A GUIDE TO DESIGNING FUTURE GROUND-BASED COSMIC MICROWAVE BACKGROUND EXPERIMENTS

The exact configuration of CMB-S4 is still being studied. A few examples were given in the Snowmass CF5 Neutrinos report (Abazajian et al. 2013a). Here we extend the experiment input grid (sensitivity, beam size, sky coverage f_sky) to understand how constraints for each parameter vary across the experiment design space.

The experimental sensitivity in μK arcmin is derived from the combination of the total number of detectors and the observed fraction of sky. Table 1 presents the sensitivity of an experiment given a total number of detectors N_det and a sky coverage f_sky. We consider N_det ranging from 10⁴ to 10⁶ and f_sky from 1% to 75%. We assume a noise-equivalent temperature (NET) of 350 μK $\sqrt{s}$ per detector (achieved by current generation experiments) and a yield (Y) of 25% that combines both the focal plane yield and the operation efficiency over an observing period of five years (ΔT). To get from NET per detector to the sensitivity s of an experiment, we use

$$\begin{eqnarray} &&s\ [\mu {\rm K\, arcmin}] \equiv \frac{ {\rm NET}\ [\mu {\rm K\,}\sqrt{s}] \times \sqrt{f_{{\rm sky}} \ [{\rm arcmin}^2] }}{ \sqrt{N_{\rm det} \times Y \times \Delta T\ [{\rm s }]}}. \end{eqnarray} \tag{ 1 }$$

For each combination of N_det and f_sky, we cover the following beam sizes: 1', 2', 3', and 4'—and in some cases up to 6' and 8'.

The Fisher matrix formalism is a simple way to forecast how well future CMB experiments can constrain parameters in the cosmological parameter space of interest. By definition, the Fisher matrix assumes a Gaussian likelihood and may not represent the true likelihood distribution in some cases (Wolz et al. 2012). This approach has been scrutinized and other methods for forecast have been proposed (Perotto et al. 2006; Wolz et al. 2012). However, for a large grid of experimental inputs, it is computationally efficient to arrive at constraints through Fisher information. It is also useful for comparison with previous work, e.g., Albrecht et al. (2006) and Font-Ribera et al. (2013), and within the grid.

Assuming the likelihood function $\mathcal {L}$ of the parameters $\boldsymbol{\theta }$ given data $\boldsymbol {d}$ to be Gaussian, it can be written as

$$\begin{equation} \mathcal {L}(\boldsymbol{\theta }| \boldsymbol{d}) \propto \frac{1}{\sqrt{| \mathbf {C}(\boldsymbol{\theta })|}} \exp \left(-\frac{1}{2} \boldsymbol{d^{\dagger }}\left[ \mathbf {C}(\boldsymbol{\theta })\right]^{-1} \boldsymbol{d}\right), \end{equation} \tag{ 2 }$$

where C is the covariance matrix of the modeled data. In our case, $\boldsymbol{d}= \lbrace a_{\ell m}^T, a_{\ell m}^E, a_{\ell m}^d \rbrace$ for temperature, E-mode polarization, and lensing-induced deflection. The vector $\boldsymbol{\theta }$ contains the model parameters of the cosmology described in Section 2.4.

The Fisher matrix is constructed from the curvature of the likelihood function at the fiducial values of the parameters:

$$\begin{equation} F_{ij} \equiv - \left\langle \frac{\partial ^2 \log \mathcal {L}}{\partial \theta _i \partial \theta _j} \left |_{\boldsymbol{\theta } = \boldsymbol{\theta _0}}\right\rangle,\right. \end{equation} \tag{ 3 }$$

where $\boldsymbol{\theta _0}$ is a vector of the model parameters evaluated at their fiducial values, which are chosen to maximize the likelihood.

From Equations (2) and (3), we can write the Fisher components F_ij for CMB spectra as

$$\begin{equation} F_{ij} = \sum _\ell \frac{2\ell +1}{2} f_{{\rm sky}} {\rm Tr} \left(\boldsymbol{C}^{-1}_\ell (\theta) \frac{\partial \boldsymbol{C}_\ell }{\partial \theta _i} \boldsymbol{C}^{-1}_\ell (\theta) \frac{\partial \boldsymbol{C}_\ell }{\partial \theta _j} \right)\protect, \protect \end{equation} \tag{ 4 }$$

where ℓ is the multipole of the angular power spectra and

$$\begin{eqnarray} \mathbf {C}_\ell \equiv \left(\begin{array}{@{}ccc@{}}C_\ell ^{TT} + N_\ell ^{TT} & C_\ell ^{TE} & C_\ell ^{Td} \\ C_\ell ^{TE} & C_\ell ^{EE} + N_\ell ^{EE} & 0 \\ C_\ell ^{Td} & 0 & C_\ell ^{dd} + N_\ell ^{dd}\end{array}\right). \end{eqnarray} \tag{ 5 }$$

Similar to Smith et al. (2009), we do not consider a B-mode power spectrum signal in this covariance matrix since we assume that the fields are unlensed and that there are no primordial B modes. We set $C_{\ell }^{Ed}$ to 0 because its effect is small (<1% improvement on constraints in M_ν for most cases). The $C_{\ell }^{dd}$ is related to the lensing power spectrum $C_{\ell }^{\phi \phi }$ by $C_{\ell }^{dd} = \ell (\ell +1)C_{\ell }^{\phi \phi }$.

The 1σ uncertainties of each parameter are obtained by marginalizing over the other parameters. For a parameter θ_i, the marginalized error is given by

$$\begin{equation} \sigma _i \equiv \sigma (\theta _i) = \sqrt{(\mathbf { F^{-1}})_{ii}}. \end{equation} \tag{ 6 }$$

The power spectra include Gaussian noise $N^{XX^{\prime }}_\ell$ defined as

$$\begin{equation} N^{ XX^{\prime } }_\ell = s^{\, 2} \exp \left(\ell (\ell +1) \frac{\theta ^{\ 2}_{\scriptsize {\rm FWHM}}}{8\log 2}\right)\protect, \protect \end{equation} \tag{ 7 }$$

where s is the total intensity instrumental noise in μK rad, defined in Equation (1), and $\theta ^{\ 2}_{\scriptsize {\rm FWHM}}$ is the full width at half maximum beam size in radians. Note that $s \rightarrow s\times \sqrt{2}$ in the case of XX' = {EE, BB}.

We calculate $C_{\ell }^{dd}$ using the iterative method outlined in Smith et al. (2012), based on Hirata & Seljak (2003) and Okamoto & Hu (2003). We estimate $N^{dd}_\ell$ from lensing reconstruction using $C^{EE}_\ell$ and $C^{BB}_\ell$ only. Figure 1 shows the $N_{\ell }^{dd}$ level for three N_det at f_sky = 0.75 and 1' beam size. We use the EB estimator because it has the lowest noise for the range of sensitivities relevant for CMB-S4 (10⁵ ≲ N_det ≲ 10⁶). For reference, the range of the $\ell (\ell +1) N^{dd}_\ell /2\pi$ noise floor for 10⁵ < N_det < 10⁶ from the EB estimator is between 10⁻⁸ and 10⁻¹⁰, while Planck's $\ell (\ell +1) N^{dd}_\ell /2\pi$ noise floor is ∼2 × 10⁻⁷ from the 143 GHz and 217 GHz channels (Planck Collaboration et al. 2013b). We note that for a couple of cases with $\mathcal {O} \left(10^4 \right)$ detectors, the TT estimator gives lower reconstruction noise for sensitivity s ≳ 4 μK arcmin and the beam sizes we consider in this paper. To demonstrate the effect of using the TT estimator instead of the EB estimator, the constraint for M_ν, which relies most heavily on lensing information, improves from 49 meV to 47 meV for the case with 10⁴ detectors, 1' beam size, and f_sky = 0.75. One can also combine the estimators from all reconstruction channels to get a minimum variance $N^{dd}_\ell$ to further improve the constraints if the systematics and foregrounds from the other channels are well understood. The iterative delensing method improves on the quadratic method once the instrumental noise is below the lensing B-mode level (s ∼ 3.5μK arcmin, i.e., polarization noise ∼5 μK arcmin; Seljak & Hirata 2004).

