Power Spectrum of CO Foreground

INTENSITY MAPPING EXPERIMENTS USING GALAXIES SELECTED BY STELLAR MASS AND REDSHIFT

3.2 Methods for Modeling Infrared Galaxies as CO Proxies

3.3.2 Power Spectrum of CO Foreground

CO emission is derived for each object in the catalog by converting infrared luminos- ity to CO line strength with the well-established 𝐿_IR–𝐿^′

CO correlation (e.g., Carilli

& Walter 2013, hereafter CW13; Greve et al. 2014, hereafter G14; Dessauges- Zavadsky et al. 2015). We can then use the measured stellar mass functions of galaxies at 0 < 𝑧 < 2 to calculate the power of CO line foregrounds, with the ability of monitoring different subsets (e.g., different stellar mass bins, quiescent vs star-forming galaxies, etc.) Now the total mean intensity of CO contamination can be expressed as

¯

𝐼_CO =∑︁

𝐽

∫ 𝑀_∗^max(𝑧) 𝑀_∗^min

d𝑀_∗Φ(𝑀_∗, 𝑧)

× 𝐿^𝐽

CO(𝐿_IR(𝑀_∗, 𝑧)) 4𝜋 𝐷²

𝐿

𝑦^𝐽(𝑧)𝐷²

𝐴, (3.7)

where 𝑀_∗^min =1.0×10⁸𝑀_⊙, 3 ≤ 𝐽_upp ≤ 6, representing all CO transitions acting as foregrounds andΦ(𝑀_∗, 𝑧)being the stellar mass function measured by the COS- MOS/UltraVISTA Survey (Muzzin et al. 2013a). 𝑀_∗^max(𝑧) represents an evolving mass cut that measures the depth of foreground masking (see Section 4.3 for a de-

0.1 1.0

k [h/Mpc]

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹ 10² 10³

k

3

P

CO(1−0)

( k ) / 2 π

2

[ µ K

2

]

z = 1

Carilli & Walter (2013) Greve et al.(2013) Padmanabhan (2018)

Figure 3.5: CO(1–0) power spectra predicted by our models. Predictions assuming prescriptions of CW13 and G14 are compared with that of the best-fit model cali- brated to the observed CO luminosity function by Padmanabhan (2018).

tailed discussion) and is set to 𝑀^max

∗,0 =1.0×10¹³𝑀_⊙ when no masking is applied.

The factor 𝑦^𝐽(𝑧) = d𝜒/d𝜈^𝐽

obs = 𝜆^𝐽

rf(1+𝑧)²/𝐻(𝑧) accounts for the mapping of fre- quency into distance along the line of sight (Visbal & Loeb 2010). The comoving radial distance 𝜒, the comoving angular diameter distance 𝐷_𝐴 and the luminosity distance 𝐷_𝐿 are related by 𝜒 = 𝐷_𝐴 = 𝐷_𝐿/(1+𝑧). In the presence of scatter (as is always the case), the expectation value of a functionF of CO luminosity at a best-fit 𝐿_IR(𝑀_∗, 𝑧) given by SIMSTACK can be written as

𝐸 h

F

𝐿^𝐽

CO

i

=

∫ ∞

−∞

d𝑥F (10^𝑥)𝑃(𝑥|log𝐿_IR), (3.8) and

𝑃(𝑥|log𝐿_IR) = 1 𝜎_tot

√ 2𝜋

exp −(𝑥− 𝜇)² 2𝜎²

tot

!

, (3.9)

10^-2 10^-1 10⁰ 10¹

k[CII][h/Mpc]

10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10¹²

kP(k)[(Jy/sr)2(Mpc/h)2]

G14 CW13

z = 6.5 unmasked

COtot

zCO(3−2)= 0.36 zCO(4−3)= 0.82 zCO(5−4)= 1.27 zCO(6−5)= 1.73 [CII], Silva+2015

[CII], Gong+2012 COsim, Silva+2015 COobs, Silva+2015

10^-2 10^-1 10⁰ 10¹

k[CII][h/Mpc]

z = 6.5 masked

G14 CW13

Figure 3.6: Comparison of masked and unmasked projected CO power spectra. Left:

a comparison of the projected power spectra of unmasked CO emission lines and [C ii] at𝜈_obs ∼ 250 GHz (𝑧_{C II} ∼6.5). Our model illustrates the relative contribution to the total CO power spectrum (gray and black solid lines, respectively, for the two 𝐿_IR–𝐿^′

CO prescriptions, CW13 & G14) from different CO transitions (green lines, just for G14), with a simulated input scatter of 𝜎_tot = 0.5 dex. Also shown are two CO models from Silva et al. (2015): the first based on simulated 𝐿^𝐽

