write down an equation similar to (29), where now
Using (30) and (31), we can write
In order to illustrate the main results obtained in the previous section we now discuss the spectral distortions arising due to the sum and difference of two pure blackbodies with different temperatures T1 and T2 and the corresponding intensities and in the presence of a reference blackbody with temperature . In principle one can directly apply the results obtained in Sect. 3 and easily derive the equations describing this situation, but in order to check the derivations of the previous section we here choose an approach starting from the expansion (9) taking only the first order corrections into account. Since the CMB temperature fluctuations are very small this approximation is sufficient.
The results obtained for the sum of two blackbodies apply to the case of an absolute measurement of the CMB sky, where the beam contains only blackbodies with two equally weighted temperatures. In the limit of narrow beams the results obtained for the difference of two blackbodies can be directly used to discuss the effects arising in differential measurements. In Sect. 8 we will use some of the result of this section in the discussion about cross calibration issues.
We want to describe the difference
function of frequency. Here we defined the sum of the two blackbodies
The easiest is to first calculate the difference
If we define
we can make use of Eq. (9) and write the
relative difference of
|Figure 4: Dependence of the inferred y-parameter on the chosen reference temperature for two blackbodies with temperatures T1=2.72162 K and T2=2.72838 K, corresponding to the minimum and maximum of the CMB dipole. In this case the mean RJ temperature is K and . - Sum of two blackbodies: total y-parameter as given by Eq. (38b) (dashed line). - Difference of two blackbodies: the absolute value of the inferred y-parameter as given in Eq. (41b) (solid line). The dotted vertical lines show the location of and respectively.|
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The dependence of the inferred y-parameter on the chosen reference temperature is illustrated in Fig. 4. As mentioned above the y-parameter reaches a minimum for . It is important to note that the minimal value of the y-parameter does not depend on the reference temperature but only on the second beam moment of the temperature distribution function. Changing from the inferred y-parameter monotonically increases being a factor of 2 larger than for . This can be understood as follows: if we set we obtain , with . Therefore the ratio .
Outside the region the inferred y-parameter increases further. It is in principle possible to gain large factors in the inferred y-parameter by going very far away from the RJ temperature of the SB, but at some point next order corrections will become important and therefore limit to the region where the approximation of small spectral distortions is still valid. For the CMB the most natural choices of the reference temperature are the whole sky mean temperature T0 and the maximum or minimum of the CMB dipole. For the SB the inferred y-parameter varies from for to for . The behavior of the inferred y-parameter shows that in order to minimize the arising spectral distortion for absolute measurements it is important to use an internal calibrator with temperature close to the beam RJ temperature .
In the limit we are in principle comparing one pure Planck spectrum with a reference blackbody. This case applies for an absolute measurement of the CMB sky in the limit of very narrow beams. In this case it follows that and in addition for . Except for some changes in scales the behavior of the curve shown in Fig. 4 is unaffected.
applying Eq. (9) one can easily
As in the case of the sum of two Plancks for outside the region the inferred y-parameter increases strongly. Again there is a limit set by the approximations of small spectral distortions. The behavior of the inferred y-parameter shows that in order to minimize the spectral distortion arising in differential measurements it is important to choose a reference temperature close to the RJ temperature of the observed region.
The largest temperature fluctuation on the observed CMB sky is
connected with the CMB dipole. Its amplitude has been accurately
measured by COBE/FIRAS (Fixsen & Mather 2002):
mK, corresponding to
very large angular scales. Assuming that it is only due to the motion
of the solar system with respect to the CMB rest frame
it implies a velocity of
km s-1 in the direction
to understand the spectral distortions arising due to the dipole, we
start with the direction dependent temperature of the CMB, neglecting
any intrinsic anisotropy:
Below we now discuss the spectral distortions arising due to the CMB dipole in the context of finite angular resolution. Since only the second moment of the temperature distribution is important, we will use the second order expansion of in as given in Eq. (42b). In the last part of this section we show that starting with Eq. (42a) all the results of this section can also be directly obtained by expansion of the blackbody in terms of small . This shows the equivalence of both approaches. Nevertheless, the big advantage of the treatment developed in Sect. 3 is that it can be applied to general temperature distributions and that the source of the y-distortion can be directly related to the second moment of the temperature distribution.
is given by Eq. (43), the full
sky moments can be calculated by the integrals
The first three moments may be found as
Here we want to note that this full sky distortions arises due to the existence of the dipole anisotropy. In absolute measurements of the CMB places a lower limit on the full sky y-parameter (see Sect. 3.1).
