Removing Interlopers From Intensity Mapping Probes Of Primordial Non-Gaussianity

Local shape PNG is produced by super-horizon non-linear evolution of primordial curvature perturbations. In the absence of PNG, on large scales the halo bias $b_{h}$ is assumed to be constant. Local shape PNG, parameterized by $f_{NL}^{\rm loc}$, leads to a scale-dependent correction to the linear halo bias (Dalal et al., 2008; Matarrese & Verde, 2008; Afshordi & Tolley, 2008).

where $\delta_{c}=1.686$ is the threshold of spherical collapse at $z=0$, $T(k)$ is the matter linear transfer function $T(k\rightarrow 0)=1$ and $D(z)$ is the normalized linear growth factor [$D(z=0)=1$].

On large scales this $k^{-2}$ scale-dependent correction has been used to constrain $f_{\rm NL}^{\rm loc}$ from power spectrum detections of LSS biased tracers due to its clean signal (Slosar et al., 2008; Leistedt et al., 2014; Ho et al., 2015). Alternatively, line intensity mapping can provide a more economical survey over larger volume to probe power spectrum on large scales.

Orthogonal shape PNG can be produced by models with higher-derivative interactions and Galilean inflation (Senatore et al., 2010). Parameterized by $f_{NL}^{orth}$, on large scales, orthogonal shape PNG leads to a scale-dependent correction to the linear halo bias (Desjacques et al., 2011)

$\alpha=(n_{s}-4)/6$, $\mathcal{M}_{R_{s}}(k,z)=W_{R_{s}}(k)\mathcal{M}(k,z)$, where $\mathcal{M}(k,z)=\frac{2}{5}\frac{k^{2}T(k)D(z)}{\Omega_{m}H_{0}^{2}}$, and we choose the window function $W_{R_{s}}(k)$ as the Fourier transform of a spherical top-hat filter with smoothing length $R_{s}=(3M/4\pi\bar{\rho})^{1/3}$,

the general spectrum moment is

Here we model the $[\rm CII]$ line intensity power spectrum to predict how well it could constrain PNG. The relation between the mean intensity of the emission line and the luminosity of $[\rm CII]$-luminous galaxies in their host halos (Visbal & Loeb, 2010; Moradinezhad Dizgah & Keating, 2019) at redshift z is expressed as

where $M_{\rm min}$ and $M_{\rm max}$ are the minimum and maximum mass of $[\rm CII]$-emitting halos, $f_{\rm duty}$ is the duty cycle of line emitting halos, (i.e. the fraction of halos in the given mass range which emit $[\rm CII]$ line at redshift z), $dn/dM$ is the halo mass function, $L(M,z)$ is the luminosity of the $[\rm CII]$-luminous galaxies from dark matter halo of mass M at redshift $z$, ${\mathcal{D}}_{L}$ is the luminosity distance. We set $M_{\rm min}=10^{9}M_{\odot}$ and $M_{\rm max}=10^{14.5}M_{\odot}$. We replace $f_{\rm duty}$ with $p_{n,\sigma}$ as given in (Moradinezhad Dizgah & Keating, 2019). We use the Tinker halo mass function (Tinker et al., 2008). $dl/d\theta$ and $dl/d\nu$ convert comoving lengths $l$ to frequency $\nu$ and angular size $\theta$, $dl/d\theta=D_{\rm A,co}(z)$, $\frac{dl}{d\nu}=\frac{c(1+z)}{\nu_{\rm obs}H(z)}$, where $D_{\rm A,co}(z)$ is the comoving angular diameter distance (For a flat universe, $D_{\rm A,co}(z)=\chi(z)$ where $\chi(z)$ is the co-moving distance to redshift $z$.) and $H(z)$ is the Hubble parameter at redshift z. For our fiducial model we use the result of the $m1$ model of Silva et al. (2015) to relate the $[\rm CII]$ luminosity to the average star formation rate $\overline{\rm SFR}(M,z)$ of Behroozi et al. (2013)

$[\rm CII]$ line bias is related to halo bias $b_{h}(M,z)$ as (Moradinezhad Dizgah & Keating, 2019)

We use halo bias $b_{h}(M,z)$ of (Tinker et al., 2010).

Setting $z=3.60$, we get $\langle I_{\rm CII}(3.6)\rangle=0.30\mu K$ from (7) and $b_{\rm line}(3.6)=3.47$ from (9).

