Teasing bits of information out of the CMB energy spectrum

Chluba, Jens; Jeong, Donghui

doi:10.1093/mnras/stt2327

Abstract

Departures of the cosmic microwave background (CMB) frequency spectrum from a blackbody – commonly referred to as spectral distortions – encode information about the thermal history of the early Universe (redshift z ≲ few × 10⁶). While the signal is usually characterized as μ- and y-type distortion, a smaller residual (non-y/non-μ) distortion can also be created at intermediate redshifts 10⁴ ≲ z ≲ 3 × 10⁵. Here, we construct a new set of observables, μ_k, that describes the principal components of this residual distortion. The principal components are orthogonal to temperature shift, y- and μ-type distortion, and ranked by their detectability, thereby delivering a compression of all valuable information offered by the CMB spectrum. This method provides an efficient way of analysing the spectral distortion for given experimental settings, and can be applied to a wide range of energy-release scenarios. As an illustration, we discuss the analysis of the spectral distortion signatures caused by dissipation of small-scale acoustic waves and decaying/annihilating particles for a PIXIE-type experiments. We provide forecasts for the expected measurement uncertainties of model parameters and detections limits in each case. We furthermore show that a PIXIE-type experiments can in principle distinguish dissipative energy release from particle decays for a nearly scale-invariant primordial power spectrum with small running. Future CMB spectroscopy thus offers a unique probe of physical processes in the primordial Universe.

cosmology: observations, cosmology: theory

INTRODUCTION

Energy release in the early Universe causes deviations of the cosmic microwave background (CMB) frequency spectrum from a pure blackbody (Zeldovich & Sunyaev 1969; Sunyaev & Zeldovich 1970; Illarionov & Sunyaev 1975a,b; Danese & de Zotti 1977; Burigana, Danese & de Zotti 1991; Hu & Silk 1993a), which we henceforth refer to as spectral distortion (SD). Thus, far no primordial SD was found (Mather et al. 1994; Fixsen et al. 1996; Fixsen & Mather 2002; Kogut et al. 2006; Zannoni et al. 2008; Seiffert et al. 2011), but technological advances over the past quarter-century since COBE/Far Infrared Absolute Spectrophotometer (FIRAS) may soon allow much more precise (at least three orders of magnitudes improvement in sensitivity) characterization of the CMB spectrum (e.g. Fixsen & Mather 2002; Kogut et al. 2011). This is especially interesting because even for the standard cosmological model, several processes exist that imprint distortion signals at a level within reach of present-day technology (see Chluba & Sunyaev 2012; Chluba 2013a; Sunyaev & Khatri 2013, for broader overview). PIXIE (Kogut et al. 2011) provides one very promising experimental concept for measuring these distortion signals, and more recently PRISM, an L-class satellite mission with about 10 times the spectral sensitivity of PIXIE, was put forward (PRISM Collaboration et al. 2013). These prospects motivated us to further elaborate on what could be learned from measurements of the CMB spectrum, taking another step forward towards the analysis of future distortion data.

Previous works primarily used distortions to rule out various energy-release scenarios (ERSs) on a model-by-model basis. These studies include discussion of decaying or annihilating particles (Hu & Silk 1993b; McDonald, Scherrer & Walker 2001), the dissipation of primordial density fluctuations on small scales (Barrow & Coles 1991; Daly 1991; Hu & Sugiyama 1994; Hu, Scott & Silk 1994a; Chluba, Erickcek & Ben-Dayan 2012a; Chluba, Khatri & Sunyaev 2012b; Dent, Easson & Tashiro 2012; Ganc & Komatsu 2012; Pajer & Zaldarriaga 2012; Powell 2012; Biagetti et al. 2013; Chluba & Grin 2013; Khatri & Sunyaev 2013), cosmic strings (Ostriker & Thompson 1987; Tashiro, Sabancilar & Vachaspati 2012, 2013), primordial black holes (Carr et al. 2010), small-scale magnetic fields (Jedamzik, Katalinić & Olinto 2000) and some new physics examples (Lochan, Das & Bassi 2012; Brax et al. 2013; Bull & Kamionkowski 2013).

