SEARCH FOR THE SUNYAEV–ZEL'DOVICH EFFECT IN A GIANT RADIO GALAXY B1358+305

Our basic plan of observation in search of the SZE signal in a radio galaxy and the feasibility evaluation are provided below. Compared to the case of galaxy clusters, the spatial extent of the cocoon relative to the radio lobes is less understood both theoretically and observationally (see McNamara & Nulsen 2007, and references therein). We, therefore, develop the basic observational plan paying a special attention to the extent of the cocoon as follows.

• 1.  
  The amplitude of the intensity decrement of SZE is proportional to the length along the line of sight (Equation (6)). Hence, a large radio galaxy should be selected as a target.
• 2.  
  In order to reduce the cancellation of the intensity decrement of SZE by the emission from the radio lobes, the observation frequency should be higher than that used in the observations of the synchrotron emission of the radio lobes. Synchrotron emission has a power-law spectral energy distribution ∝ν^−α with a typical α about 0.7, and its intensity decreases with ν in the GHz regime.
• 3.  
  Taking into account the potentially wider spatial extent of the cocoon in comparison to the radio lobes that is assumed in the classical hydrodynamic models (e.g., Begelman & Cioffi 1989), a wider region than the radio lobes should be mapped.
• 4.  
  Since the expected signal of the SZE is weak and the cocoon structure is not well understood, substructures and emission sources within the mapped region should be carefully resolved and removed. To resolve the substructures and to take account of the proportionality of the y-parameter to the length scale along the line of sight (Equation (6)), a giant radio galaxy whose angular size is also large should be selected as a target.

In addition to the point 2, the amplitude of the thermal SZE is known to be maximum at ν ≃ 90 GHz (Birkinshaw 1999), hence the first choice for the observation frequency is 90 GHz. However, the beam size decreases with the observation wavelength, and thus the observation frequency cannot be too high in order to perform the imaging observation of a wide region (the point 4) in a reasonable timescale. The beam size at 90 GHz of the NRO 45 m telescope is about 20'', which is too small to image the large central region of B1358+305 (≳5') to achieve a reasonable signal-to-noise ratio within a realistic timescale (see discussion on the feasibility in Section 3.1.1 below). Additionally, the atmospheric stability becomes worse for higher frequencies in this waveband, or the typical system temperature T_sys≈ 250–900 K at 80–90 GHz is much higher than that at 21GHz (T_sys≈ 100–150 K). Hence, the required observational time becomes unrealistically long.

We chose the single-dish observation as the first trial rather than interferometric observation. This follows from the fact that the hydrodynamic models predict a smoother distribution of SZE compared to galaxy clusters (Begelman & Cioffi 1989; Yamada et al. 1999). In other words, the smooth SZE signal would be resolved out if an interferometric observation were carried out. However, interferometric observations would be useful for detecting the SZE for more compact objects with known structures. In addition, high angular resolution observation with interferometers can also be used to remove emission source contaminations within a field of view. Both single-dish and interferometric observations have their advantages and disadvantages. In order to avoid the risk of resolving out of the possibly extended SZE signal, we use the single-dish observation strategy to measure the total flux.

Given these considerations and the low luminosity at 10 GHz, we decided to observe the central region of a giant radio galaxy B1358+305 with the NRO 45 m telescope at 21 GHz. The typical beam size of the 45 m telescope is ∼80'' at this frequency. The low system temperature (typically ∼100–150 K) and a wide bandwidth (2 GHz) of the HEMT 22 receiver installed on the 45 m telescope led to a decrease in the necessary observation time for an SZE search in a faint giant radio galaxy. The detailed setups and observation method are described in Section 3.3 below.