Here we are interested in the angular pattern of the y-distortion
induced by the SB over the dipole for an observation with a finite
angular resolution and in particular in the location of the maximal
y-distortion, when we compare the beam flux at different frequencies
to a reference blackbody with beam RJ temperature. To model the beam
we use a simple top-hat filter function
The kth moment of some variable X over the beam is then given by
Using formula (42b) and Eqs. (47) and (49) the mean RJ temperature inside the beam in some
relative to the dipole axis is given as
|Figure 5: Angular distribution of as given by Eq. (52) for . The maximal spectral distortion is expected to appear in a ring perpendicular to the dipole axis and in this case has a value . The minimal y-parameter has a value of and is located around the maximum and minimum of the dipole. The galactic plane was cut out using the kp0-mask of the WMAP data base.|
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Using Eq. (51) the position of the maximum can be
This suggests that the
location of the maximal distortion is where the derivative of the
temperature distribution is extremal. The maximal y-parameter is given
|Figure 6: Dependence of the y-parameter in units of on the beam radius in degree for a circular beam: for as given by Eq. (52) (dash-dot-dotted), according to Eq. (53) (solid). For values the curves lie between these two extremes.|
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We have seen in Sect. 3.1 that the inferred y-parameter for the comparison of the SB with some reference blackbody with temperature has one contribution from the second beam moment and another due to the dispersion of the beam RJ temperature relative to the temperature of the chosen reference blackbody (cf. Eq. (27b)). Therefore one can immediately write down the relative difference of the beam intensity and the intensity of the chosen reference blackbody making use of Eqs. (27), (28), (50) and (51). In the limit we can apply the results obtained in Sect. 4.1 for the sum of two blackbodies, if we set T1=T2. Here we want to discuss the case in more detail. For this choice the residual full sky y-distortion (which may be obtained by averaging the intensity data over the whole sky) is minimal.
In the picture of the SB it is easily understandable that there is no difference in the resulting spectral distortion whether the dipole anisotropy is intrinsic or due to motion. Therefore, as was noted earlier by Kamionkowski & Knox (2003), it is impossible to distinguish the intrinsic dipole from a motion-induced dipole by measurement of the frequency-dependent temperature quadrupole.
First we want to calculate the SB in a circular beam and compare it to the blackbody I0 of temperature T0, i.e. we want to derive . Using Eq. (47) and since only and depend on and this integration with Eq. (49) immediately leads to Eq. (57).
Next we want to derive the spectral distortion relative to the
reference blackbody with beam RJ temperature starting with Eq. (58). For this we rearrange Eq. (58) in
The largest temperature anisotropy on the CMB sky is connected with the CMB dipole which was discussed in detail above (Sect. 5). In what follows here we are only concerned with spectral distortions arising from multipoles with . As has been shown above, the spectral distortions induced by the CMB dipole are indistinguishable from a y-distortion. Since the typical amplitude of the temperature fluctuations for is a factor of 100 less than the dipole anisotropy ( ), the spectral distortion arising from higher multipoles will also be indistinguishable from a y-distortion.
Using the generated CMB maps we extract the temperature distribution function R(T) inside a circular beam of given radius in different directions on the sky. We then calculate the spectrum and the spectral distortion inside the beam using Eq. (12) and setting the reference temperature to the beam RJ temperature . As expected, we found that the spectral distortions are indistinguishable from a y-distortion with y-parameter given by Eq. (18). As discussed above this follows from the fact that the temperature fluctuations for are extremely small (10-5).
|Figure 7: Top: probability density p(y) for different aperture radii: (solid), (dashed) and (dashed-dotted). - Bottom: dependence of the average y-parameter on the aperture radius in degrees: for the beam spectral distortion (solid) and for the y-parameter resulting from the dispersion of the beam RJ temperatures in different directions on the sky relative to T0 (dashed). Also shown is the maximal beam spectral distortion arising due to the dipole, (dotted), according to Eq. (53).|
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Figure 7 also shows the y-parameter resulting from the dispersion of the beam RJ temperatures in different directions on the sky relative to T0. It drops with increasing beam radius, because the average beam RJ temperature becomes closer to T0 for bigger beam radius. Looking at the sum of the beam distortion and the distortion due to the dispersion of the beam RJ temperatures shows that the averaged whole sky distortion is independent of the angular resolution.