The shot noise power spectrum is

Including redshift space distortions, the line clustering power spectrum can be expressed as

The factor of $\left[1+\mu^{2}\beta(k,z)\right]^{2}$ comes from the Kaiser effect (Kaiser, 1987), $\beta(k,z)=f/b_{\rm line}(z)$,where $f$ is the logarithmic derivative of the growth factor, $\mu=k_{\parallel}/k$ is the cosine of the angle between the k and the line of sight direction. The factor of ${\rm exp}\left(-\frac{k^{2}\mu^{2}\sigma_{v}^{2}}{H^{2}(z)}\right)$ is from the finger-of-god effect (Jackson, 1972) with the pairwise velocity dispersion approximated as $\sigma_{v}^{2}=(1+z)^{2}\left[\frac{\sigma_{\rm FOG}^{2}(z)}{2}+c^{2}\sigma_{z}^{2}\right]$, where $\sigma_{\rm FOG}(z)=\sigma_{{\rm FOG},0}\sqrt{1+z}$, $\sigma_{z}=0.001(1+z)$ and $\sigma_{{\rm FOG},0}=250km.s^{-1}$. According to Eqs. (9), (2) and (3) will manifest as a scale-dependent correction to the $b_{\rm line}$ and therefore to the line power spectrum (11).

Line emission of CO interloper line at redshift $z_{i}$ confuses with target $[\rm CII]$ line by residing at the same observed frequency $\nu_{obs}$ as the $[\rm CII]$ line. When we convert the observed frequency interval $\Delta\nu_{\rm obs}$ and angular separation $\Delta\theta_{obs}$ in a data cube to co-moving coordinates to calculate the power spectrum, if we adopt the target line redshift $z_{t}$, we will end up with wrong wavenumbers (Lidz & Taylor, 2016). The relation between the true wavenumbers $k_{\parallel}$ and $\textbf{k}_{\perp}$ and these apparent wavenumbers $\tilde{k}_{\parallel}$ and $\tilde{\textbf{k}}_{\perp}$, calculated by wrongly adopting the $z_{t}$ for coordinate conversion, is

where $\alpha_{\parallel}$ and $\alpha_{\perp}$ are distortion factors caused by this incorrect redshift adoption. Therefore for the multiple interloper line emission case with N interloper lines, the total power spectrum in the data cube is

where $z_{j}$ is the redshift of the $j$th interloper line. In this paper we consider up to 7 CO interloper lines, with $J$ ranging from 4 to 10 in $\rm CO\left(J\rightarrow J-1\right)$ molecule transitioning between rotational states $J$ and $J-1$.

For $\rm CO\left(1\rightarrow 0\right)$, expressed in units of solar luminosity, the luminosity is

Using the fit from (Carilli & Walter, 2013), CO line luminosity $L^{\prime}_{\rm CO}$, which is expressed in units of $\rm Kkms^{-1}pc^{2}$, is related to the far-infrared luminosity $L_{IR}$, which is in the unit of $L_{\odot}$ as

We use the base model of Li et al. (2016), where $L_{\rm IR}$ is related to the $\overline{\rm SFR}$ in units of $M_{\odot}{\rm yr}^{-1}$ via Kennicut relation (Kennicutt, 1998) using the results of Behroozi et al. (2013)

We use $\delta_{\rm MF}=1$ (Behroozi et al., 2013; Li et al., 2016).

Using Eq. (7) we get intensities for $\rm CO\left(1\rightarrow 0\right)$ at different redshifts $z_{j}$ that will interlope with the $[\rm CII]$ target line at $z=3.6$, i.e. $\frac{\rm J\times 115\rm GHz}{z_{j}+1}=\frac{1901\rm GHz}{3.6+1}$. In order to construct a rough estimate of the interloper luminosities, we use the ratio between the luminosity of 7 CO interloper lines with $J$ ranging from 4 to 10 and that for the $\rm CO\left(1\rightarrow 0\right)$ (Visbal & Loeb, 2010) to get the intensities for these 7 interlopers, we also calculate the line biases for these interloper lines according to Eq. (9); these values for interlopers are listed in Table 1.

According to the values in Table 1, we plot spherically averaged $k^{3}P(k)/(2\pi)^{2}$ of the $[\rm CII]$ line at $z=3.6$ and the 7 CO lines that interlope with it in Figure 1.

As in previous work (Lidz & Taylor, 2016; Gong et al., 2012), we plot contours of constant power in the $k_{\perp}-k_{\parallel}$ plane in Figure 3 and 3 to visualize the anisotropy of the sum of the seven interloper power spectrum. Figure 3 characterizes the redshift space distortion in the target $[\rm CII]$ line power spectrum. The contours in Figure 3 illustrate the anisotropy of the power spectrum summation of the interloper CO lines from the coordinate mapping distortion, with the $k_{\parallel}$ strongly elongated.

We use the Fisher matrix methodology to forecast how well a hypothetical $[\rm CII]$ survey will constrain $f_{NL}$. We first consider the simple case with only 1 CO interloper $\rm CO\left(4\rightarrow 3\right)$. We consider a parameter space q. Our forecast is conducted at $z_{t}=3.6$.