3.1.1. Feasibility Estimation

B1358+305 is one of the largest radio galaxies of FR-II type at a moderate redshift (z = 0.206: Parma et al. 1996; Saripalli et al. 1996). Employing cosmological parameters for the Λ-dominated flat universe (Spergel et al. 2003), its projected size amounts to ∼926 kpc, extending over ∼10' in the plane of the sky (Parma et al. 1996).⁷ It has a relatively low radio luminosity (P_1.4 GHz = 1.9 × 10²⁵ W Hz⁻¹), and the spectral age was estimated to about ∼2.5–7.5 × 10⁷ yr from the multi-band imaging observations (Parma et al. 1996). Parma et al. (1996) further calculated the advance speed of the shock head of B1358+305 as R/τ_spec ≃ 0.02c–0.03c (R is the lobe length, τ_spec is the spectral age, c is the speed of light, respectively). From the balance of the external ram pressure and the jet ram pressure, they concluded that the environment density of B1358+305 is quite low (n_e ≲ 10⁻⁶ cm⁻³), and B1358+305 is overpressured even for transverse direction, for an ambient temperature as high as that of a typical poor cluster environment (∼10⁷ K). This makes B1358+305 a good target in the sense that it resembles the classical model.

B1358+305 has an FR-II morphology with two aligned lobes with a radio core, which resides closer to the northern lobe. As SZE appears as the decrement of the CMB in the frequency range ν≲ 200 GHz, any compensating radio emission should be avoided. Since the lobe intensity rapidly decreases toward the region slightly south of the core, we focused on that region in order to avoid the lobe emission contamination as much as possible (Figure 1).

Observations of B1358+305 were performed with the HEMT 22 receiver on the Nobeyama 45 m telescope from 2001 February 28 to March 18. The receiver has 2 GHz bandwidth, and is equipped with dual channels which can simultaneously receive two circular polarization components, which enables the reduction of the required observation time. The near-central region in B1358+305 of size 67 × 67 around (α₅₀, δ₅₀) = (13^h58^m250, + 30°32'00) was raster-scanned with 40'' spacing, which created a map composed of 11 × 11 pixels. Since the jet axis is nearly along the declination, raster-scans were performed along the R.A.–decl. coordinates in turn. The scan speed is about 25'' s⁻¹, resulting in the acquisition of one map in ∼180 s. The exposure time was about 74.8 ks in total, corresponding to 618 s for each pixel. The system noise temperature was about T_sys = 140 K on average, ranging from about just below 130 K to nearly 150 K. The noise produced by the fluctuations in the atmosphere was reduced by the simultaneous observation of an off-source point via the beam switching technique. The off-source point is about 400'' away in azimuth direction to the east and was simultaneously observed at 15 Hz. The antenna temperature was calibrated using the chopper-wheel method, while the optical depths of the atmosphere were measured by elevation scans. Since the atmospheric optical depths were about 0.04 throughout the observing period, the flux was not corrected for atmospheric absorption. The main beam size θ_HPBW was estimated to be 812 by observing 3C273. With this beam size we can image a large (67 × 67) area in a reasonable timescale, simultaneously resolving the substructures in the mapped region. For the pointing and flux calibrator, we used 3C286 about 75 away from the center of the image. The pointing offset was monitored about every hour, and was typically smaller than 6'' throughout the observing period. The flux of 3C286 was assumed to be 2.56 ± 0.2 Jy (Ott et al. 1994), and it was used to evaluate the aperture efficiency of the telescope. Due to the good weather conditions, the aperture efficiency was stable during the latter half of the observation period (rms of the flux variation is about 5%). In the former period, the stability was slightly worse (the flux variation reached about 10%), and we carefully corrected each observation using the efficiency measured just before the observation.

Raster-scans were performed in two orthogonal directions (along the right ascension and the declination coordinates) in turn in order to obtain pairs of orthogonally scanned maps taken as close together in time as possible. Within a raster-scanned map, antenna temperature fluctuations induced by the time variation of the atmospheric conditions, instability of the detectors or other instruments and so on, appeared along the scanning direction (scanning noise). The scanning noise in the data was reduced by combining the two orthogonally scanned maps in the Fourier space via the basket-weaving method (Sieber et al. 1979; Kuno 1993).

Although we tried to avoid contamination from lobe emissions into the mapped region, some of the scans suffered from lobe emissions in the operation of beam switching. Due to the limited throw amplitude (∼400'') and the rotation of the position angle of the beam switch, emissions from one of the lobes entered into the reference field, resulting in over-subtraction in a part of the mapped region (less than 1/4 of the mapped area at most). This over-subtraction generated artificial intensity decrements. We identified the affected pixels in each scan and removed them in the integration. In the identification process, we made use of the 10.6 GHz map of Saripalli et al. (1996; also see Figure 1 below), and set the position of the reference beam 400'' away from the main beam. Since the spectrum is a decreasing function of frequency at 21 GHz, both lobes are expected to be dimmer than displayed in the map in Saripalli et al. (1996); the procedure above does not overlook the affected pixels. Due to this partial masking in the integration, the noise level is not uniform in the obtained integrated map, ranging from 0.72 mJy beam⁻¹ to 1.2 mJy beam⁻¹ (1σ level): in spite of the loss of some pixels, however, the root mean square of the intensity in the integrated map reached 1.22 mJy beam⁻¹ with emission sources.