CO− 𝑀_halo relations (purple line labeled ‘sim’); and the second from scaling the observed infrared luminosity function (red line labeled ‘obs’). [C ii] signals predicted by Gong et al. (2012) and Silva et al. (2015, “m2” model), shown as blue and yellow lines, respectively, highlight the large range of existing [C ii] predictions. Right:

the same comparison as in the left panel, but after masking bright galaxies down to an evolving stellar mass cut (see Section 3.3.3), which results in a∼8% (G14) and

∼18% (CW13) loss of the total survey volume.

where

𝜇=log𝐿^𝐽

CO(log𝐿_IR) =𝛼⁻¹(log𝐿_IR− 𝛽) +log𝑟_𝐽 (3.10) is derived from the best-fit 𝐿^′

CO–𝐿_IR correlation, with𝑟_𝐽 being some scaling factor for different 𝐽’s. Consequently, in the presence of scatter, Equation 3.7 becomes

⟨𝐼¯_CO⟩ ≡ 𝐸 𝐼¯_CO(𝐿^𝐽

CO)

, which describes the expectation value of the total CO mean intensity, averaged over the probability distribution of 𝐿^𝐽

CO as specified by 𝜇 and 𝜎_tot. In our calculation, we consider two prescriptions: (i) CW13, who give 𝛼 = 1.37 ± 0.04, 𝛽 = −1.74± 0.40, and scaling relations appropriate for sub- millimeter galaxies which are used to convert to transitions higher than𝐽 =1→ 0, and (ii) G14, who provide𝛼and𝛽coefficients for each individual𝐽transition (i.e., 𝑟_𝐽 = 1) based on samples of low-𝑧 ultra luminous infrared galaxies (ULIRGs) and high-𝑧 dusty star-forming galaxies (DSFGs) comparable to CW13. Since the total

CO foreground consists of multiple 𝐽 transitions, we deem the G14 prescription more appropriate for our purposes, because it treats both the slope and intercept as free parameters when fitting to galaxies observed in different 𝐽’s. Henceforth, we present our results based on the G14 model unless otherwise stated.

It is worth noting that the𝐿^′

CO–𝐿_IRrelation is usually determined using a compilation of galaxy samples which collectively spans the stellar mass range log₁₀(𝑀_∗) ∼9.5- 11.5. Simply extrapolating this relation to lower stellar masses without considering possible changes in the ISM in this regime likely overestimates the predicted CO emission, as observations suggest that local galaxies with𝑀_∗ <10⁹𝑀_⊙are deficient in CO, due to lower molecular gas contents and low metallicities (e.g., Bothwell et al.

2014). Dessauges-Zavadsky et al. (2015) find a 0.38 dex scatter in 𝐿_IR for a given 𝐿^′

CO(1−0), which corresponds to 0.32 dex scatter when converting infrared luminosity into CO luminosity. Comparable levels of scatter have also been identified by CW13 and G14 using galaxy samples of similar types. We note that different from our assumption in Equation 3.9, Li et al. (2016) re-normalize the log-normal distribution so that thelinear mean remains constant and that the level of scatter only affects the shot-noise component of the power spectrum. Instead, we choose to fix the logarithmic mean in this work to best represent the distribution about the best-fit line for the observed log𝐿_IR-log𝐿^′

CO correlation. Also, for simplicity, we ignore any potential correlation between the∼0.3 dex scatter intrinsic to the total infrared luminosity and the comparable ∼0.3 dex scatter in the IR-to-CO conversion2 and combine them orthogonally (i.e., adding in quadrature, see also Li et al. 2016), yielding a total scatter of𝜎_tot∼ 0.5 dex, which is what our reference model assumes hereafter.

Figure 3.5 shows the CO(1–0) power spectra predicted by our CO model at 𝑧 = 1, compared with the best-fit model from Padmanabhan (2018) which is derived from abundance matching the halo mass function to the CO luminosity function observed at 0 < 𝑧 < 3. The overall power spectrum of the CO foreground can be written as the sum of the clustering and shot noise terms

𝑃^tot

CO(𝑧_𝑓, 𝑘_𝑓) =𝑃^clust

CO (𝑧_𝑓, 𝑘_𝑓) +𝑃^shot

CO (𝑧_𝑓, 𝑘_𝑓), (3.11) where the clustering component can be derived from the mean intensity 𝐼_CO, the average bias ¯𝑏_CO(𝑧)(Visbal & Loeb 2010) and the nonlinear matter power spectrum

2This is a somewhat arbitrary choice given the potentially similar physics (star formation, dust attenuation, etc.) that leads to the observed scatters in both cases. As it is difficult to accurately determine this potential correlation, we simply assume here that 0.5 dex is a relatively conservative estimate of the total scatter.