Until recently, all constraints were based on simple estimates for the chemical potential, μ, and Compton y-parameter (Zeldovich & Sunyaev 1969; Sunyaev & Zeldovich 1970). It was, however, shown that the distortion signature from different ERSs generally is not just given by a superposition of pure μ- and y-distortion (Chluba & Sunyaev 2012; Khatri & Sunyaev 2012b; Chluba 2013b). The small residual beyond μ- and y-distortion contains information about the time-dependence of the energy-release history, which in principle can be used to directly constrain, for instance, the shape of the small-scale power spectrum, measure the lifetime of decaying relic particles, or simply to discern between different energy-release mechanisms (Chluba 2013a). In particular, Chluba (2013a) demonstrated that CMB spectrum measurement with a PIXIE-type experiment provide a sensitive probe for long-lived particles with lifetimes t_X ≃ 10⁹–10¹⁰ s. Similarly, the shape of the small-scale power spectrum can be directly probed with PIXIE's sensitivity if the amplitude of primordial curvature perturbations exceeds A_ζ ≃ few × 10^{− 8} at wavenumber k ≃ 45 Mpc⁻¹ (Chluba 2013a). Future CMB distortion measurements thus provide a unique avenue for studying early-universe models and particle physics.

In Chluba (2013a), model parameters (e.g. abundance and lifetime of a decaying particle) were directly translated into the SD signal (the photon intensity in different frequency channels) using a Green's function method (Chluba 2013b), which was recently added to the cosmological thermalization code cosmotherm¹ (Chluba & Sunyaev 2012). Even when explicitly knowing the relation between ERS and SDs, model comparison and forecasts of uncertainties (or detection limits) are still rather involved. This is because (i) different energy-release mechanisms can cause very similar SDs, (ii) the parameters in different models are often unrelated and (iii) in general, the parameter space is non-linear especially close to the detection limit. One natural question therefore is whether the information contained by the CMB spectrum (the intensity in each frequency channel) could be further compressed and described in a model-independent way (μ, y, plus additional distortion parameters).

The precise shape of the resulting SD directly depends on the underlying energy-release history. Model-dependence is only introduced when asking which physical process caused a specific energy-release history, but this step can be separated from measuring the energy-release history itself. We thus ask, how well future CMB SDs can constrain different energy-release histories, independent of the responsible physical mechanism. For this we perform a principal component analysis (see Mortonson & Hu 2008; Finkbeiner et al. 2012; Farhang, Bond & Chluba 2012; Shaw et al. 2013, for other cosmology-related applications of this method) of the residual (non-μ/non-y) distortion signal, in order to identify spectral shapes and their associated energy-release histories that can be best constrained by future distortion data. The amplitudes, μ_k, of the signal eigenmodes then define a set of parameters that describes all information encoded by the residual distortion signal. These observables can be measured in a model-independent manner with predictable uncertainties. The mode amplitudes, by construction, are uncorrelated and the parameter dependence is linear, which greatly simplifies further analysis in this new parameter space.

The principal components depend on experimental setting (number of channels, distribution over frequency, noise in each channel and its correlations; see Section 2.1) as well as foregrounds and systematic effects. Here, we do not consider the effect of foreground contamination, and therefore only focus on what the minimal instrumental sensitivity should be in order to constrain or detect the signatures of different energy-injection scenarios. Generalization is straightforward, but we leave a more detailed investigation of foreground issues to future work. Along similar lines, we plan on investigating the optimization of experimental settings for various ERSs using the principal component analysis.