In order to evaluate the noise level without emission sources, we constructed two independent maps by dividing the maps taken in the former and the latter halves of the observing period, and subtracted the latter from the former. In this process, emission from true sources should cancel out and the resultant map should contain only random noise. The rms fluctuation of the differential map constructed in this way is 1.01 mJy beam⁻¹.

In order to improve the effective signal-to-noise ratio, we constructed projections of the two-dimensional map along two orthogonal directions. We can also expect that these projections reflect the structure of the cocoon more clearly than the original two-dimensional map if the data contains the SZE signal from the cocoon of B1358+305. As can be seen in Figure 1, the jet axis is almost perpendicular to the right ascension. Thus, we can examine the structure perpendicular (parallel) to the jet axis by projecting the map along the declination (right ascension) coordinates. In the process of projection, we average the pixels along each column or row without using the masked pixels corresponding to the emission sources that are indicated with the white lines in Figure 3.

Figure 4(a) shows the projection along the jet axis (the declination coordinate). A relatively dark region appears at about +2' west of the central column. Comparing Figure 4(a) with the original two-dimensional map (Figure 2), one can see the decrement appearing in Figure 4(a) mainly picks up the dark region between the unidentified source labeled "D" in the map and the AGN (labeled "A"). However, as the statistical errors are large (error bars in Figure 4 correspond to 1σ) even though they are reduced by ${\sim} 1/\sqrt{N_p}$ in averaging over each column (N_p is the number of pixels included), we cannot rule out the possibility that the projection obtained here is consistent with a flat distribution (no SZE signal accompanied B1358+305). As shown in Figure 1, B1358+305 has a relatively large axis ratio, and it is reasonable to approximate it as a cylinder with an axis along the jet axis. Hydrodynamic simulations (e.g., Loken et al. 1992; Kaiser & Alexander 1999; Scheck et al. 2002) show that the pressure p_e is nearly uniform inside a cocoon when the density distribution of the surrounding IGM is nearly flat (there will be a difference in ξ_e at internal and external shocks, and accordingly p_e would not be spatially uniform as is suggested by observation, but this assumption is discussed separately in Section 5.2 below). Since y ∝ p_el (Equation (2), where l is the length cut through the cocoon along the line of sight), the projection along the jet axis is expected to present l(θ) in the isobaric cocoon, where θ is the angular distance projected onto the sky from the map center. For a cylindrical cocoon, therefore, the intensity decrement ΔI_ν is proportional to $-\sqrt{1-\theta ^2}$. We overplot the expected profile ($\Delta I_{\nu }\propto -\sqrt{1-\theta ^2}$) based on Equation (6) with fiducial values on Figure 4(a) as a dashed line, offset so that the average intensity is equal to zero. The obtained distribution is not inconsistent with the expected one or with the flat distribution. This implies the dominance of the statistical errors over the signal reflecting the possible cocoon structure in the observed region. We estimate the standard deviation from the flat distribution,

$$\begin{equation} |\Delta I_{\nu }| = 0.28 \;{\rm mJy \;beam}^{-1}, \end{equation} \tag{ 7 }$$

which sets the upper limit y_upp ≲ 1.04 × 10⁻⁴ at the 95% confidence level (Equation (1)).

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We further examine the allowed upper limit for a Compton y-parameter by a chi-square test of the data for the projection along the jet axis and the expected profile from a cylindrical cocoon model $\Delta I_{\nu }\propto -\sqrt{1-\theta ^2}$. We find that as long as ΔI_ν⩽ 0.32 mJy beam⁻¹ (or y ⩽ 1.19 × 10⁻⁴), these two distributions can agree at the 95% confidence level. Herewith, we denote these as |ΔI_ν|^χ and y^χ_upp.