|Figure 8: Top: cumulative probability for different aperture radii (solid), (dashed) and (dashed-dotted). Bottom: dependence of the cumulative probability on the aperture for (solid), (dashed) and y0=10-9 (dashed-dotted).|
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In Fig. 8 the cumulative probability of finding a spectral distortion with is shown for different apertures. Fixing the value y0, there is a maximum probability of to measure for an aperture radius during the mapping or scanning of extended regions of the sky. This corresponds to 500 sources on the whole sky. In Fig. 9 the dependence of the number of sources with y above y0 for a given radius is illustrated. It peaks around degree, corresponding to the first acoustic peak. The maxima of the y-distortion do not coincide with the maxima of , but that they both should have similar statistical properties. We expect that the maxima of the y-parameter are there, where the derivatives of the averaged temperature field are large.
|Figure 9: Number of regions on the sky with for different aperture radii in degree: for (solid), (dashed) and y0=10-9 (dashed-dotted).|
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In Sect. 5 we discussed spectral distortions due to the motion of the solar system relative to the CMB restframe up to the second order in . On the level of third order corrections should also start to contribute. In this order of not only the motion-induced dipole and quadrupole lead to spectral distortions but also terms connected to the products of the intrinsic dipole and quadrupole with the motion-induced dipole and quadrupole. Here we now only discuss the motion-induced CMB third order terms.
Expanding Eq. (42a) up to third order in
the beam averaged RJ temperature for the top-hat beam using Eq. (47) leads to
In Sect. 5 we discussed the spectral distortion arising
due to the CMB dipole inside a single circular beam in
comparison to a reference blackbody with temperature T0.
In this chapter we address the spectral distortion arising in differential measurements of the CMB fluctuations, where two beam
intensities I1 and I2 are directly compared with each other and
the intensity difference
is measured. Since in
Sect. 6 we have shown that y-distortion arising due to
higher multipoles have corresponding y-parameters
here we are only taking distortions arising due to the CMB dipole into
If we assume that the both beams are circular and have the same radius
(see Fig. 10), we may define the beam RJ temperatures as
of each beam is given by Eq. (50).
Now, defining the reference blackbody
using Eqs. (7) and (29) we find for the inferred temperature difference
|Figure 10: Illustration of a differential observing strategy: the maximum of the dipole lies in the direction of the z-axis. Both observed patches have the same radius and the observed intensity difference is .|
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Now one can write for the relative temperature difference of the two
beams at two frequency
On the CMB sky the most natural choices for the reference temperature is the full sky mean temperature T0 and the temperature of the maximum or the minimum of the CMB dipole anisotropy, whereas the last two are equivalent. Below we now discuss the dependence of the inferred y-parameter on the angle between the beams for these two choices of the reference temperature. In the limit we can apply the results obtained in Sect. 4.2 for the difference of two blackbodies.
|Figure 11: Dependence of the y-parameter in units of on the beam radius in degree for differential measurements: as given by Eq. (73) (dashed/left ordinate), as given by Eq. (76) (solid/right ordinate) and as given by Eq. (77) (dotted/left ordinate).|
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Let us note here that in the case , i.e. when we are directly comparing the maximum and the minimum of the CMB dipole, the spectral distortion vanishes. This can be understood as follows: As was argued in Sect. 3.2, if the chosen reference temperature is equal to Ty=0 as given by Eq. (34) the total y-parameter is zero. For and the beam moments are both equal and therefore do not contribute to the total inferred y-parameter (68b). In this case we obtain . Therefore, consistent in the second order of the total inferred y-parameter vanishes. This conclusion can also be directly drawn from Eq. (36).
Table 1: y-distortion: in K for y as given in the left column in some of the PLANCK spectral channels. Here , and .In this case, again making use of Eqs. (50) and (68c) one may find
The dependence of on the beam radius is shown in Fig. 6. As in the previous case, the y-distortion does not vanish for high resolution ( ) and in the maximum it is even 12 times bigger than , corresponding to . This is only 5 times below the current upper limit on the mean y-parameter given by COBE/FIRAS. The corresponding absolute temperature difference in some of the PLANCK frequency channels are given in Table 1.
