Astrophysics with the Spatially and Spectrally Resolved Sunyaev-Zeldovich Effects

Abstract

In recent years, observations of the Sunyaev-Zeldovich (SZ) effect have had significant cosmological implications and have begun to serve as a powerful and independent probe of the warm and hot gas that pervades the Universe. As a few pioneering studies have already shown, SZ observations both complement X-ray observations—the traditional tool for studying the intra-cluster medium—and bring unique capabilities for probing astrophysical processes at high redshifts and out to the low-density regions in the outskirts of galaxy clusters. Advances in SZ observations have largely been driven by developments in centimetre-, millimetre-, and submillimetre-wave instrumentation on ground-based facilities, with notable exceptions including results from the Planck satellite. Here we review the utility of the thermal, kinematic, relativistic, non-thermal, and polarised SZ effects for studies of galaxy clusters and other large scale structures, incorporating the many advances over the past two decades that have impacted SZ theory, simulations, and observations. We also discuss observational results, techniques, and challenges, and aim to give an overview and perspective on emerging opportunities, with the goal of highlighting some of the exciting new directions in this field.

Appendix

This appendix covers some of the practical aspects common to observations of the SZ effect. What follows are observational considerations like beam size, beam solid angle, and how to compare sensitivity in common units like $\mbox{Jy}/\mbox{beam}$, $\upmu\mbox{K}$, and $\upmu\mbox{K}_{\mbox{ CMB}}\mbox{-arcmin}$.

1.1 A.1 Beam Size

After frequency and sensitivity, the primary observational considerations are resolution and scales recovered. The scales recovered differ for observations with an interferometric array and for photometric measurements with a bolometric array on a single dish, and are considered in Sect. 5.1 and Sect. 5.2, respectively.

The full width at half maximum $\theta_{\mbox{ FWHM}}$ for the main beam of a diffraction limited telescope can be approximated as a Gaussian,

$$ \theta_{\mbox{ FWHM}} \approx1.22 \frac{\lambda}{D}, $$

(18)

for wavelength $\lambda$ and dish diameter $D$. Any power outside of this main beam, such as that due to ‘side lobes’ (see e.g. Thompson et al. 2017), is often referred to as the ‘error beam,’ and can often be characterised with a wider Gaussian of lower amplitude. In practice, $\theta_{\mbox{ FWHM}}$ is often slightly larger due to under-illumination of the primary and imperfect focusing. The solid angle $\varOmega_{\mbox{ bm}}$ subtended by a beam with $\theta_{\mbox{ FWHM}}$ is

$$ \varOmega_{\mbox{ bm}} = \frac{\pi(\theta_{\mbox{ FWHM}}/2)^{2}}{ \ln(2)}. $$

(19)

For instance, the 100-m Green Bank Telescope operating at 90 GHz (3.3 mm) would have a diffraction limited FWHM of $8.38^{\prime\prime}$ and a corresponding beam volume $\varOmega_{\mbox{ bm}} = 80$ square arcseconds; however, in practice the under-illumination yields a main beam which, if fitted by single Gaussian would have $\theta_{\mathrm{FWHM}} \approx9.5^{\prime\prime}$. Surface imperfections create an error beam, such that the total beam volume is better estimated as $\varOmega_{\mbox{ bm}} = 120~\mbox{square arcseconds}$.

1.2 A.2 Surface Brightness

The surface brightness $S_{\nu}$ in units of flux density per beam (in $\mbox{Jy}/\mbox{bm}$) is related to intensity $\Delta I_{\nu}$ (Eqs. (2), (7), & (9)) by integrating over the beam solid angle:

$$ \Delta S_{\nu}= \int \Delta I_{\nu}\, {\mathrm{d}}\varOmega= \langle \Delta I_{\nu}\rangle\, \varOmega_{\mbox{ bm}}. $$

(20)

Generically, one can convert between intensity ${\Delta I}$ and a change in the CMB temperature ${\Delta T_{\mbox{ CMB}}}$ using the derivative of the blackbody function. The ratio is

$$ \frac{\Delta I}{\Delta T_{\mbox{ CMB}}} = \frac{I_{0}}{T _{\mbox{ CMB}}} \frac{x^{4} {\mathrm{e}^{x}} }{({\mathrm{e} ^{x}}-1)^{2}}, $$

(21)

where the primary CMB intensity normalisation $I_{0}$ was defined in Eq. (3).

1.3 A.3 CMB Survey Noise

The RMS noise in maps made using arcminute-resolution CMB instruments, such as those from ACT and SPT, are often compared in units of $\upmu\mbox{K}_{\mbox{ CMB}}\mbox{-arcmin}$, defined as the RMS of the CMB temperature fluctuations $\Delta T_{\mbox{ CMB}}$ within a map created with pixels that each subtend a solid angle of $1~\mbox{arcmin}^{2}$. To convert this figure to the RMS of CMB temperature fluctuations within a given instrument’s beam, one would divide by the square root of the beam solid angle in square arcminutes (Eq. (19)). For example, a map made from an instrument with $\varOmega_{\mbox{ bm}} = 120$ square arcseconds ($0.033~\mbox{arcmin}^{2}$) with an RMS noise of $10~\upmu\mbox{K}_{\mbox{ CMB}}$ per beam would correspond to $1.8~\upmu\mbox{K}_{\mbox{ CMB}}\mbox{-arcmin}$. This conversion assumes the noise properties in the maps are Gaussian and uncorrelated on the scales being binned, which is a simplification that is not generically applicable.

1.4 A.4 Rayleigh-Jeans Brightness Temperature

Many instruments report sensitivities in Rayleigh-Jeans brightness temperature

$$ \Delta T_{\mathrm{b}} \equiv\Delta I_{\nu} \lambda^{2} / 2 k _{\mbox{ B}}, $$

(22)

which at $\nu> 40\mbox{ GHz}$ can diverge significantly from the temperature decrement in the CMB, $\Delta T$, in units of $K_{ \mbox{ CMB}}$. This relation can also be expressed in surface brightness units (e.g. using Eq. (20)), as

$$ \Delta S_{\nu}= 2 k_{\mbox{ B}}\Delta T_{\mathrm{b}} \varOmega_{\mbox{ bm}} / \lambda^{2}. $$

(23)

Brightness temperature can be trivially converted to units more directly applicable to CMB and SZ measurements. Following Finkbeiner et al. (1999), this conversion, called the ‘Planck correction factor’, is

$$ \frac{\Delta T_{\mbox{ CMB}}}{\Delta T_{\mathrm{b}}} = \frac{( {\mathrm{e}^{x}}-1)^{2}}{x^{2} {\mathrm{e}^{x}}}, $$

(24)

where $x$ is defined as it was for Eq. (2). In Table 2, we provide computations of the ratio of $\Delta T_{\mbox{ CMB}}/T_{\mathrm{b}}$ for a few representative frequencies.

Table 2 Table correction factors for converting surface brightness temperature $T_{\mathrm{b}}$ to $\Delta T_{\mbox{ CMB}}$ and the classical (non-relativistic) tSZ functions $g(x)$ and $f(x)$ in Eqs. (2) and (4)