We have also constructed the projection onto the jet axis. If the cocoon has a prolate figure and lies in the plane of the sky, the pressure scale length along the jet axis is larger than the declination range of the observed field (see Figure 1). Hence, the projection onto the jet axis would display a more moderate decrement in contrast with the projection along the jet axis.⁸ Figure 4(b) shows the projection onto the jet axis. There clearly exist a peak about 07 north of the central row and two decrements toward both edges, which are not consistent with the expectation from the isobaric cocoon model filled with thermal electrons (Section 2). These decrements are due to the two dark regions north and south of the unidentified source "D" (at around α₅₀ ∼ 13^h58^m20^s; see Figure 2). The standard deviation from a flat distribution derived from the projection in this direction (onto the jet axis) is |ΔI_ν| = 0.44 mJy beam⁻¹, which is larger than the statistical error estimated from the noise level of the original two-dimensional map in the process of the projection ($1.01/\sqrt{N_p}$, ∼0.27 mJy beam⁻¹ on average). Even though the dark regions appearing in Figure 4 may indeed suggest the existence of SZE signals, we cannot rule out the possibility that these are not due to B1358+305.

In order to examine the dominant source of the fluctuations in Figures 4(a) and (b), we analyzed the differential map in Section 3.4 in the same manner as the total map. Specifically, the differential map was constructed by dividing the observation period into the former half and the latter half. Figure 5 shows the fluctuations in the projections of the differential map overplotted with dashed lines on the total map (solid lines, the same as Figure 4). The projections on the both axes have similar amplitudes (the rms of the projection along the jet axis is 0.27 mJy beam⁻¹ and the projection onto the jet axis is 0.43 mJy beam⁻¹) as those of the total map, but their distributions do not resemble the total map. Taking into account the fact that real signals, including foreground or background emission sources in the field of view, do not appear in the differential map, we conclude that the fluctuations in Figure 5 are largely caused by the excess atmospheric noises at 21 GHz.

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The expected amplitude of SZE is estimated using a steady mean acceleration efficiency 〈ξ_e〉 (Equation (6)). If 〈ξ_e〉 is significantly different from assumed, the expected value of y is also affected. In particular, different cooling timescales for the thermal and non-thermal electrons may alter 〈ξ_e〉 from the fiducial value (=0.05 in Equation (6)) in time, especially in the central region where the oldest population is contained. Since an underestimate of 〈ξ_e〉 leads to the overestimate of the expected y-amplitude (Equation (6)), we examine whether the different cooling timescales for the thermal and non-thermal electrons can significantly alter 〈ξ_e〉 using a simple argument about exhaustion of the internal energy. Since the cooling timescale is strongly dependent on the electron energy γ, we discuss synchrotron electrons of high γ (γ ≳ 10³ or more) and thermal electrons (γ ≲ 10 at T_e ≲ 100 keV) separately.

Parma et al. (1996) derived the spectral ages of the non-thermal electrons for both the northern and southern lobes using the compiled multi-frequency observation of B1358+305 from 325 MHz to 10550 MHz, finding the break frequency to be 2.4 GHz from the integrated spectral energy distribution. Further fitting of the multi-frequency images with the energy loss formula including synchrotron and CMB inverse Compton losses leads to spectral ages of τ_spec for B1358+305 in the range 2.5 × 10⁷ yr ≲τ_spec ≲ 7.5 × 10⁷ yr. Since the synchrotron cooling timescale and inverse Compton cooling timescales are proportional to γ⁻¹, lower energy electrons have a longer cooling timescale. The simple scaling of the above estimation by Parma et al. (1996) indicates that an electron that emits 325 MHz synchrotron light has a spectral age of about a few times of ∼10⁸ yr, or even in the extreme case ∼10⁹ yr at best. We adopt 10⁸ yr as the typical cooling timescale for the non-thermal electrons.