Another case may be interesting for CMB missions with partial sky
coverage, but which have access to the maximum or the minimum of the
and a region in the directions perpendicular to
the dipole axis (): using Eq. (75) with
The cross calibration of the different frequency channels of CMB experiments is crucial for the detection of any frequency-dependent signal. As has been mentioned earlier usually the dipole and its annual modulation due to the motion of the earth around the sun is used for calibration issues. The amplitude of the dipole is known with a precision of on a level of mK. But the sensitivities of future experiments will be significantly higher than previous missions and request a possibility for cross calibration down to the level of tens of nK.
In this section we now discuss two alternative ways to cross calibrate the frequency channels of future CMB experiments. Most importantly we show that making use of the superposition of blackbodies and the spectral distortions induced by the CMB dipole should open a way to cross calibrate down to the level necessary to detect signals from the dark ages as proposed by Basu et al. (2004).
It is obvious that the brightest relaxed clusters of galaxies on the sky open additional way to cross calibrate the different frequency channels of future CMB experiments like PLANCK and ground based experiments like ACT. The CMB signal of the majority of rich clusters has a very distinct y-type spectrum due to the thermal SZE. Multifrequency measurements allow one to determine their y-parameters and thereby open a way to use them as sources for cross calibration. This cross calibration will then enable us to detect the effects of relativistic correction for very hot clusters and thereby to measure the temperatures of clusters independent of X-ray observations.
Clusters such as Coma, where existing X-ray data provide excellent measurements of the electron densities and temperatures inside the cluster and where we know the contamination of the CMB brightness by radiosources and the dust in galaxies very well, even may be used for absolute calibration of CMB experiments. For the Coma cluster today X-ray data allow us to predict the CMB surface brightness with precision of the order of a few percent.
Due to the redshift independence of the SZ signal, distant clusters have similar surface brightness but much smaller angular diameters. Therefore they are good sources for calibration issues for experiments covering only limited parts of the sky and having high angular resolution like ACT (for example ACT is planing to investigate 200 square degree of the sky (Kosowsky 2003)).
One disadvantage of clusters for cross calibration purposes is that they are too bright: typically clusters have y-parameter of the order of 10-5-10-4 and therefore are only a few times weaker than the temperature signal of the CMB dipole (3 mK). In comparison to the level of calibration and cross calibration achieved directly using the dipole this is not much of an improvement.
In addition planets or galactic and extragalactic radiosources might serve for cross calibration purposes, if we know their spectral features with very high precision. Quasars and Active Galactic Nuclei are not good candidates for this issue, since they are highly variable in the spectral band of interest.
All the sources mentioned in Sect. 8.1 do not allow us to cross calibrate down to the level of the sensitivity of future CMB experiments and do not enable us to detect signals from the dark ages as predicted by Basu et al. (2004). Therefore here we want to discuss a method to cross calibrate the frequency channel using the spectral distortions induced by the CMB dipole (see Sect. 5). These spectral distortions can be predicted with very high accuracy on a level of K. This is 1000 times lower than the dipole signal and should therefore allow us the cross calibration to a very high precision.
Each CMB experiment measures the fluctuations of the CMB intensity on the sky. Due to the nature of the physical processes producing these fluctuations at each point they are related to the fluctuations of the radiation temperature. In a map-making procedure the measured fluctuations in the intensity are translated into the fluctuations in the radiation temperature. Usually the aim of any map-making procedure is to reduce statistic and systematic error in order to obtain a clean signal from these CMB temperature fluctuations. As has been discussed in Sect. 3 the superposition of blackbodies with slightly different temperature in the second order induces y-distortions in the inferred temperature differences. The amplitude of these distortions depends on the temperature difference and the chosen reference temperature. In Sects. 5 and 7 it has been shown that due to the two basic observing strategies, absolute and differential measurements, the dipole can induce y-distortions with y-parameters up to . It was also shown in Sect. 6 that contributions from higher multipoles are much smaller ( ).