This paper is organized as follows: we start by decomposing the SD signal into temperature shift, μ, y and residual distortion (Section 2). This decomposition already depends on the experimental settings (we envision a PIXIE-like experiment), which determines the level to which different spectral shapes are distinguishable. This allows us to obtain visibility functions in redshift for the different distortion components (Fig. 1), providing a generalization of the SD visibility function,²|$\mathcal {J}_{\rm bb}(z)$| (see Section 2.3 for more details), used in earlier works to account for the suppression of distortions by the efficient thermalization process at redshift z ≳ few × 10⁶ (e.g. Burigana et al. 1991; Hu & Silk 1993a). We then construct the energy-release and signal eigenmodes (Section 3), and illustrate how they can be used for simple parameter estimation (Section 4). In Section 5, we demonstrate how constraints on different ERSs can be derived, with particular attention to detectability, errors and model comparison.

QUASI-ORTHOGONAL DECOMPOSITION OF THE THERMALIZATION GREEN'S FUNCTION

The average CMB frequency spectrum, |$I_{\nu }^{\rm CMB}$| (≡ spectral intensity in units W m⁻² Hz⁻¹ sr⁻¹ as a function of frequency ν), can be broken down as follows:

\begin{eqnarray} I_{\nu }^{\rm CMB}=B_\nu (T_0) + \Delta I_\nu ^{T} + \Delta I^{y}_{\nu }+\Delta I^{\rm prim}_\nu . \end{eqnarray}

(1)

The main theme of this paper is to develop an analysis tool for the primordial, pre-recombination distortion signal, |$\Delta I^{\rm prim}_\nu$|⁠, introduced by different energy-release mechanisms at early times, z ≳ 10³ (see Section 2.2). Because this term is usually small compared to the other contributions to |$I_{\nu }^{\rm CMB}$|⁠, we seek a scheme to optimize the search for this signal. The first term in equation (1) describes the CMB blackbody part, |$B_\nu (T_0)=\frac{2h\nu ^3}{c^2}/({{\rm e}^{x}}-1)$|⁠, where T₀ is the CMB monopole temperature T₀ = (2.726 ± 0.001) K (Fixsen et al. 1996; Fixsen 2009) and x ≡ hν/kT₀. The exact value of the CMB monopole temperature, T, is not known down to the accuracy that can be reached by future experiments (ΔT ≃ few × nK). It thus has to be determined in the analysis. This is captured by the second term in equation (1), which is obtained by shifting a blackbody from one temperature T₀ to T, causing a signal

\begin{eqnarray} &\Delta I_\nu ^{T}= G_{T}(\nu )\,\Delta _{T} [1+\Delta _{T}]+Y_{\rm SZ}(\nu )\,\Delta _{T}^2/2 +\mathcal {O}\left(\Delta ^3_T\right), \end{eqnarray}

(2)

where Δ_T = (T − T₀)/T₀ ≪ 1. Here, we defined the spectrum of a temperature shift |$G_{T}(\nu )=[T\,\mathrm{\partial} _T B_\nu (T)]|_{T=T_0}\equiv \frac{2h\nu ^3}{c^2} \frac{x{{\rm e}^{x}}}{({{\rm e}^{x}}-1)^2}$| at lowest order in Δ_T. At second order in Δ_T, a correction related to the superposition of blackbodies (Zeldovich, Illarionov & Syunyaev 1972; Chluba & Sunyaev 2004) appears, having a spectrum that is similar to a Compton y-distortion, |$Y_{\rm SZ}(\nu )\equiv G_{\rm T}\left[x\coth (x/2)-4\right]$|⁠, also known in connection with the thermal Sunyaev–Zeldovich effect caused by galaxy clusters (Zeldovich & Sunyaev 1969).