On the other hand, a relativistic Maxwellian distribution of high temperature thermal electrons,

$$\begin{equation} P_e(\gamma)d\gamma = \frac{\gamma ^5\beta ^2\exp (-\gamma /\Theta)}{\Theta K_2(\Theta ^{-1})} d\gamma, \end{equation} \tag{ 8 }$$

spans over γ ≲ 10 for k_BT_e≲100 keV (where K₂(x) is the second-order modified Bessel function, β = v/c, and Θ ≡ k_BT_e/m_ec² is the non-dimensional electron temperature). We estimate the cooling timescales following Sarazin (1999). Inverse Compton scattering of CMB photons, synchrotron self-Compton scattering, Coulomb cooling, and bremsstrahlung are taken into account as cooling processes. The density of the IGM and in B1358+305 is quite low (n_e ≲ 10⁻⁶ cm⁻³; Parma et al. 1996), and thus bremsstrahlung and line cooling are negligibly small compared with the other processes. In this low-energy regime (γ ≲ 10) and in the very poor environment (the density of the surrounding IGM was estimated to be 1.4 × 10⁻⁷ cm⁻³ ≲ n_IGM ≲ 8.4 × 10⁻⁷ cm⁻³; see Table 3 of Parma et al. 1996), the major cooling mechanisms are inverse Compton cooling and the Coulomb cooling, but the cooling time is longer than the Hubble time (τ_cool ≳ 10¹⁰ yr). Therefore, the thermal energy of electrons is conserved for a typical cooling time of the synchrotron electron (∼10⁸ yr). Even if we adopt a cooling timescale for electrons that radiate low frequency 325 MHz photons (τ_cool ≲ 10⁹ yr), it is unlikely that the cooling time of thermal electrons is lower than that of non-thermal electrons. In other words, as long as the radio lobes of B1358+305 are bright in synchrotron emission, the thermal energy of electrons in B1358+305 does not change in time. Therefore, it is unlikely that the lack of the detection of an intensity decrement in B1358+305 is attributable to the loss of thermal electrons by cooling.

In addition to the cooling processes described above, some models emphasize the importance of adiabatic expansion (Ito et al. 2008) or backflows of jet matter that escapes thermalization at the shock interaction (Scheck et al. 2002). These models predict a lower thermal pressure of electrons than the model of Begelman & Cioffi (1989). Quantitative evaluation of these effects has yet to be determined, but it is to be noted that our model tends to overestimate the thermal pressure of the electrons in a cocoon. If these effects that reduce the thermal pressure of electrons are significant in the evolution of a radio galaxy, much more sensitive observations will be necessary.

In this paper, we adopted a model based on the classical overpressured cocoon model with a constant and uniform mean shock acceleration efficiency 〈ξ_e〉, and a uniform distribution of thermal electrons within a cocoon as the simplest and most optimistic interpretation of the observation (Section 2). However, in spite of the search for X-rays in radio galaxies in galaxy clusters, the clear evidence for non-thermal electron acceleration at the external shock has not been found except for objects such as Cen A (Croston et al. 2009). Although this lack of clear evidence may reflect the limited observational sensitivity or angular resolution, the uncertainty in determining 〈ξ_e〉 is also likely to contribute. Especially, at the internal shock or termination shock of the relativistic jets, ξ_e can be as large as close to unity, while ξ_e at the non-relativistic external shock is much smaller. In this case, the radio lobes within the shocked shell would have only a small population of thermal electrons, whereas the shocked shell has a large fraction of thermal electrons as in the first adopted model (Section 2).

The spatial SZE profile of this model would differ significantly from that adopted (see Section 4.1). The amplitude of SZE caused by non-thermal power-law electrons depends on the parameters describing the electron energy distributions (e.g., the power-law index, the lower and upper cutoff energies, and the break energy if any). The ratio of thermal and non-thermal SZE amplitudes is consequently dependent on these parameters. If the non-thermal SZE amplitude is much smaller than thermal SZE in the radio lobes, the SZE spatial profile for this case would not reflect the length passing through the cocoon, but the length passing through the shocked IGM shell. Therefore, it may have a flatter SZE distribution as compared to that in Section 4.1 ($\propto -\sqrt{1-\theta ^2}$). This possibility makes the detection of SZE much more difficult (see, e.g., Pfrommer et al. 2005; Colafrancesco 2008; Hardcastle & Looney 2008). On the contrary, it may also be possible that the non-thermal SZE amplitude is similar to the thermal SZE. In this case, the spatial profile does not differ from that in Section 4.1, and detection of SZE is unlikely to be significantly more difficult compared with the cocoon model filled with thermal electrons (Section 2). Multi-frequency observation will be necessary to discriminate these distributions (see Section 5.4 below). Additionally, it is possible that a mature giant radio galaxy does not expand supersonically in the lateral direction (e.g., Konar et al. 2009). In this case, the SZE signal peaks at the head of the external shock and decreases toward the central region where we observe (see below). Since the diffuse shock acceleration efficiency problem remains to be solved, we cannot exclude either of these scenarios with different acceleration efficiencies. In order to solve this problem and to obtain a deeper understanding of the subsequent dynamical and thermal evolution of a radio galaxy, progress in both theoretical studies of shock acceleration efficiency and higher sensitivity observations will be required.