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For parameter constraints that could benefit from low-redshift information, we use the BAO signal forecasted from the DESI and put 1% priors on H₀, which is achievable in the next decade.

DESI will measure the spectra of approximately 24 million objects in the redshift range 0 < z < 1.9 (we do not include the Lyα forest at z > 1.9 for simplicity). We limit ourselves to use only the DESI BAO signal, because it is the most systematically robust and time-tested measurement—constraints will be limited by statistical errors even at the high level of precision of DESI (Levi et al. 2013).

While constraints from the DESI broadband power spectrum and redshift space distortion measurements will be competitive for many of the models discussed in this work, we do not include them because modeling uncertainties related to galaxy bias make projections for them less certain and we want to focus on the effect of CMB lensing. This way, we will have strong independent measurements from both CMB lensing and optical surveys for these parameters in the coming decade.

Because the BAO signal is completely independent from CMB measurements including the CMB lensing signal, we add the two Fisher matrices,

$$\begin{equation} F_{{\rm total}} = F_{\rm CMB} + F_{\rm BAO}, \end{equation} \tag{ 8 }$$

when BAO is included in the forecast. The independence holds even when the observing patch overlaps, as one cannot measure the BAO signal from CMB lensing maps. Details of DESI modeling can be found in Font-Ribera et al. (2013). On the practical level, when we use the matrices from that paper, we transformed them to the parameter space of this paper.

To add a 1% H₀ prior to the Fisher matrix, we set

$$\begin{equation} F_{H_0 H_0} \rightarrow F_{H_0 H_0} + \frac{1}{(1\% \times H_{0,{\rm fid}})^2}, \end{equation} \tag{ 9 }$$

where $F_{H_0 H_0}$ is the diagonal entry for H₀ in the Fisher matrix, and H_{0, fid} is the fiducial value of H₀.

The fiducial cosmology is a flat νΛCDM universe, with parameter values from Table 2 of the Planck best-fit value (Planck Collaboration et al. 2013c), i.e., Ω_ch² = 0.12029, Ω_bh² = 0.022068, A_s = 2.215 × 10⁻⁹ at k₀ = 0.05 Mpc⁻¹, n_s = 0.9624, τ = 0.0925, and H₀ = 67.11 km s⁻¹ Mpc⁻¹. The fiducial Ω_νh² is set at 0.0009, which corresponds to M_ν ≃ 85 meV.

Unless otherwise stated, for all extensions of the minimal model, we vary all of the above parameters (including neutrino mass) and marginalize over them to obtain the parameter constraints. The parameters investigated in this work are:

• 1.  
  M_ν and N_eff, see Section 3;
• 2.  
  w₀ and w_a, see Section 4;
• 3.  
  p_ann in Section 5;
• 4.  
  Ω_K, n_s, α_s, r, and n_t, see Section 6.

The ℓ_max used throughout this paper is 5000 in all the auto- and cross-spectra. Note that high-ℓ polarized radio sources are expected to contribute to the polarization power beyond ℓ ∼ 5000 (Smith et al. 2009). Upper limits on the mean-squared polarization fraction of radio sources and dusty galaxies have been placed by previous studies (Battye et al. 2011; Seiffert et al. 2007). Given knowledge of their millimeter-wave flux density distribution (Vieira et al. 2010; Marriage et al. 2011; Planck Collaboration et al. 2011), we can identify and mask the extragalactic polarized sources with a small data loss. For polarized galactic foregrounds, we can avoid them by scanning patches that are outside of the Milky Way.

Foregrounds in the temperature power spectrum, however, start to dominate over the signal around ℓ = 3000 (Reichardt et al. 2012; Dunkley et al. 2013; Crawford et al. 2014; Planck Collaboration et al. 2013d). The different sources of high-ℓ foregrounds in the temperature are thermal and kinetic Sunyaev–Zel'dovich effects, radio galaxies, and cosmic infrared background. Progress is being made toward an understanding of these foregrounds, as many studies are currently underway (Reichardt et al. 2012; Dunkley et al. 2013; Crawford et al. 2014; Planck Collaboration et al. 2013d). For parameters that rely heavily on high-ℓ information, we make a conservative estimate on parameter constraints by imposing an ℓ = 3000 cut in the temperature spectrum.

The standard hot big bang model predicts a relic sea of neutrinos—the cosmic neutrino background, whose density is the second highest of all species. Neutrinos decoupled from the primordial plasma at a temperature of T ∼ 1 MeV (Lesgourgues & Pastor 2006) but maintained temperature equilibrium with photons as the universe expanded. Once the temperature of the photons dropped below the mass of the electrons, electron–positron annihilation transferred heat to the photons. Using entropy conservation, and assuming neutrinos had completely decoupled from the plasma by that time, we can relate the temperatures of neutrinos and photons by

$$\begin{eqnarray} &&T_{\nu } = \left(\frac{4}{11} \right)^{1/3} T_{\gamma }\protect, \protect \end{eqnarray} \tag{ 10 }$$

where the factor of 4/11 comes from the effective degrees of freedom of positrons, electrons, and photons before and after electron–positron annihilation is complete (Steigman 2002).

To account for the radiation-like behavior that neutrinos have in the primordial plasma, it is conventional to parameterize the relativistic energy density ρ_R as a function of photon energy density ρ_γ and N_eff, the effective number of relativistic species, as

$$\begin{equation} \rho _R = \left(1 + N_{{\rm eff}} \frac{7}{8} \left(\frac{4}{11} \right)^{4/3} \right)\rho _{\gamma }, \end{equation} \tag{ 11 }$$

where the factor of 7/8 accounts for the fermionic degrees of freedom of neutrinos. Given three neutrinos in the Standard Model of particle physics, one would naively expect N_eff = 3. However, QED corrections in the primordial plasma and spectral distortion due to electron–positron annihilation raise the value of N_eff to 3.046 (Mangano et al. 2005). Because of the generic parameterization, N_eff encapsulates radiation-like behavior from any other relativistic species as well. For example, sterile neutrinos, a candidate for nonstandard radiation content, could change N_eff on the basis of their masses and mixing angles with active neutrinos (Fuller et al. 2011). Therefore, tight constraints on N_eff from the CMB and LSS, along with big bang nucleosynthesis (BBN), would be essential to either confirm the Standard Cosmological Model or discover new physics relating to extra relativistic species.