The y-distortions induced by the dipole can contaminate the maps produced in the map-making procedure on a level higher than the sensitivity of the experiment. Therefore it is important to chose the map-making procedure such that spectral distortions are minimized. The discussion in Sect. 4 has shown that for this purpose in absolute measurements of the CMB sky it is the best to choose the temperature of the reference blackbody close to the beam RJ temperature. This implies that for CMB experiment the best choice for the temperature of the internal calibrator is the full sky mean temperature T0. In order to minimize the spectral distortions arising due to the superposition of blackbodies in differential measurements the best choice for the reference temperature used to relate the intensity difference maps to the temperature difference maps is the RJ temperature of the combined temperature distribution of both beams. This optimal reference temperature will be time dependent due to the various scanning strategies and orientations of the spinning axis relative to the dipole during observations.
The purpose of this section is not to discuss the details about map-making procedures but to show that there are ways to manipulate these CMB maps in order to make the spectral distortions arising due to the superposition of blackbodies become useful for cross calibration purposes. As the discussion in Sect. 4 has shown, for this issue it is better to move as far as possible away from T0. The optimal choice of depends on the sensitivity of the experiment and on the observing strategy: for absolute measurements should be as close as possible to T0 in order to minimize the induced y-distortions but on the other hand it should be chosen such that the induced y-distortion due to the CMB dipole is still measurable within the sensitivity of the experiment. But in principle there is no strong constraint, since the induced y-distortions can again eliminated afterwards. For differential measurements it is possible to chose the optimal reference temperature for calibration issues independent of the best map-making reference temperature, but here it is important to compare regions on the sky with maximal temperature difference, i.e. with maximal angular separation in the observed field. Below we now separately discuss CMB experiments using differential measurements with full and partial sky coverage in more detail.
For full sky missions like PLANCK the regions with maximal temperature difference are located around the extrema of the CMB dipole. Using Eq. (75) we see that they correspond to and the resulting maximal spectral distortion is characterized by , if we set or , i.e. to the maximum or minimum of the temperature on the CMB sky arising due to the dipole.
One attractive procedure to compare the maximum and minimum of the CMB dipole is to take the CMB intensity maps of each spectral channel and to calculate the difference between each map and the map obtained by remapping the value of the intensity at position to the value at position , i.e rotating the initial map by 180 degrees around any axis, which is perpendicular to the CMB dipole axis and is crossing the origin or equivalently setting in Eq. (67).
Afterwards these artificial intensity maps for each spectral channel
are converted into temperature maps using Eq. (7)
and setting the reference temperature at each point to
of the intrinsic
map. Neglecting the contributions from intrinsic multipoles with
the difference map in some frequency channel
will be given by
Now, using the artificial map of the lowest frequency channel as a reference and taking the differences between this reference map and the artificial maps in higher spectral channels we can eliminate the frequency-independent term corresponding to twice the dipole in Eq. (78). This then opens a way to cross calibrate all the channels to very high precision, since the quadrupole component in these artificial maps will be larger in higher spectral channels, following the behavior of gy with .
The signal will include both statistical and systematic errors for the
average temperature of the sky and the dipole amplitude
uncertainties are mainly influencing the frequency-independent dipole
term but for the frequency-dependent monopole and quadrupole they will
become important only in next order:
Applying the procedure as described above we are adding signals that have statistically independent noise. Therefore the statistical noise of the artificial maps should be a factor of stronger than the initial maps. On the other hand we are increasing the amplitude of the y-distortion quadrupole by a factor of 12, resulting in strong gain.
The maxima of the quadrupole component in the artificial maps will be rather broad and correspond to thousands of square degrees on the sky, allowing us to average the signal and thereby increasing the sensitivity. For a circular beam average the formula (78) will be applicable. The statistical sensitivity of the PLANCK experiment will permit us to find these spectral distortions in the artificial maps down to the level necessary to detect the effects of reionization as discussed by Basu et al. (2004). Applying the above procedure to the 33 GHz and 93 GHz frequency channels of WMAP will lead to a maximal temperature difference of K, whereas the maximal difference will be K for the 353 GHz channel of PLANCK (see Table 1). Nevertheless, even in the case of WMAP the very high precision of the experiment might permit the detection of this quadrupole and thereby open the way to cross calibrate its different spectral channels.