Finally, in equation (1) we also added a y-distortion, |$\Delta I^{y}_{\nu }=y\,Y_{\rm SZ}(\nu )$|⁠, that is created at low redshifts (z ≲ 10³) but is not directly accounted for by the primordial distortion, |$\Delta I^{\rm prim}_\nu$|⁠. One strong source of late-time y-distortions stems from reionization and structure formation, giving rise to an effective y-parameter y_re ≃ 10^{− 7}-10^{− 6} (Sunyaev & Zeldovich 1972; Hu, Scott & Silk 1994b; Cen & Ostriker 1999; Miniati et al. 2000; Refregier et al. 2000; Oh, Cooray & Kamionkowski 2003; Zhang, Pen & Trac 2004). The aim of this section is to find an operational decomposition of the spectral signal caused by early energy release (z ≳ 10³) from the non-primordial signatures such as temperature shift and late-time y-distortion.

Instrumental aspects

For our analysis, we envision an experiment similar to PIXIE, which is based on a Fourier transform spectrometer (Kogut et al. 2011). PIXIE covers the frequency range ν = 30 GHz–6 THz, with synthesized channels of constant frequency resolution Δν_c = 15 GHz, depending on the mirror stroke.³ The noise in each channel over the mission's duration is ΔI_c ≃ 5 × 10⁻²⁶ W m⁻² Hz⁻¹ sr⁻¹. We assume the noise to be constant and uncorrelated between channels (diagonal covariance matrix |$C_{ij}=\Delta I_{\rm c}^2\,\delta _{ij}$|⁠), with bandpass given by top-hat functions. The SD signal we are after is important only at ν ≃ 30 GHz–1 THz, which for Δν_c = 15 GHz means about 65 channels. The remaining ≃335 channels at ν ≳ 1 THz are used to construct a detailed model for the dust and cosmic infrared background (CIB) component, which we assume is subtracted down to the noise level for the lower frequency channels. In the text, we refer to these specifications as PIXIE-settings. We also consider cases with improved channel noise ΔI_c, as specified.

Detailed foreground modelling could make use of the high-resolution maps obtained with Planck (Planck Collaboration et al. 2013b), allowing to separate bright clusters (Planck Collaboration et al. 2013f), and providing spatial templates for the CO emission (Planck Collaboration et al. 2013a), the CIB (Planck Collaboration et al. 2013e) and Zodiacal light (Planck Collaboration et al. 2013c), but a more in depth analysis is left to future work.

Defining the residual distortion

Information about the thermal history before recombination is encoded by |$\Delta I^{\rm prim}_\nu$| in equation (1). The problem is to disentangle all spectral functions in of equation (1), with the aim to isolate the primordial signal. A small⁴ primordial distortion, |$\Delta I^{\rm prim}_\nu$|⁠, caused by some energy-release history, |${\,\rm d}(Q/\rho _\gamma )/{\,\rm d}z$|⁠, can be computed using a Green's function method⁵ (Chluba 2013b):

\begin{equation} \Delta I^{\rm prim}_\nu (z=0) \equiv \int G_{\rm th}(\nu , z^{\prime }) \, \frac{{\,\rm d}(Q/\rho _\gamma )}{{\,\rm d}z^{\prime }} {\,\rm d}z^{\prime }. \end{equation}

(3)

Here, ρ_γ ≃ 0.26 (1 + z)⁴ eV cm⁻³ is the CMB blackbody energy density and Q has dimensions of energy density. The Green's function, G_th(ν, z), contains all the physics of the thermalization problem. The accuracy of the Green's function method simply relies on the condition that the thermalization problem can be linearized, i.e. that the distortion remains small. It describes the observed SD response for single energy injection at 10³ ≲ z, and can be tabulated prior to the computation to accelerate the calculation.