Total neutrino mass, M_ν, contributes to the critical density of the universe as (Mangano et al. 2005)

$$\begin{eqnarray} &&\Omega _{\nu }h^2 \simeq \frac{M_{\nu }}{94\, {\rm eV}}. \end{eqnarray} \tag{ 12 }$$

Main Effects of M_ν and N_eff on CMB Spectra:

• 1.  
  Suppression on small angular scale lensing power spectrum for nonzero M_ν. Neutrinos' large thermal velocities allow them to free stream on scales smaller than ∼(T_ν/m_ν) × (1/H) (Abazajian et al. 2011), where m_ν is the mass of an individual neutrino and H is the Hubble expansion rate, instead of falling into gravitational wells. As a result, structure formation below this scale is suppressed (Kaplinghat et al. 2003). Figure 2 shows the effect of neutrino free streaming on the lensing potential power spectrum $C_\ell ^{\phi \phi }$ for various total neutrino masses.
• 2.  
  Increasing N_eff increases Silk damping. The main effects of N_eff on the CMB temperature and E-mode polarization power spectra is increased Silk damping at small scales (Hou et al. 2013).

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The current constraint on N_eff is $3.30^{+0.54}_{-0.51}$ (95% C.L.) from Planck+WMAP polarization+small-scale CMB+BAO data, and the current limit on total neutrino mass is M_ν < 0.23 eV (95% C.L.) from Planck+BAO data (Planck Collaboration et al. 2013c). From solar and atmospheric neutrino oscillation experiments, we know the mass differences between each mass eigenstate satisfy

$$\begin{eqnarray} m_2^2 - m_1^2 &=& 7.54^{+0.26}_{-0.22} \times 10^{-5}\ {\rm eV}^2 \nonumber\\ m_3^2 - \frac{m_1^2+m_2^2}{2} &=& \pm 2.43^{+0.06}_{-0.10} \times 10^{-3}\ {\rm eV}^2. \end{eqnarray} \tag{ 13 }$$

From these equalities, the lower bound for the sum of neutrino masses is ∼58 meV for the normal hierarchy, where the smallest mass is set to zero, and ≳ 100 meV for the inverted hierarchy, in which case m₃ is set to have the smallest mass (Lesgourgues & Pastor 2006).

The next qualitatively interesting result for neutrino mass through cosmological probes will be to detect the total neutrino mass with significance and potentially distinguish normal from inverted hierarchies if the smallest mass is <100 meV. In addition, a precise measurement of N_eff would determine whether the three active neutrinos scenario accurately describes the thermal content of the early universe. An order of magnitude improvement from current constraints is likely to be needed to achieve this goal.

Experiments like Katrin¹¹ directly probe the effective electron neutrino mass. Coupled with neutrino mixing angle measurements like Noνa,¹² T2K,¹³ or Double Chooz,¹⁴ one can pin down the masses of each neutrino species. Agreement will confirm our understanding of cosmology and particle physics, and disagreement will initiate search for new physics.

3.2.1. N_eff Forecast

Table 2 presents a sample of N_eff constraints given different beam sizes, sky coverage, and detector counts in an experiment. The best constraint we get in this grid is σ(N_eff) = 0.016 from the 10⁶ detectors, 1' beam, f_sky = 0.75 case (equivalent sensitivity is 0.58 μK arcmin), which distinguishes 3.046 from 3 at the 3σ level. We observe that increasing N_det from 10⁴ to 10⁵ improves constraints in N_eff by about 35%, while increasing N_det from 10⁵ to 10⁶ improves constraints in N_eff by about 30%. Increasing f_sky also improves the constraints even though the sensitivity of the experiment decreases. The constraints improve by about 10% for each arcminute decrease in beam size. Out of all the parameters studied in this work, N_eff improves most with decreasing beam size. This is partly due to the high constraining power on N_eff from high ℓ multipoles in the $C_\ell ^{TT}$ and $C_\ell ^{EE}$ spectra, which are sensitive to effects from Silk damping. Figure 3 shows the trend of N_eff constraints for increasing f_sky, N_det, and beam sizes.

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We know that for multipoles ℓ > 3000, the temperature power spectrum is also contaminated by foregrounds. For this reason, we run the forecast up to ℓ = 3000 in $C_\ell ^{TT}$, while keeping the other spectra the same. In this case, we get the N_eff 1σ constraint at 0.021 with 10⁶ detectors, f_sky = 0.75, and 1' beam size. The set of experiments that could constrain N_eff to better than 0.025 shrinks to {N_det, beam, f_sky} = {10⁶, 0.75, 1'}, {5 × 10⁵, 0.75, 1'}, {2 × 10⁵, 0.75, 1'}, {10⁶, 0.5, 1'}, {10⁶, 0.75, 2'}, and {5 × 10⁵, 0.75, 2'}.

Relationship with relic helium abundance Y_P. In our analysis, we set the relic helium abundance Y_P = 0.248, assuming that its value is measured externally. However, if we assume $N_{\rm eff}^{\rm BBN}$ is the same as $N_{\rm eff}^{\rm CMB}$, we can improve the constraints of N_eff at recombination by varying the value of Y_P self-consistently for each $N_{\rm eff}^{\rm CMB}$ (N_ν in the expression below) and Ω_bh² using the relation (Simha & Steigman 2008)

$$\begin{equation} Y_P = 0.2486 + 0.0016 [(\eta _{10} - 6)+ 100(S-1)]\protect, \protect \end{equation} \tag{ 14 }$$

where η₁₀ = 273.9 Ω_Bh², S = (1 + 7ΔN_ν/43)^1/2, and ΔN_ν = N_ν − 3. With the imposed relation, we improve the constraints over the entire parameter space by 15%–20%. In particular, the best constraint for N_eff is 0.013, in the range of experimental configurations we consider in this work.

On the other hand, if $N_{\rm eff}^{\rm BBN}$ is different from $N_{\rm eff}^{\rm CMB}$ because of new physics, we can use CMB to constrain N_eff and Y_P independently. In this case, the constraints on N_eff degrade from those listed in Table 2 by about a factor of two, and the constraints on Y_P is on the order of 10⁻³.

3.2.2. M_ν Forecast

Table 3 presents a sample of M_ν constraints given different beam sizes, sky coverage, and detector counts in an experiment. The best constraint we get for CMB only is M_ν = 34.1 meV from the 10⁶ detector (0.58 μK arcmin equivalent), f_sky = 0.75, 1' beam case. This constraint improves to 15.1 meV when BAO is added. The top row of Figure 4 shows the trend of how neutrino constraints vary as a function of detector count, sky fraction, and beam size.

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Table 3. 1σ Constraints on M_ν, in units of meV, from CMB and from CMB+BAO

	CMB				CMB+BAO
	1'	2'	3'	4'	1'	2'	3'	4'
104 Detectors
fsky = 0.25	71.7	72.8	74.4	76.6	22.8	23.0	23.4	23.9
fsky = 0.50	54.7	55.7	57.2	59.2	20.6	20.9	21.3	21.9
fsky = 0.75	48.1	49.0	50.5	52.5	20.1	20.4	20.9	21.5
105 detectors
fsky = 0.25	63.3	65.2	66.7	68.3	19.7	19.8	19.9	20.1
fsky = 0.50	46.4	47.2	48.2	49.4	16.9	17.0	17.1	17.2
fsky = 0.75	38.5	39.2	40.0	41.0	15.7	15.8	15.9	16.0
106 detectors
fsky = 0.25	54.9	58.1	62.2	64.7	19.1	19.2	19.3	19.4
fsky = 0.50	40.8	42.7	45.1	46.5	16.4	16.4	16.5	16.6
fsky = 0.75	34.1	35.7	37.2	38.3	15.1	15.2	15.3	15.3

Note. "CMB" includes lensing.