For COBE/FIRAS the proposed method permits us to check the precision of the internal calibration. The internal calibrator was measured and tested with some finite precision and there is the possibility that the internal calibration may be better than the precision at which it was tested. The proposed method has distinct spectral and angular properties, making it possible to improve the cross calibration of the spectral channels and possibly this will open a way to further improve the great results this experiment already gave us.
Surveys with partial sky coverage may not simultaneously have access to regions around the maximum and the minimum of the dipole. Therefore here one should choose two regions in the accessible field of view such that the mean temperature difference between these regions is as large a possible. The dependence of the signal on the separation angle has been discussed in detail in Sect. 7.
Before cross calibration all the bright sources detected such as bright clusters of galaxies and point sources inside the chosen areas should be extracted. Typically the expected amount of SZ sources will be of the order of a few tens per square degrees with corresponding angular extensions less than 1' (see Carlstrom & Holder 2002; Rubiño-Martín & Sunyaev 2003). Afterwards the difference of the signals in the two patches can be taken and compared to the signals in the other spectral channels.
For example choosing one area centered on the minimum or maximum of the CMB dipole as a reference ( ) and choosing the second area in the ring perpendicular to the dipole axis the maximal y-parameter is (cf. Eq. (77)). If we instead set we obtain the same maximal y-parameter but a stronger dependence on the beam radius or size of the regions we average (cf. Eq. (73)). This small example shows that for any experiment with partial sky coverage a separate analysis of the optimal choice of the reference temperature and the regions on the sky has to be done.
Injection of energy into the CMB prior to recombination leads to spectral distortions of the background radiation. Given the WMAP best fit parameters, before redshift all distortions are efficiently wiped out, whereas energy injection in the redshift range , with , leads to a -distortion and injection in the range to a y-distortion, with (Hu & Silk 1993; Illarionov & Sunyaev 1975; Burigana et al. 1991; Sunyaev & Zeldovich 1970b).
As was shown by Silk (1968) photon diffusion and thermal viscosity lead to the dissipation of small scale density perturbations before recombination. This damping of acoustic waves will contribute to a - and y-distortion because it was leading to (i) energy release (Hu et al. 1994a; Daly 1991; Sunyaev & Zeldovich 1970b) and (ii) mixing of photons from regions having different temperatures (Zeldovich et al. 1972).
Now, the WMAP data implies that the initial spectrum of perturbations is close to a scale-invariant Harrison-Zeldovich spectrum with spectral index (Bennett et al. 2003). Different estimates for the spectral distortions arising from the dissipation of acoustic waves in the early universe give a chemical potential of the order of and y-distortions with (Hu et al. 1994a; Daly 1991).
Another contribution to the y-distortion arises from the epoch of
reionization (Zeldovich & Sunyaev 1969; Hu et al. 1994b) and from the sum of the SZE of
clusters of galaxies (Markevitch et al. 1991). The WMAP results for the TE power spectrum point towards an early reionization of the universe
with corresponding optical depth
(Spergel et al. 2003; Kogut et al. 2003). Here two main effects are important: (i) the photoionized gas typically has temperatures T of the order of 104 K. Therefore the diffuse gas after reionization will
produce a y-distortion in the CMB spectrum with
All the effects mentioned in this section lead to y-distortions with corresponding y-parameter in the range . As has been shown in Sects. 5 and 7 the y-distortions arising due to the CMB dipole are of the same order. For example the CMB dipole induces a full sky y-distortion with . Therefore, to measure any of the effects discussed above it is necessary to take spectral distortions associated with the CMB dipole into account. But since the angular distribution and the amplitude of these distortions can be accurately predicted they can be easily extracted. As has been shown in Sect. 6 spectral distortions arising from higher multipoles typically have and therefore do not play an important role in this context. But even for these distortions the locations and amplitudes can be accurately predicted with measured CMB maps and therefore offer the possibility to eliminate these distortions from the maps.
The results of this derivation were then applied to measurements of the CMB temperature anisotropies with finite angular resolution and in particular to the CMB dipole and its associated spectral distortions, but in principle the method developed here can be applied whenever one is dealing with the superposition of blackbodies with similar temperatures. We have shown that taking the difference of the CMB intensities in the direction of the maximum and the minimum of the CMB dipole leads to a y-distortion with y-parameter . This value is 12 times higher than the y-type monopole, . Since the amplitude of this distortion can be calculated with the same precision as the CMB dipole, i.e. today (Fixsen & Mather 2002), it opens a way to cross calibrate the different frequency channels of CMB experiments down to the level of a few tens of nK (for more details see Sect. 8).