We investigate whether the $f_{NL}$ and $[\rm CII]$ target line, which is characterized by $I_{\rm CII}$ and $b_{\rm CII}$ can be forecasted precisely with the interloper contamination which is characterized by $\langle I_{\rm CO\left(4\rightarrow 3\right)}\rangle$ and $b_{\rm CO\left(4\rightarrow 3\right)}$ by calculating the Fisher matrix with component for the $i\rm th$ and $j\rm th$ of q

where the total power spectrum including the $[\rm CII]$ and $\rm CO\left(4\rightarrow 3\right)$ power is given as Eq. (2.3). $V_{\rm eff}$ is the effective volume of the redshift bin at $z_{t}=3.6$, $V_{\rm eff}=\left[\frac{P_{\rm tot}(k,\mu,z_{t})}{P_{\rm tot}(k,\mu,z_{t})+P_{\rm shot}(z_{t})+\tilde{P}_{N}(k,\mu)}\right]^{2}V_{i}$, $P_{\rm shot}(z_{t})$ is given as Eq. (10), $\tilde{P}_{N}(k,\mu)$ is the effective instrumental noise power spectrum given later as Eq. (27), $V_{i}$ is the volume of the redshift bin between $z_{min}$ and $z_{max}$ with a fraction of the sky $f_{sky}$; specifically, $V_{i}=\dfrac{4\pi}{3}f_{sky}[\chi^{3}(z_{max})-\chi^{3}(z_{min})]$, where $\chi(z)$ is the co-moving distance to redshift z.

The bias from interloper contamination in power spectrum detection can be calculated using the formalism constructed in Pullen et al. (2016). The parameter bias is

where F is the Fisher matrix with the parameter space of q. The $j$th component of $\Delta\textbf{D}$ is given as

where $\Delta P_{t}(k,\mu,z_{t})$ is the bias the interlopers contribute to the target power spectrum $P_{t}(k,\mu,z_{t})$, i.e. the second term in Eq. (10).

As in Moradinezhad Dizgah & Keating (2019), we consider an Planck-like telescope having an aperture with dish length of $D_{ant}=1.5$ m. We assume a frequency range of 310-620 GHz, corresponding to a [CII] redshift range of $z=2.06-5.13$, and a frequency resolution $\Delta\nu=0.4$ GHz, aperture temperature $T_{\rm aperture}=40$ K, and approximately $N_{\rm det}=10^{4}$ detectors working for $\tau_{\rm tot}=4\times 10^{4}$ hours. We assume a survey sky coverage fraction $f_{\rm sky}=0.34$, which leads to an angular limit on the sky of $\theta_{max}=\sqrt{\Omega_{\rm surv}}=\sqrt{4\pi f_{\rm sky}}=118^{\circ}$.

We include in our noise model photon noise from the CMB ($T_{\rm B}=2.725$ K), galactic dust ($T_{\rm B}=18$ K) and zodiacal dust ($T_{\rm B}=240$ K) contributions. The emissivity of the dust is $\varepsilon=\varepsilon_{\circ}(\nu/\nu_{\circ})^{\beta}$, where for galactic dust emission $\varepsilon_{\circ}=2\times 10^{-4}$, $\nu_{\circ}=3$ THz, $\beta=2$; while for zodiacal dust emission $\varepsilon_{\circ}=3\times 10^{-7}$, $\nu_{\circ}=2$ THz, $\beta=2$. The radiation emitted at frequency $\nu$ is

The Rayleigh-Jeans Law yields

From Eq. (22) we get that at the observed frequency for the $[\rm CII]$ line, $T_{\rm galactic\>\rm dust}=46\mu K$ and $T_{\rm zodiacal\>\rm dust}=53\mu K$; therefore the systematic temperature is

The instrumental noise for each k-mode is (Moradinezhad Dizgah et al., 2019)

The scale-independent correction to the clustering bias is relatively negligible at small scales, so we set $k_{\rm max}=0.1h\>Mpc^{-1}$.

We consider the largest scales recoverable from foreground contamination. We model this with the parameter $\eta_{\rm min}$, defined as the ratio of the observed frequency divided by the maximum bandwidth over which the frequency dependent response of the instrument is assumed to be smooth, as in Moradinezhad Dizgah & Keating (2019). We model the recoverable largest-scale per-parallel mode from foregrounds as $k_{\parallel,\rm min}=2\pi\eta_{\rm min}[\nu_{obs}dl/dv]^{-1}$. Setting $\eta_{\rm min}=4.5$, which is 3$\times$ the frequency-bandwidth ratio of the fiducial instrument, we have $k_{\parallel,\rm min}=0.0077$ Mpc^-1. The survey area sets the largest-scale perpendicular mode, given by $k_{\bot,\rm min}=2\pi[2\sin(\theta_{\rm max}/2)\frac{dl}{d\theta}]^{-1}=0.0005$ Mpc^-1. The smallest-scale modes are generated by the spatial and spectral resolutions respectively, $k_{\bot,max}\approx 2\pi[\dfrac{c/\nu_{obs}}{D_{ant}}\frac{dl}{d\theta}]^{-1}$, $k_{\parallel,\rm max}\approx 2\pi[\delta\nu\frac{dl}{dv}]^{-1}$.