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For the levels of sensitivity achieved with N_det ⩾ 10⁵, constraints on M_ν are sample variance limited as opposed to sensitivity limited. We see from Figure 4 that as we increase detector counts, the constraints on M_ν plateau. For CMB alone, increasing N_det from 10⁴ to 10⁵ improves constraints for f_sky = 0.5 cases by about 15%, while increasing N_det from 10⁵ to 10⁶ improves constraints for the same cases by 6%–12% (smaller beam sizes give more improvements). In contrast, with CMB and BAO increasing N_det from 10⁴ to 10⁵ improves constraints for f_sky = 0.5 cases by about 20%, while increasing N_det from 10⁵ to 10⁶ improves constraints by only 3%. This shows that increasing N_det beyond 10⁵ does not improve the constraints on M_ν much once BAO is included. Decreasing beam sizes help the CMB-only cases more when N_det is high, e.g., in the 10⁶ N_det case, the improvement is almost 10% for each decreased arcminute. However, this effect is washed away when BAO is added. For the lower N_det regime, the improvement over decreasing beam size is a few percent.

CMB lensing significantly improves the constraints on M_ν obtained using primary spectra alone. Figure 5 compares the constraints on M_ν with and without using CMB lensing from an experiment with 10⁵ detectors and 4' beam. This is because the lensing spectrum breaks the degeneracy that neutrino mass has with Ω_ch² in the primary spectra. Adding BAO information further breaks degeneracies as seen in the right column of Table 3. The bottom row of Figure 4 shows that a wide range of experimental configurations can lead to tight neutrino mass constraints. Specifically, constraints go below 25 meV for f_sky ⩾ 0.125 and for any number of detectors and beam sizes. These constraints are similar to those estimated by Font-Ribera et al. (2013) for Planck CMB+redshift-space distortions+DESI BAO and/or lensing measurements based on galaxies, e.g., DES/LSST.

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The observed acceleration in the expansion of the universe is currently explained phenomenologically by dark energy. In this section, we focus on the constraints for a homogeneous dark energy with the equation of state parameterized as

$$\begin{eqnarray} &&w(a) = w_0 + (1-a)\,w_a, \end{eqnarray} \tag{ 15 }$$

allowing for a time-varying dark energy field. In this parameterization, a cosmological constant would be the same as dark energy in its functional form when the equation of state w(a) is constant and set to −1, i.e., w₀ = −1 and w_a = 0. Two questions ensue: (1) how well could future experiments differentiate a cosmological constant from dark energy? and (2) how does dark energy evolve, if it does? This later question is the first step to capturing the much broader class of possible dark energy models. Once we precisely measure the equation of state, we will have a much better sense of directions in the theoretical modeling.

The effects of dark energy on cosmological observations are encapsulated in the Hubble parameter $H \equiv \dot{a}/{a}$, where a = a(t) is the scale factor. The Friedmann equation relates the expansion rate of the universe with the energy densities of the contents in it:

$$\begin{eqnarray} H(a)^2 &=& H_0^2 \left(\Omega _r a^{-4} + \Omega _M a^{-3} + \Omega _K a^{-2}\right. \nonumber\\ &&\left. + \Omega _{{\rm DE}} \exp \left(3 \int _a^1 \frac{da^{\prime } }{a^{\prime }} [1+w(a^{\prime })] \right)\right), \end{eqnarray} \tag{ 16 }$$

where Ω_r, Ω_M, Ω_DE, and Ω_K are the density of radiation, matter, dark energy, and curvature, respectively, and w(a) is the specific parameterization of the equation of state for dark energy.

The Hubble expansion rate H also governs the growth of linear perturbations, δ, in structure formation:

$$\begin{equation} \ddot{\delta } + 2H \dot{\delta } - 4 \pi G \rho _M \delta = 0. \end{equation} \tag{ 17 }$$

Primordial CMB fluctuations constrain dark energy parameters through the effects H have on them at recombination, i.e., z ≃ 1100—acoustic peaks positions are sensitive to the value of w (Huterer & Turner 2001). If w < −1, both $C_\ell ^{TT}$ and $C_\ell ^{EE}$ would shift toward higher ℓ, and if w > −1 they would shift toward lower ℓ. The w_a parameter gives the same effects as w₀ but at a smaller amplitude. One gets the same qualitative feature when H₀ (value of H(z) at z = 0) is decreased or increased, respectively. Therefore, when we only use information from the CMB (z ≃ 1100), dark energy equation of state parameters are degenerate with geometrical parameters, like H₀ and Ω_K.

To break this degeneracy, low redshift probes like weak lensing (CMB and optical), BAO, and H₀ measurements are essential. Here we focus on CMB lensing in relation to dark energy. CMB lensing is sensitive to structures at various redshifts (the lenses) and the CMB (the source), see Hanson et al. (2010). Because they are both a function of H(z), we can observe the effects of dark energy through them. The effect of the equation of state w on the lensing power spectrum is an overall enhancement or suppression in power across all scales with a minor scale dependence. Because CMB lensing probes redshifts higher than optical surveys, it is particularly useful for studying structure formation beyond z > 2—a direct observatory of the early history of dark energy. Specifically, if effects of a nonstandard dark energy manifest at high z (e.g., z ∼ 5), then structure formation could be significantly suppressed, which CMB lensing can uniquely constrain.

As discussed in the previous section, to constrain the dark energy equation of state, it is essential to have observations from various redshifts and multiple probes. In this section, we present the constraints on w₀, w_a, and the figure of merit (FoM) as defined by the Dark Energy Task Force (DETF; Albrecht et al. 2006), from the CMB, with and without H₀ prior and with and without BAO measurements from DESI.

In addition to constraints on w₀ and w_a, and the DETF FoM, we quote a constraint on w_p at a_p, where a_p is the value of the scale factor a when the uncertainties of w(a) are minimized (Albrecht et al. 2006). To be explicit, FoM = 1/(σ(w_a)σ(w_p)). Table 10 in the Appendix lists the constraints from CMB, CMB+BAO, and CMB+BAO+1% H₀ prior. CMB includes CMB lensing. The best FoM is 303. As defined earlier, this Fisher matrix parameter space includes massive neutrinos. For reference to the original DETF FoM, we include the constraints with fixed sum of neutrino mass in Table 11 in the Appendix. In this scenario, the best FoM is 576.

In contrast to the rest of the models investigated in this work, the DETF FoM is sensitivity limited. For example, for CMB+BAO+H₀ prior 1' beam cases, increasing N_det from 10⁴ to 10⁵ improves the FoM by 10%–19%, while increasing N_det from 10⁵ to 10⁶ improves the FoM by 21%–29% (bigger f_sky gives more improvement). For CMB alone and a 1' beam, increasing N_det from 10⁴ to 10⁵ and from 10⁵ to 10⁶ improves the FoM by more than double, respectively. Improvements from decreasing beam sizes become more noticeable at high N_det, and the relative improvement per arcminute decreases as the beam size decreases. The improvement ranges from a few percent to tens of percent. While f_sky does not drive the improvement of the FoM as much, the FoM does improve for increasing f_sky for all cases in the table though the sensitivity of the experiment decreases.

Figure 6 shows the trend of how the FoM improves with increasing detector counts and sky fraction. We note that BAO and H₀ prior improve the FoM from using CMB alone from a factor of several to an order of magnitude. Also, with BAO and H₀ prior, increasing the detector count does not improve the FoM much at f_sky smaller than 0.2.