We discussed another possibility to check the zero levels of different frequency channels by observing the difference of the brightness in the direction of the dipole maximum and in the direction perpendicular to the dipole axis: the dipole-induced spectral distortion in this case is 4 times weaker than for the difference of the maximum and minimum and corresponds to . Nevertheless, it is still 3 times stronger than the dipole-induced whole sky y-distortion with y-parameter .
The value of is only 5 times lower than the upper limit on the whole sky y-parameter obtained by the COBE/FIRAS experiment Fixsen & Mather (2002) and is orders of magnitudes above the sensitivity of PLANCK and CMBPOL in each of their spectral channels. Therefore this signal might become useful for both cross calibration and even absolute calibration of different frequency channels of these experiments to very high precision, in order to permit detection of the small signals of reionization as for example discussed by Basu et al. (2004), and to study frequency-dependent foregrounds with much higher sensitivity. We should emphasize that on this level we are only dealing with the distortions arising from the CMB dipole as a result of the comparison of Planck spectra with different temperatures close to the maximum and minimum of the CMB dipole, i.e. due to the superposition of blackbodies. The distortions are introduced due to the processing of the data and become most important in the high frequency channels.
The distortions discussed in this work have both well known spectral and angular dependence and that they are connected with the much stronger signal of the CMB dipole. The amplitude of the dipole is known to a relative precision and therefore the induced spectral distortions can be calculated to the same accuracy. The great properties of the calibration source discussed above might even be detectable in the high frequency channels () of the existing COBE/FIRAS data and thereby help to further improve its calibration.
The development of the CMB experimental technology is repeating with a 20 year delay the progress that was made in the astronomical observations with X-ray grazing incidence mirrors and CCD detectors. Due to the big efforts of many spacecraft teams, the X-ray background today is resolved to more than 85% (Rosati et al. 2002). During the next decade, CMB observers will be able to pick up all rich clusters of galaxies and all bright y-distorted features connected with supernovae in the early universe, groups of galaxies and even down to patchy reionization in the CMB maps. Deep source counts may permit the separation of their contribution to the y-parameter and in principle might bring us close to the level of the y-distortions mentioned in Sect. 9. As has been argued, at this stage the contributions to the y-parameter arising from the superposition of Planck spectra with different temperatures corresponding to the observed CMB temperature fluctuations will be the easiest to separate. Again, the main contribution will arise from the CMB dipole. On the full sky the corresponding y-parameter will be . But even the distortions arising from the primordial temperature fluctuations with multipoles can lead to distortions of the order of in significant parts of the sky. In the future this type of spectral distortion can be taken into account making it possible to enter the era of high precision CMB spectral measurements.
The main obstacle for measuring and utilizing the effects connected with the superposition of blackbodies for calibration issues will be the lack of knowledge about both the spectral and spacial distribution of foregrounds. Given the fast progress in experimental technology and the increasing amount of data available there may still be a way to separate all these signals in the future. In any case, for upcoming CMB experiments spectral distortions arising due to the superposition of blackbodies, the effects of foregrounds and correlated noise should be taken into account simultaneously and self-consistently.
Some of the results in Sect. 6 have been obtained using the HEALPIX distribution (Gorski et al. 1999). J.C. would like to thank C. Hernández-Monteagudo, S. Yu. Sazonov, J. A. Rubiño-Martín, G. Huetsi and K. Basu for very valuable discussions and suggestions. R.S. wants to acknowledge support from the Gordon Moore distinguished scholar fellowship which allowed working on this paper during Dec. 2003 and Jan. 2004 at the California Institute of Technology. R.S. also is grateful to Marc Kamionkowski for hospitality at Caltech, useful discussions and the information about the paper of Kamionkowski & Knox (2003). R.S. wants to acknowledge stimulating conversations with Lyman Page, Dale Fixsen, John Mather, Charles Lawrence, Rüdiger Kneissl and Tony Readhead especially about future experiments.