Radio images and some X-ray images of radio galaxies suggest that the central region close to the nucleus lack the energetic electrons that many hydrodynamic models suggest. If this applies to B1358+305, our results would correspond to genuine atmospheric noise. However, the X-ray images of an FR-II radio galaxy MS0735+74221 and Cygnus A show extended diffuse X-ray emissions that fill the central regions (for a review, see McNamara & Nulsen 2007, and references therein). In addition, Hardcastle & Looney (2008) observed bright FR-II galaxies at 90 GHz with the BIMA, and found that some of their sample (3C 20 and 3C 388) have extended emissions that cover the central regions. Although there are ambiguities in the distributions of the thermal and synchrotron electrons, these results imply that some FR-II radio galaxies seem to have electron populations at the central regions away from the jets and hotspots. We cannot tell if B1358+305 is deficient of electrons because of a small number of past observations or limited sensitivity of our results. Hence, we conclude that our results are largely due to atmospheric noise, but we cannot completely rule out the small SZE signal in B1358+305, and much higher sensitivity is required in future observations.

Since only upper limits on the y-parameter were obtained, we compare our results with past studies of B1358+305 in the case where (1) the true pressure is close to that derived from the upper limit, and (2) the true pressure is significantly lower than the obtained upper limit.

Consider the first case. Since the observed region does not extend over the entire cocoon, the perpendicular projection onto the jet axis will show a flatter intensity distribution than the parallel projection (i.e., the pressure scale length along the jet axis is longer than the declination range of the observed field; see Figure 1). Hence, it is more easily subject to the contamination from the background emission sources and the excess atmospheric noise. On the other hand, the right ascension range of the observation is expected to be comparable to the cocoon width and the parallel projection along the jet axis is expected to represent the structure of the cocoon (Section 4.1). Therefore, we employ the upper limit of the y-parameter calculated with the projection along the jet axis (|ΔI_ν|_2σ = 0.56 mJy beam⁻¹, or y_upp ≲ 1.04 × 10⁻⁴) in the discussion below. If we assume that the obtained intensity fluctuation accompanies B1358+305, then the electron pressure derives p_e = 2.20 × 10⁻¹² dyne cm⁻² (Equation (6)) for l = 300 kpc. If we take values obtained by the chi-square test y^χ_upp, the equivalent electron pressure becomes p_e = 2.51 × 10⁻¹² dyne cm⁻². On the other hand, Parma et al. (1996) deduced the jet ram pressure to be ⩾1.2 × 10⁻¹² dyne cm⁻², which is comparable to the equipartition value under the assumption that B1358+305 expands supersonically. This result is quantitatively consistent with Parma et al. (1996) in the sense that B1358+305 expands supersonically against the surrounding IGM. However, the results obtained using the upper limit in our observation (2.20–2.51 × 10⁻¹² dyne cm⁻²) are almost double the result of Parma et al. (1996; ⩾1.2 × 10⁻¹² dyne cm⁻²). This difference would come from the assumed parameter value 〈ξ_e〉 = 0.05 (Equation (6)), and may imply a larger mean acceleration efficiency (〈ξ_e〉 ≳ 0.1) in B1358+305 and significant contribution from non-thermal SZE of smaller amplitude than thermal SZE.