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Figure 7 illustrates how much a CMB-S4 experiment improves on the constraints on w₀ and w_a over Planck including DESI BAO and marginalizing over neutrino masses. The FoM from Planck (no lensing)+DESI BAO is 22 and from CMB-S4+DESI BAO is 141. In this figure, the assumed CMB-S4 configuration has 500,000 detectors, 3' beam, and covers half of the sky. The improvement comes mainly from the tight constraints one gets for neutrino masses from CMB-S4 and DESI BAO.

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To further constraint dark energy parameters, adding extra probes like SNe and optical weak lensing, as suggested by the DETF report, will certainly help.

If DM is a weakly interacting massive particle and it is a thermal relic, then its self-annihilation cross section can be determined by its relic density today through the Boltzmann equation (Scherrer & Turner 1986; Gondolo & Gelmini 1991; Steigman et al. 2012).

Depending on the model, DM particles can annihilate into gauge bosons, charged leptons, neutrinos, hadrons, or more exotic states. These annihilation products then decay or interact with the photon–baryon fluid and produce electrons, positrons, protons, photons, and neutrinos. Neutrinos do not further interact with the photon–baryon fluid, but they interact gravitationally and their effects can be observed through the lensing of the CMB. A proton's energy deposition to the fluid is inefficient because of their high penetration length. Hence, the main channels for energy injection are through electrons, positrons, and photons. At high energies, positrons lose energy through the same mechanisms as electrons. High energy electrons lose energy through inverse Compton scattering of the CMB photons, while low energy electrons lose energy through collisional heating, excitation, and ionization. Photons lose energy through photoionization, Compton scattering, pair-production off nuclei and atoms, photon–photon scattering, and pair-production through CMB photons (Zdziarski & Svensson 1989). The rate of energy release per unit volume by a self-annihilating DM particle is given by (Galli et al. 2009)

$$\begin{equation} \frac{dE}{dt}(z) = \rho ^2_c c^2 \Omega ^2_{{\rm DM}} (1+z)^6 f \frac{\langle \sigma v \rangle }{m_{{\rm DM}}}, \end{equation} \tag{ 18 }$$

where ρ_c is the critical density of the universe today, Ω_DM is the DM density, f is the energy deposition efficiency factor, 〈σv〉 is the velocity-weighted annihilation cross section, and m_DM is the mass of the DM particle, assumed to be a Majorana particle in this work.

Because of these processes, the photon–baryon plasma is heated and the ionization fraction is modified. This leads to modifications of the recombination history (Padmanabhan & Finkbeiner 2005; Galli et al. 2009) and, consequently, of the CMB spectra. For details on the energy injection processes, see Slatyer et al. (2009). The energy injection due to DM annihilation broadens the surface of last scattering but does not slow recombination (Padmanabhan & Finkbeiner 2005). The extra scattering of photons at redshift z ≲ 1000 damps power in the CMB temperature and polarization fluctuations at small angular scales (ℓ ≳ 100) and adds power in the E-mode polarization signal at large scales (ℓ ≲ 100). The "screening" effect at ℓ ≳ 100 goes as an exponential suppression factor C_ℓ → e^−2ΔτC_ℓ, where Δτ is the excess optical depth due to DM annihilation. This exponential factor is partially degenerate with the amplitude of the scalar perturbation power spectrum A_s, and polarization data help to break this partial degeneracy.

The CMB is sensitive to the parameter p_ann, defined as

$$\begin{equation} p_{{\rm ann}} \equiv f \frac{\langle \sigma v \rangle }{m_{{\rm DM}}}. \end{equation} \tag{ 19 }$$

In this work, we take a thermal s-wave annihilation cross section 〈σv〉 = 3 × 10⁻²⁶ cm³ s⁻¹ and f = 1, and we will present the 95% C.L. upper limit of the DM particle mass, m_DM, that a CMB Stage IV experiment could reach. We note that 〈σv〉 could vary for different ranges of m_DM (Steigman et al. 2012) and that the value of f is a function of redshift and the interaction of annihilation products (Slatyer 2013). The reader can easily scale our results for any other models considered. If, instead, DM has a p-wave annihilation cross section, with a v₂ dependence, we do not expect any relevant bound from the CMB, since velocities at DM freeze out and recombination can be orders of magnitude different.

In particular, we expect most of the constraining power of CMB-S4 to come from its polarization spectrum $C_\ell ^{EE}$ and the cross spectrum $C_\ell ^{TE}$. Figure 8 illustrates the percentage deviation from the fiducial cosmology in $C_\ell ^{EE}$ at the 95% C.L. value of p_ann that Planck and CMB-S4 would be able to differentiate, respectively. The CMB-S4 configuration chosen for this figure has 10⁶ detectors, f_sky = 0.75, and 1' beam size.

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In the range of sensitivities and beam sizes considered, we found that the main factor that improves the limit in m_DM is sky coverage f_sky. This means that the constraints are largely sample variance limited.

Figure 9 illustrates how steep the dependence is on f_sky (it goes as ${\sim} f_{{\rm sky}}^{1/2}$) and the mild dependence on detector number and beam size. It also shows how much more many of the possible CMB-S4 configurations improve over limits from Planck.

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At perfect energy deposition f = 1, Planck could exclude m_DM < 114 GeV at 95% C.L. for 〈σv〉 = 3 × 10⁻²⁶ cm³ s⁻¹, when including polarization and lensing information. Any CMB-S4 configuration with beams smaller than 4', minimum of 10⁴ detector, covering half the sky could exclude m_DM < 190 GeV at the 95% C.L. or better. In particular, the best exclusion limit considered here is 255 GeV with 10⁶ detectors, 1' beam size, and f_sky = 0.75. It remains exciting to see significant improvement on m_DM limits from Planck.

The theory of cosmic inflation was proposed to solve the missing monopole problem, the flatness problem, and the horizon problem from the standard big bang theory (Guth 1981). Inflation invokes a period of rapid expansion of the universe before the standard big bang expansion phase. Because of the rapid expansion, any relics would be diluted to a point where they would be extremely rare in our observable universe. Similarly, the curvature of the universe could be diluted as well. For the horizon problem, modes that would not have been in causal contact in the standard big bang theory were in fact once in causal contact in the inflationary framework before their horizon exit (Dodelson 2003). Outside the horizon, their amplitudes were frozen. Therefore, after they reentered the horizon, modes that would not have been in causal contact if not for inflation appear to have equilibrated with each other. Not only does inflation solve these problems, it sets the initial conditions for LSS, thus providing a physical foundation for the observed fluctuations in the CMB and LSS. During inflation, quantum fluctuations were stretched and became classical perturbations. Scalar perturbation of the metric seeded the formation of LSS. Tensor perturbation of the metric created primordial gravitational waves. The observables caused by these perturbations provide definite signatures to confirm or falsify the theory of inflation. To study them, we parameterize the perturbation spectra using a power law in wavenumber k, as follows.

The scalar spectrum is

$$\begin{equation} P_s(k) = A_s(k_*)\left(\frac{k}{k_*}\right)^{n_s(k_*)-1+\frac{1}{2}\alpha _s(k_*)\log (k/k_*)}, \end{equation} \tag{ 20 }$$

where A_s is the amplitude of the scalar spectrum, n_s is the scalar spectral index, k_* is the pivot scale, and α_s is the running of n_s. The tensor spectrum can be written as

$$\begin{equation} P_t(k) = A_t(k_*)\left(\frac{k}{k_*}\right)^{n_t(k_*)}, \end{equation} \tag{ 21 }$$

where A_t is the amplitude of the tensor spectrum, and n_t is the tensor spectral index.

