In this section we show how we model the line and dust emission from star formation rate inside galaxies. All these correlations are taken from existing literature, we merely apply the appropriate normalization for massive star formation within halos, and rearrange or re-group them to find the luminosity which interests us. All the references are cited in individual sections.
7.3.1
Emission from C
+fine-structure line
The CII 158µ FS line is the most extensively studied FIR line that has been used as a star- formation tracer in galaxies, and it the dominant coolant of inter-steller gas. This radiation is generated both in diffuse inter-stellar medium, and in photo-dissociation regions, at the interface between molecular clouds and HII regions. The photo-ionizing radiation is dominated by B3 and B0 stars with 5 ≤ M ≤ 20 M¯ (e.g. Xu et al. 1994), but of course hotter and more massive
stars contribute. However, the intensity in CII line can in a galaxy not be expected to be directly proportional to the star formation rate, mainly because of the variety of the line sources and their physical condition. Also the excitation of the upper fine-structure level of CII fine-structure doublet saturates at high temperatures and high densities (e.g. see Kaufman et al. 1999). However the emission in this line is strong, and future sum-mm and far-IR experiments like ALMA and Herschell will be able to pick up galaxies in the CII line easily in wide range of redshifts.
To find the brightness temperature from the emission of CII line in a star-forming halo of mass Mhalo at redshiftz, we start with the correlation between total CII luminosity of a galaxy and its
star formation rate, ˙M∗ (Boselli, Gavazzi, Lequeux et al. 2002)
˙ M∗= 5.953×10−33 × 100.8×logLCII M¯ yr−1 (7.12) which gives logLCII = 1.25 h log ˙M∗ + 32.225 i (7.13) The slightly non-linear correlation given above is also supported by the observations from Stacey et al. 1991. This CII luminosity (in erg s−1) is integrated over the entire line profile. To convert it into spectral luminosity at the line center, we divide it by the line-width,Lν ≈LCII/∆ν. This
spectral luminosity is then converted to the brightness temperature with the standard formalism for conversion between flux and temperature.
7.3 Modeling the line emission
7.3.2
Emission from dust
Although we shall not be presenting the results for temperature fluctuations due to dust emission, we show below that the dust emission from a halo can be correlated with its star formation rate in a similar manner. The correlation between the total FIR emission and star-formation rate is well established. From Kennicutt 1998, we have for starburst galaxies
˙
M∗ (M¯ yr−1) = 4.5×10−44LF IR (erg s−1) (7.14)
Here LF IR refers to the full integrated IR luminosity in the 8−1000µ range. However, most of
this emission comes from the wavelength range 20−200µ, with the peak of emission near 100µ, and hence the spectral luminosity of dust at 100µshows good correlation with the star-formation rate (Buat & Xu 1996, Misiriotis et al. 2004)
logL100 = 29.03 + log ˙M∗ (7.15)
whereL100 is the spectral luminosity (in ergs s−1Hz−1 sr−1) for the dust spectrum at 100µ. The advantage of this formulation is thatL100is a directly observable quantity, and we need no detailed dust SED modeling and dependence on dust temperature.
Forbroad-band CMB experiments like PLANCK, the flux incident in each frequency channels will be the integrated flux over a band-width ∆ν ≈0.25ν. In such case the detailed modeling of dust SED is necessary. In star-forming galaxies like M 82, the dust SED is modeled with a ν1.5 emissivity law
Fν= (const.)ν1.5Bν(Td) (7.16)
We try to use this relation, normalized with the 100µ flux from eqn.(7.15) to get the SED for dust emission. Our modeling is based on the observations of Colbert et al. (1999). As the first approximation, we again model the spectrum of the star-forming galaxy M 82, and assume that similar spectrum will be obtained from halos at all redshifts. Surely this will give us an upper limit on the contribution from dust emission. Under such assumption, the formula for the observed flux density from dust emission becomes
Fνobs= 8.22 µ L100 D2 L ¶ · νobs(1 +z) 3×1012Hz ¸4.5 · exp µ hνobs(1 +z) kTd ¶ −1 ¸−1 (7.17) in ergs s−1 cm−2 Hz−1, whereνobs is the observing frequency, νobs=νem(1 +z), andDL is the
luminosity distance (in cm). Integrating this flux we immediately get the dust contribution in each broad-band observing channel.
The correlation between the IR-luminosity and dust temperature is very weak, as shown by Blain, 1999. He has given a simple relation between bolometric luminosity and Td for luminous
infrared galaxies for the same emissivity indexβ = 1.5 and a single population of isothermal dust Td/K'40¡LF IR/1010L¯¢
0.03
7. EMISSION FROM DENSER REGIONS
For further correction, we should extrapolate the dust temperature at high redshifts taking into account the increased CMB temperature, to get appropriate dust temperature at that redshift. This can be done approximately by using the relation
Td(z) =
h
Td4+β+TCM B4+β ¡
(1 +z)4+β−1¢i1/(4+β)
(7.19) whereTCM B is today’s CMB temperature.