𝑃^nl

𝛿𝛿(computed with theCAMB-basedHMFcalccode, Murray et al. 2013) as 𝑃^clust

CO (𝑧_𝑓, 𝑘_𝑓) =∑︁

𝐽

¯ 𝑏²

CO

𝐼¯^𝐽

CO

2

𝑃^nl

𝛿𝛿(𝑧_𝑓, 𝑘_𝑓), (3.12) and the shot noise or Poisson component is given by

𝑃^shot

CO (𝑧_𝑓) =∑︁

𝐽

∫ 𝑀_∗^max(𝑧) 𝑀_∗^min

d𝑀_∗Φ(𝑀_∗, 𝑧_𝑓)

× (

𝐿^𝐽

CO 𝐿_IR(𝑀_∗, 𝑧_𝑓) 4𝜋 𝐷²

𝐿

𝑦^𝐽(𝑧_𝑓)𝐷²

𝐴

)2

. (3.13)

Estimating the CO contamination for any given observed [C ii] power spectrum also requires scaling (i.e., projecting) the corresponding CO comoving power spectrum at low redshift to the redshift of [C ii]. Following Visbal & Loeb (2010) and Gong et al. (2014), the projected CO power spectrum can be written as

𝑃^𝐽

obs,CO(𝑧_𝑠,𝒌𝑠) =𝑃^𝐽

CO(𝑧_𝑓,𝒌𝑓) ×

𝜒(𝑧_𝑠) 𝜒(𝑧_𝑓)

2

𝑦^𝐽(𝑧_𝑠)

𝑦^𝐽(𝑧_𝑓), (3.14) where |𝒌𝑓| =√︃

(𝜒(𝑧_𝑠)/𝜒(𝑧_𝑓))²𝑘²

⊥+ (𝑦^𝐽(𝑧_𝑠)/𝑦^𝐽(𝑧_𝑓))²𝑘²

∥ is the 3D𝑘 vector at the redshift of CO foreground. Here we assume 𝑘_⊥ =

√︃

𝑘²

1+𝑘²

2 and 𝑘₁ = 𝑘₂ = 𝑘_∥ for the 3D 𝑘vector |𝒌𝑠|=√︃

𝑘²

⊥+𝑘²

∥ at the redshift of [C ii] signal.

The left panel of Figure 3.6 shows our predicted CO power spectra projected into the frame of [C ii] at redshift 𝑧 = 6.5 or an equivalent observing frequency of 𝜈_obs =250 GHz. Contributions from different CO transitions (green curves) to the total CO power (gray and black solid curves) are shown by different line styles.

For comparison, we also show two alternative CO models from Silva et al. (2015).

The simulation-based model (purple line) is derived from fitting to the simulated 𝐿^𝐽

CO− 𝑀_halo relations (Obreschkow et al. 2009a,b), whereas the observational CO model (red line) is based on rescaling the observed infrared luminosity function (Sargent et al. 2012) with the ratios given by CW13. Finally, we note that Breysse et al. (2015) assume “Model A” of Pullen et al. (2013), which models the CO luminosity at a given dark matter halo mass with a simple scaling relation and predict a much higher level of CO foreground for the [C ii] signal given by the “m2”

model of Silva et al. (2015). However, we note that the Pullen et al. (2013) “Model A” is only optimized for observations at𝑧 ∼ 2 and fails to capture the transition to sub-linear scaling of the𝐿_CO–𝑀_halorelation at halo masses 𝑀_halo> 10¹¹𝑀_⊙.

Figure 3.7: 𝐿_IR(𝑀_∗, 𝑧) model predictions in five narrow redshift intervals.

𝐿_IR(𝑀_∗, 𝑧) predictions are color-coded by the derived CO(4–3) flux in [W/m²], assuming the G14 prescription. Each (𝑀_∗, 𝑧) point is taken directly from the UVISTA-DR2 catalog. Note that some bands are shifted vertically for visual clarity (the multiplicative factors are reported in the legend). The magenta curves are two examples of constant CO flux, or equivalently evolving stellar mass cut, correspond- ing to a total masked fraction of 8% (Case A, extensive) and 4% (Case B, moderate), respectively.

The variation in modeling the [C ii] intensity is illustrated by the predicted signals from Gong et al. (2012) and Silva et al. (2015, “m2” model), shown as blue and yel- low lines, respectively. Recent ALMA observations of several typical star-forming galaxies at 5 ≲ 𝑧 ≲ 8 have tentatively suggested a high [C ii]-to-IR luminosity ratio (Capak et al. 2015; Aravena et al. 2016). The [C ii] luminosity function derived from these observations is similar to that of Gong et al. (2012), implying a high clustering amplitude. In terms of the cumulative number density of𝑧 ∼ 6 galaxies, the Gong et al. (2012) model is also supported by recent observations (Aravena et al.

2016; Hayatsu et al. 2017), which suggest a cumulative number density more than 10 times higher for galaxies with𝐿_[CII] > 2×10⁸𝐿_⊙compared to Silva et al. (2015).