We can measure A_s, n_s, α_s, A_t, and n_t using the CMB. One of the most sought-after parameters, the tensor-to-scalar ratio r ≡ P_t/P_s, tells us the energy scale of inflation. Specifically,

$$\begin{equation} V^{1/4} = 1.06 \times 10^{16}\, {\rm GeV} \left(\frac{r}{0.01} \right)^{1/4}\protect. \protect \end{equation} \tag{ 22 }$$

Therefore, a value of r larger than 0.01 would confirm that inflation happened at an energy scale comparable to that at the Grand Unified Theory, at which the strong, weak, and electromagnetic forces are unified. The CMB provides a unique window to the HEP that cannot be attained in terrestrial accelerator experiments.

The value of r is also directly related to how big the field excursion Δϕ is from when fluctuations seen in the CMB were created to the end of inflation. Given the number of e-folds N_e, in single-field slow-roll and single-field Dirac-Born-Infeld inflation (Silverstein & Tong 2004; Lyth 1997), we can write (Lyth 1997)

$$\begin{equation} N_e = \int \frac{da}{a} = \int H dt = \int \frac{HM_{{\rm pl}}}{\dot{\phi }}\frac{d\phi }{M_{{\rm pl}}} = \sqrt{8} r^{-1/2} \frac{\Delta \phi }{M_{{\rm pl}}}, \end{equation} \tag{ 23 }$$

where M_pl is the Planck mass, r = P_t/P_s, $P_t \propto H^2/M_{{\rm pl}}^2$, and $P_s \propto H^4/\dot{\phi }^2$. For N_e ∼ 50, a Planck mass field range corresponds to r ∼ 0.003.¹⁵ (Similarly, for N_e ∼ 60, Δϕ = M_pl implies r ∼ 0.002.) For super-Planckian field excursion, Δϕ > M_pl implies that the inflation field is sensitive to an infinite series of operators of arbitrary dimensions (Baumann et al. 2009). In order for a large-field inflation model to be UV complete, effective field theory requires a shift symmetry in the field, which protects the flatness of the potential over a large field range. Differentiating between sub-Planckian and super-Planckian field excursion during inflation would rule out different classes of inflationary models.

Here we use the simplest class of models—single-field slow-roll inflation—to illustrate how observations can falsify inflationary models. The conditions for slow-roll require the inflationary potential to be flat enough such that the slow-roll parameters and η are small. They are defined as follows:

$$\begin{eqnarray} \epsilon &\equiv & \frac{M^2_{{\rm pl}}}{2}\left(\frac{V_{,\phi }}{V}\right)^2 \ll 1 \end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray} \eta &\equiv & M^2_{{\rm pl}} \frac{V_{,\phi \phi }}{V} \ll 1\protect, \protect \end{eqnarray} \tag{ 25 }$$

where V is the inflaton potential, and the subscript, ϕ denotes the partial derivative with respect to ϕ.

We can write n_s, α_s, n_t, and r, to leading order, in terms of the slow-roll parameters:

$$\begin{eqnarray} n_s &=&1- 6\epsilon +2\eta \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} n_t &=& -2\epsilon \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} \alpha _s &= & -16\epsilon \eta + 24\epsilon ^2 + 2\xi ^2 \end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray} r &=& 16\epsilon, \end{eqnarray} \tag{ 29 }$$

where $\xi ^2 = M^4_{{\rm pl}} V_{,\phi } V_{,\phi \phi \phi }/V^2$.

Therefore, for each specific single-field slow-roll model, there exists a unique set of predictions on each of these parameters. This allows us to rule out models by measuring the values of the parameters.

Inflation predicts an almost but not exactly flat universe, where |Ω_K| is on the order of 10⁻⁵ (Planck Collaboration et al. 2013e; Abazajian et al. 2013b). This small spatial curvature is a consequence of the curvature fluctuations of the large-scale modes at the horizon. A statistically significant deviation from the expectation of inflation for the curvature would give us information on the process of inflation. For example, if a significant departure from flatness is measured, it can mean that inflation was not slow-rolling when perturbations of the scales just larger than our observable universe exited the inflationary horizon.

The current best constraints for these parameters are from the first release of Planck data (Planck Collaboration et al. 2013e). It is measured that n_s = 0.9603 ± 0.0073 (and n_s = 0.9629 ± 0.0057 when combined with BAO), α_s = −0.0134 ± 0.0090, and $\Omega _K = -0.058^{+0.046}_{-0.026}$ with Planck and WMAP polarization data (and Ω_K = −0.0004 ± 0.00036 when combined with BAO). An upper bound for r is set at r < 0.11 at 95% C.L.

To get to the next qualitatively significant level of constraints for these parameters or confirm null results, we need to at least constrain r to 0.0006 at 1σ. This will confirm a discovery for large-field inflation in the case where r ≳ 0.003 if N_e ∼ 50. Alternatively, a null result at this level will rule out large field inflation. For curvature Ω_K, it would be very interesting to get 1σ constraints at the level of 10⁻⁵. While n_s is constrained to a level where we confirm inflation predictions, more precise constraints will be needed to rule out models. As for α_s, most inflationary models predict it to be undetectable, so any detection would be interesting.

In this section, we present the 1σ constraints of Ω_K, n_s, and α_s that are close to getting to evidence (3σ) and/or discovery (5σ) regimes for the grid of experimental setups. The treatment for constraining the tensor-to-scalar ratio r is different from that for the other three parameters, so we devote a separate section for r.

6.2.1. Ω_K Forecast

Table 4 lists the constraints for Ω_K using CMB combined with BAO measurements and 1% H₀ prior. CMB experimental inputs listed have 10⁴, 10⁵, and 10⁶ detectors, 1'–4' beams, and a fixed sky fraction of 0.75.

Table 4. Ω_K 1σ Constraints in Units of 10⁻³ for Five Combinations of CMB with DESI BAO and 1% H₀ Prior at f_sky = 0.75

	1'	2'	3'	4'
104 Detectors
CMB (no lens)	8.03	8.42	9.02	9.77
CMB	5.29	5.41	5.55	5.70
CMB+H0	1.30	1.31	1.33	1.36
CMB+BAO	0.94	0.96	0.98	1.01
CMB+BAO+H0	0.93	0.95	0.97	1.00
105 detectors
CMB (no lens)	6.26	6.61	7.09	7.71
CMB	4.24	4.31	4.39	4.52
CMB+H0	1.21	1.23	1.24	1.26
CMB+BAO	0.84	0.86	0.88	0.90
CMB+BAO+H0	0.84	0.85	0.87	0.89
106 detectors
CMB (no lens)	4.89	5.38	5.96	6.47
CMB	3.37	3.65	3.84	4.03
CMB+H0	1.13	1.16	1.19	1.22
CMB+BAO	0.75	0.79	0.83	0.85
CMB+BAO+H0	0.74	0.78	0.82	0.84

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Constraints on Ω_K are sample variance limited in the sensitivity range we consider for CMB-S4 in this study. Increasing N_det by an order of magnitude in the CMB only improves Ω_K constraints by about 20%, while adding BAO and/or H₀ improves the constraints by factor of about five. Decreasing beam sizes improve the constraints at the percent level.