Since the intensity fluctuations obtained turned out to be dominated by the excess atmospheric fluctuations, however, it is also possible that the true thermal electron pressure in B1358+305 is much lower than that calculated with y_upp or y^χ_upp (case (2)). Because B1358+305 resides in a poor environment, it is difficult to directly measure the IGM pressure around it. Instead, we estimate the external pressure under the assumption that the environment resembles an extremely poor galaxy cluster or a group of galaxies, and assume the IGM temperature to be ≲10⁷ K. For the density, we can take the cosmological baryon density at the redshift z = 0.206 as the reference value, and describe the IGM density with a density excess parameter (⩾1), n_IGM =  4.3 × 10⁻⁷ cm⁻³ (Spergel et al. 2003), which sets the lower limit to the IGM pressure to be ≳ 5.9 × 10⁻¹⁶ dyne cm⁻². In this case, if the true thermal electron pressure in B1358+305 is less than 10⁻⁴ of that inferred from our upper limit, it indicates that thermal electrons do not play a major role in supporting the radio lobe against the external pressure. We note that this conclusion strongly depends on the estimate of the IGM pressure and the SZE measurement. Deeper X-ray observations for the pressure determination of the IGM and much sensitive SZE measurement are required to be more conclusive.

Finally, we discuss the X-ray observations in conjunction with the SZE in radio galaxies as effective means of probing the pressure of a cocoon (also see Pfrommer et al. 2005). The observations by ROSAT and Chandra satellites have revealed that radio lobes of radio galaxies centered at the cluster of galaxies generate cavities in the surface brightness distribution of the X-ray emitting ICM (e.g., Böhringer et al. 1993; Carilli et al. 1994; McNamara et al. 2000; Fabian et al. 2000; McNamara & Nulsen 2007). These are considered to be due to the pressure in the lobe overwhelming that of the surrounding hot ICM. Hardcastle & Worrall (2000) and Leahy & Gizani (2002), however, discovered that the ICM pressure inferred by the X-ray emission obtained by ROSAT observation of bright FR-II radio galaxies is much higher than the minimum pressure estimated by the synchrotron luminosity in the radio band for large (≳100 kpc) radio galaxies. A detailed study by Leahy & Gizani (2001) confirmed this pressure discrepancy for 3C388. These results suggest the dominance of the other forms of pressure above that due to the synchrotron electrons (and magnetic field) in the radio lobes.

Hardcastle & Worrall (2000) argued that plausible candidates for carrying the "invisible" pressure are non-thermal protons, magnetic field with strength differing from that derived from the minimum energy condition (≃B_eq), or low-energy electrons which do not emit strong radiation in the observed energy bands. If we assume that ξ_e in the internal shock is as small as that in the external shock, the radio lobes could have a significant amount of thermal electrons (Section 2). In this case, the pressure of the thermal electrons could be one of the candidates for the origin of the "invisible" pressure. Since SZE is sensitive only to electron energy, it is a good probe of either the thermal electrons or non-thermal electrons of low energy which do not emit high frequency emission in the observation bands. The observational different signature between thermal and non-thermal SZE is the frequency of zero amplitude (e.g., Birkinshaw 1999). Specifically, the thermal SZE has the null frequency (=218 GHz) which is almost independent of the gas temperature, whereas the null frequency of non-thermal SZE changes with the electron energy distribution parameters. Hence, future multi-frequency observations that covers the null frequency of thermal SZE will be a tool to probe the electron contribution to the "invisible" pressure components in the cocoon. On the other hand, if ξ_e of the internal shock is larger than that of the external shock and close to unity, resulting in a significant difference in the fractions of thermal and non-thermal electrons in the shocked shell and the radio lobes inside, SZE studies become much more difficult. In this case, the multi-frequency observation should be carried out over more than one region. For example, multi-frequency observations of the head-top region of the bow shock will be able to determine the contribution of mainly thermal electrons formed at the external shock. This can be used to correct the observation of the central region apart from the shock head, where the thermal and non-thermal SZE coexist.⁹ In summary, the SZE can be a probe of electron contribution to the "invisible" pressure in a radio galaxy, but discrimination between thermal and non-thermal contributions requires sensitive multi-frequency and multi-region observations in the future.

For the other sources of pressure, it is unlikely that the magnetic field is significantly different from B_eq. For example, X-ray observations have shown that majority of the radio galaxies tend to have larger particle energy than that of the magnetic field (for a recent results, see Isobe et al. 2009, and references therein). Protons, as another possible candidate for the "invisible" pressure, on the other hand, would lead to an SZE much smaller than that given by Equation (6). The existence of high-energy protons should be probed in the hard X-ray and/or γ-ray regime (Scheck et al. 2002).