We introduce the attenuation of the signal due to foregrounds as in Moradinezhad Dizgah & Keating (2019)

and due to finite spectral and angular resolution

writing the effective instrumental noise as

In this section we forecast the local shape $f_{NL}$ measured from only the auto-power spectrum. The Fisher analysis results for this section are listed in Table 2.

In the first scenario we assume there is no interloper contamination. The parameter space includes four parameters $\textbf{q}=\{f_{NL}^{loc},\langle I_{\rm CII}\rangle,b_{\rm CII},P^{\rm CII}_{\rm shot}\}$. We set the fiducial value of the parameters as $f_{NL}^{loc}=0$, $\langle I_{\rm CII}\rangle=0.30\mu K$, $b_{\rm CII}=3.47$, as what was calculated before, $P^{\rm CII}_{\rm shot}=13.57\mu K^{2}(\rm Mpc/h)^{3}$. We find $\sigma(f_{NL}^{loc})=1.32$, which being of order $\mathcal{O}(1)$ is the target level of uncertainty needed to discriminate between inflation models. This result deviates from the one from Moradinezhad Dizgah & Keating (2019) because we adopted different fiducial survey parameters, and therefore yield a smaller instrumental noise. We plot the $1-\sigma$ errors in different parameter planes shown as the yellow ellipses in Figure (4).

In the second scenario we include a contamination from one interloper $\rm CO\left(4\rightarrow 3\right)$ at $z_{j}=0.113$ according to the second term in Eq. (2.3), the anisotropic contribution to the [CII] power spectrum due to $\rm CO\left(4\rightarrow 3\right)$. The parameter space now includes seven parameters $\textbf{q}=\{f_{NL}^{loc},\langle I_{\rm CII}\rangle,b_{\rm CII},\langle I_{\rm CO\left(4\rightarrow 3\right)}(z_{j})\rangle,b_{\rm CO\left(4\rightarrow 3\right)}(z_{j}),P^{\rm CII}_{\rm shot},$ $P^{\rm CO\left(4\rightarrow 3\right)}_{\rm shot}(z_{j})\}$. We set the fiducial values of the interloper parameters as $\langle I_{\rm CO\left(4\rightarrow 3\right)\rangle}(z_{j})\rangle=0.07\mu K$, $b_{\rm CO\left(4\rightarrow 3\right)}(z_{j})=0.6$ as listed in Table 1, $P^{\rm CO\left(4\rightarrow 3\right)}_{\rm shot}(z_{j})=1.35\mu K^{2}(\rm Mpc/h)^{3}$, and the remaining parameters are set as in the first scenario. We compute the marginalized $\sigma(f_{NL}^{loc})=1.64$, a 24% increase due to interloper contamination. We plot the $1-\sigma$ errors in different parameter planes shown as yellow ellipses in Figure (5). Alternatively, if we consider the interloper contamination as pure noise other than mixing signals for the target $\rm CII$ line, the parameter space becomes $\textbf{q}=\{f_{NL}^{loc},\langle I_{\rm CII}\rangle,b_{\rm CII},P^{\rm CII}_{\rm shot}\}$ and we find $\sigma(f_{NL}^{loc})=1.60$. Note that treating the CO(4-3) line as noise gives you slightly less error in $f_{NL}^{loc}$ because this model has less degrees of freedom. The uncertainty in $f_{NL}^{loc}$ becomes much larger when multiple interloper lines are introduced. If we include interloper contamination from all the 7 CO interlopers whose fiducial intensities and biases are given in Table 1 and marginalize over all these fiducial quantities, we get $\sigma(f_{NL}^{loc})=4.04$.

Considering the case where one interloper $P_{\rm CO\left(4\rightarrow 3\right)}$ biases the measured power spectrum, we get the parameter bias $\Delta\textbf{p}^{T}=[\Delta f_{NL}^{loc},\Delta b_{\rm CII},\Delta\langle I_{\rm CII}\rangle,\Delta P_{\rm shot}^{\rm CII}]=[4.54,-5.45,0.49,6.87]$. This would suggest one interloper can masquerade as a 2.8$\sigma$ detection of $f_{NL}$ while providing a false positive for non-standard inflation models. This result shows that the presence of interlopers are prohibitive when attempting to measure $f_{NL}^{loc}$ using an auto-power spectrum.