The effects of the curvature density Ω_K on the CMB temperature and E-mode polarization power spectra are degenerate with those from H₀ (Efstathiou & Bond 1999; Hu et al. 2006), because a larger correlation angle between hot spots could be due to both the curvature of space and the surface of last scattering being closer to us. Measuring Ω_K at multiple redshift slices could break this degeneracy. Therefore, we will use multiple probes—CMB lensing, BAO, and H₀ priors—for constraining Ω_K in this section.

CMB lensing provides an overall factor of ∼2 improvement from CMB (TT, EE, TE) for the smaller beam cases and about a factor of three improvement for the 8' beam. When BAO information is added to CMB lensing, there is a factor of 5–10 improvement across the whole grid. This is because the BAO signal is orthogonal to the CMB in Ω_K and H₀ space, as illustrated in Figure 10. We expect the Hubble parameter to be constrained to about or better than 1% in the coming decades. The Ω_K constraints improve by about a factor of three to five from CMB lensing given the 1% H₀ prior. Figure 11 shows the trends of the constraints in Ω_K for CMB, CMB+1% H₀ prior, and CMB+BAO, respectively (CMB includes lensing).

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From Table 4, we note that the 1σ constraints on Ω_K is 0.00075 from CMB+DESI BAO, which is very close to the regime where we want to be able to constrain Ω_K. To further shed light on the mean spatial curvature of our universe, 21 cm mapping (Mao et al. 2008), QSOs, and the Lyα forest will also be useful (Font-Ribera et al. 2013).

6.2.2. The n_s, α_s Forecast

Most of the sensitivity on r comes from the B-mode polarization, which is already known to be much smaller in power than the E modes and the temperature anisotropies. Since tensor modes are so far undetected, we set r = 0 to be the fiducial value and study the 1σ uncertainty on r for various experimental configurations. In this "discovery" phase, it is initially more advantageous to improve sensitivity on a few spatial modes (Jaffe et al. 2000). When lensing starts to dominate (r ∼ 0.01), high-resolution B-mode data can be used to reconstruct the lensing field with high fidelity for delensing. In this regime, there are complicated trade-offs between sky coverage, delensing residual, and polarized foregrounds. As a result, the optimized survey for r can be significantly different from the lensing survey, which is important for constraining the sum of the neutrino mass and the dark energy equation of state. We use this separate section to study the strategies for the tensor search, taking delensing and foregrounds into account. This deep tensor survey is assumed to have an equal observing time as the lensing survey, with enough frequency coverage to achieve the target foreground residual. If r is large (>0.01), CMB-S4 can be a powerful tool for a detailed characterization of the tensor perturbations. In this section, we also look at how well r and n_t could be constrained for larger values of r. Another potentially interesting goal is to measure the three-point functions of B modes. However, this is beyond the scope of this paper.

6.3.1. Method

We use a smaller range of sky fractions when producing constraints for r, while maintaining the range of detector numbers and beam sizes. The grid of net sensitivities on r is given in Table 7. The range of f_sky is chosen to observe the so-called recombination bump at multipole ℓ ∼ 100. The inflationary tensor signal at very large angular scales (the "reionization bump" at ℓ ∼ 8) can in principle exceed lensing even for low values of r (r < 10⁻³). It is buried deep in galactic foregrounds but could in principle be recoverable with frequency-component separation. We acknowledge this exciting possibility but choose not to include the reionization bump in the forecast for simplicity.

Table 7. Table of Experimental Sensitivities s Given Detector Count and Sky Coverage, in μK arcmin

Ndet/fsky	0.0004	0.0006	0.0025	0.01	0.0225	0.04	0.25
10000	0.13	0.17	0.33	0.67	1.00	1.34	3.34
20000	0.09	0.12	0.24	0.47	0.71	0.95	2.37
50000	0.06	0.07	0.15	0.30	0.45	0.60	1.50
100000	0.04	0.05	0.11	0.21	0.32	0.42	1.06
200000	0.03	0.04	0.07	0.15	0.22	0.30	0.75
500000	0.02	0.02	0.05	0.09	0.14	0.19	0.47
1000000	0.013	0.02	0.03	0.07	0.10	0.13	0.33

Note. This grid is used for investigating the constraints for the tensor-to-scalar ratio r.

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At r = 0.01, the magnitude of the primordial gravitational wave B-mode power is comparable to lensing at ℓ = 100. Lensing-induced B modes have the same frequency dependence as the CMB and cannot be distinguished by multi-frequency observations. The lensing contamination can be debiased (i.e., subtracted in power spectrum space) in the same way instrumental noise is removed in temperature power spectrum measurements when the expected lensing power is well known. Alternatively, the lensing deflection field can be reconstructed from arcminute-scale B-mode measurements and the expected lensing contamination to degree-scale B modes predicted and subtracted from the observed B-mode map. The quadratic estimators (Hu & Okamoto 2002) and maximum-likelihood estimators (Hirata & Seljak 2003) are techniques that have been developed to delens CMB maps. Here we use the method outlined in Smith et al. (2012) to forecast the residual noise level after iterative delensing, which converges to the maximum-likelihood solution.

Polarized synchrotron and thermal dust constitute the major sources of astrophysical foregrounds in the measurement of the B-mode power spectrum. In this study, we use the Planck Sky Model (Delabrouille et al. 2013) to estimate the level of polarized foreground we would have given that we observe the cleanest patches of the stated sky area. We compute the BB spectrum from polarized dust and synchrotron at 95 GHz, which is amongst the cleanest frequencies for small sky areas (Dunkley et al. 2009) and expect them to be cleaned to 1%–10% of the observed foreground BB power using existing component separation techniques (Planck Collaboration et al. 2013f). For synchrotron, we use a curved power law for emission with the reference frequency set at 20 GHz and curvature at −0.3. We use the model developed in Miville-Deschênes et al. (2008) for the synchrotron emission index. For polarized dust, we employ the galactic polarization model from the same paper (Miville-Deschênes et al. 2008), where the thermal dust emission in intensity is based on Model 7 of Finkbeiner et al. (1999), with the mean polarization fraction set to 5%. We then incorporate the cleaned foreground level as a noise term in the Fisher forecast.

The effect of tensor modes on the temperature and E-mode polarization power spectra is small compared to that on the B-mode power spectrum. So we can forecast constraints on r using only the B-mode spectrum to obtain almost all of the constraining power. In the B-mode spectrum, given perfect delensing and aside from the B-power enhancement at low-ℓ by τ during reionization, the constraint on r is independent of the rest of the νΛCDM parameter space. Therefore, in the forecast for r, losing the extra constraining power from τ, the Fisher matrix F_ij in Equation (3) can be reduced to

$$\begin{equation} F_{rr} = \sum _\ell \frac{(2\ell +1)f_{{\rm sky}}}{2} \frac{1}{(\delta C_\ell)^2} \left(\frac{\partial C^{B_{{\rm tens}}}_\ell }{\partial r}\right)^2 \end{equation} \tag{ 30 }$$

with

$$\begin{equation} \delta C_\ell = C^{B_{{\rm tens}}}_\ell + N^{BB}_\ell + N^{fg}_\ell + N^{{\rm res}}_\ell \protect, \protect \end{equation} \tag{ 31 }$$

where $C^{B_{{\rm tens}}}$ is the tensor contribution of the BB power spectrum, $N^{BB}_\ell$ is defined in Equation (7), $N^{{\rm fg}}_\ell$ is noise from foreground, and $N^{{\rm res}}_\ell$ is the BB residual after delensing the B-mode power spectrum using the fast algorithm developed in Appendices A and B of Smith et al. (2012).

In the scenario that delensing is imperfect, the constraints on r do depend on our knowledge of other parameters that determine the amplitude and shape of the lensing power spectrum. This is because the uncertainty in the lensing power spectrum cascades into the uncertainty in $N^{{\rm res}}_\ell$. Given a CMB-S4 like experiment with 5 × 10⁵ detectors, 3' beam, and f_sky = 0.5, we estimate a 0.5% uncertainty (95% C.L.) in the level of BB lensing amplitude averaged over ℓ range of 2–3000. If the power of $N^{{\rm res}}_\ell$ was 10% of the lensing BB power, the uncertainty in $N^{{\rm res}}_\ell$ would be 0.05%, which is small compared with other terms.

Since we take the fiducial r to be 0, the constraints in r inform us how well (in σ) we can differentiate r ≠ 0 from r = 0. For the specific experimental cases considered in this work, a comparison of Equation (30) with the formalism developed in Errard et al. (2011) shows a relative difference of ⩽1%.

6.3.2. Results and Discussion

Table 8 presents the 1σ constraints for r for a range of detector counts, sky fractions, beam sizes, and foreground residuals. The main result is that almost all of these experiments can constrain r at 1σ to 0.0006 or better if the foreground residual is at 1% level. A few 10⁶ N_det cases at 10% foreground residuals can also reach this level of constraints. This means that many of these configurations can certainly determine whether the field range is sub-Planckian or super-Planckian, and it will drive theoretical research on models of inflation. More importantly, CMB-S4 will be able to measure and characterize the B-mode power spectrum with high significance if r is large (r > 0.01).

Table 8. 1σ Constraints on r, in Units of 10⁻³

	1% Foreground				10% Foreground
	1'	2'	3'	4'	1'	2'	3'	4'
104 Detectors
fsky = 0.0025	0.50	0.53	0.59	0.66	1.76	1.82	1.91	2.03
fsky = 0.04	0.52	0.54	0.56	0.58	1.19	1.21	1.24	1.28
fsky = 0.25	0.72	0.72	0.73	0.74	1.46	1.47	1.48	1.50
105 detectors
fsky = 0.0025	0.23	0.24	0.28	0.32	1.20	1.25	1.34	1.45
fsky = 0.04	0.18	0.19	0.20	0.22	0.66	0.68	0.71	0.74
fsky = 0.25	0.22	0.22	0.23	0.24	0.72	0.73	0.75	0.77
106 detectors
fsky = 0.0025	0.15	0.16	0.18	0.21	0.96	1.01	1.08	1.19
fsky = 0.04	0.09	0.10	0.11	0.12	0.48	0.49	0.52	0.56
fsky = 0.25	0.10	0.11	0.12	0.13	0.51	0.52	0.54	0.56

Notes. For 1% foreground residuals, almost all the constraints are better than 0.6 × 10⁻³ in the wide range of experimental configurations we considered. For 10% foregrounds, the cases that have 10⁶ detectors and f_sky > 0.04 can get better than 0.6 × 10⁻³ 1σ constraints on r.

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Figure 12 shows the constraints on r as a function of detector count and sky coverage for 1'–4' beams with 1% and 10% foreground residuals in power. The constraints with 1% foreground residuals are three to six times better than those with 10% foregrounds. Under a wide range of experimental parameters, we find a broad minimum in σ(r) to lie between 0.01 < f_sky < 0.1. This is because delensing requires high enough experimental sensitivity, which one loses when scanning a larger sky area. Even for the most ambitious configuration we studied (10⁶ detectors, 1' beam), we find the optimal survey area to be around 1000 deg².

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We compared iterative and quadratic delensing methods and learned that iterative delensing helps constrain r a lot more than quadratic delensing at a small sky area compared with a large sky area. For example, it improves the constraints at smaller patches by a factor of 6–20 for f_sky = 0.01 for 10⁵ detectors depending on what beam sizes the experiment has. However, the improvement is only a factor of less than two for f_sky = 0.25. This is important at the discovery phase and when we do not observe large patches.

Beam size matters less when the sky area is large because having a larger sky area provides more modes that debias the result regardless of how well delensing is done. On the other hand, for a smaller sky fraction, one relies more heavily on delensing to provide competitive constraints, thus a smaller beam is advantageous. However, this improvement from delensing using a small beam gets greatly washed out if the foreground level is high, especially when the experiment has high sensitivity. For example, at f_sky = 0.04, for different beam sizes and detector numbers, at least a factor of two and at most an order of magnitude of constraining power on r is lost when the foreground residual is 10% and not 0.25% (a very foreground clean case).

The n_t and r constraints if r > 0.01. In the scenario that r is bigger than 0.01, the B-mode peak around ℓ of 100 is higher than the lensing B-mode power. We can therefore constrain r and n_t without delensing. We run the forecast using only $C^{BB}_{\ell }$ for the ℓ range of 10–500 for experiments with a grid shown in Table 1 and 4' beam. The reason for this ℓ range is to exclude the reionization bump at ℓ < 10 and noise above ℓ > 500. Beam size is not critical here because we are not delensing. We picked a few fiducial r values greater than 0.02 to illustrate the trend. The fiducial n_t value is n_t = 0, i.e., we are not imposing the consistency relation between r and n_t valid in single-field slow-roll inflation. Table 9 presents the constraints for r and n_t for several fiducial r values.

Table 9. 1σ Constraints for r and n_t for Various r_fid for Combinations of N_det and f_sky at 4' Beam Size

fsky	σ(r)			σ(nt)
	0.25	0.50	0.75	0.25	0.50	0.75
rfid = 0.2
104Ndet	0.030	0.026	0.025	0.09	0.08	0.07
105Ndet	0.021	0.016	0.014	0.07	0.05	0.04
106Ndet	0.020	0.015	0.012	0.06	0.05	0.04
rfid = 0.1
104Ndet	0.022	0.020	0.019	0.13	0.11	0.11
105Ndet	0.016	0.012	0.010	0.10	0.07	0.06
106Ndet	0.015	0.011	0.009	0.09	0.07	0.05
rfid = 0.05
104Ndet	0.016	0.015	0.014	0.18	0.16	0.16
105Ndet	0.012	0.009	0.007	0.14	0.10	0.08
106Ndet	0.011	0.008	0.007	0.13	0.09	0.08
rfid = 0.02
104Ndet	0.012	0.011	0.011	0.31	0.29	0.28
105Ndet	0.008	0.006	0.005	0.22	0.16	0.14
106Ndet	0.008	0.005	0.004	0.21	0.15	0.12

Notes. Constraints are derived without delensing and counting the lensing $C^{BB}_\ell$ as one of the noise terms. Foreground contributions are not included.

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We observe that for the few high fiducial r values listed, we can have 3σ–5σ constraints without delensing in the grid we consider in this work. We can also constrain n_t at the 10% level for many r_fid ≳ 0.1 cases. However, with this approach, n_t cannot be measured to a precision that allows us to verify the consistency relation between r and n_t in single-field slow-roll inflation.

After submitting this manuscript, the BICEP2 team released the first measurement of B mode at degree scales and reported r = 0.2 (BICEP2 Collaboration et al. 2014). With this new information of a high r value, we note that many configurations of CMB-S4 can constrain n_t to better than 10% without delensing and information from the reionization bump (ℓ < 10). This does not, however, include the contribution of foregrounds on the B-mode power spectrum. Projecting forward, better information on polarized foregrounds will be essential for us to extract the primordial B-mode power spectrum.