At very early times (z ≳ 2 × 10⁶), thermalization processes are extremely efficient, and the Green's function has the shape of a simple temperature shift, |$G_{\rm th}(\nu , z)\propto G_{\rm T}\equiv \frac{2h\nu ^3}{c^2} \frac{x{{\rm e}^{x}}}{({{\rm e}^{x}}-1)^2}$|⁠. Later (3 × 10⁵ ≲ z ≲ 2 × 10⁶), photon production by double Compton and Bremsstrahlung at low frequencies becomes less efficient, while redistribution of photons over frequency by Compton scattering is still very fast. In this regime, the distortion assumes the shape of a pure μ-distortion, M(ν) = G_T[x/β − 1]/x, with β = 3ζ(3)/ζ(2) ≈ 2.1923. At late times (z ≲ 10⁴), even Compton scattering becomes inefficient and the distortion is very close to a pure y-distortion, |$Y_{\rm SZ}\equiv G_{\rm T}\left[x\coth (x/2)-4\right]$|⁠.

At all intermediate redshifts, the Green's functions is given by a superposition of these extreme cases with some correction, R(ν, z), which we call residual distortion (see Chluba 2013b, for similar discussion):

\begin{eqnarray} G_{\rm th}(\nu , z) &=& \frac{G_{\rm T}(\nu )}{4} \,\mathcal {J}_{T}(z) + \frac{Y_{\rm SZ}(\nu )}{4} \,\mathcal {J}_{y}(z)\nonumber \\ && +\, \alpha \,M(\nu ) \,\mathcal {J}_{\mu }(z) + R(\nu , z). \end{eqnarray}

(4)

Here, we used the identities |$\int G_{\rm T}(\nu ){\,\rm d}\nu =\int Y_{\rm SZ}(\nu ){\,\rm d}\nu =4 \rho _\gamma$| and ∫M(ν) dν = ρ_γ/α with α = [4ζ(2)/[3ζ(3)] − ζ(3)/ζ(4)]⁻¹ ≈ 1.401 to re-normalize terms. The redshift-dependent function, |$\mathcal {J}_k(z)$|⁠, for k ∈ {T, y, μ}, define the branching ratios of energy going into different components of the signal (see Section 2.3). These ratios are not unique but depend on the experimental settings, which determine the orthogonality between different spectral components. To obtain these functions, we use PIXIE-like instrumental specification (Section 2.1), where the CMB spectrum is sampled over some range of frequencies ν ∈ [ν_min, ν_max] with constant bandwidth Δν_c and constant sensitivity ΔI_c per channel. This turns equation (4) into |$G_{i, \rm th}(z) = G_{i, \rm T} \,\mathcal {J}_{T}(z)/4 + Y_{i, \rm SZ} \,\mathcal {J}_{y}(z)/4 +\alpha \,M_i \,\mathcal {J}_{\mu }(z) + R_i(z)$|⁠, where the subscripts indicate the individual signals in the ith channel. Then, we can interpret G_{i, th}(z), G_{i, T}, Y_{i, SZ}, M_i and R_i(z) (i = 1, …, N) as N-dimensional (N ≡ number of frequency channels) vectors.⁶

In this vector space, the decomposition problem reduces to finding the residual distortion |${\bf {R}}(z)$| such that it is perpendicular to the space spanned by |${\bf {G}}_{\rm T}$|⁠, |${\bf {Y}}_{\rm SZ}$| and |${\bf {M}}$| (see Appendix A for details). Once the residual distortion is identified, we obtain all energy branching ratios, |$\mathcal {J}_{k}(z)$|⁠, of equation (4) by projecting the rest of the Green's function on to |${\bf {G}}_{\rm T}$|⁠, |${\bf {Y}}_{\rm SZ}$| and |${\bf {M}}$|⁠, respectively. The results are shown in Fig. 2. We also defined |$\mathcal {J}_{R}(z)=1-\mathcal {J}_{T}(z)-\mathcal {J}_{y}(z)-\mathcal {J}_{\mu }(z)$|⁠, which determines the amount of energy found in the residual distortion only. At redshift z ≲ 4 × 10⁴, most of the energy release produces a y-distortion, while at 4 × 10⁴ ≲ z ≲ 1.7 × 10⁶ most of the energy goes into a μ-distortion. At 1.7 × 10⁶ ≲ z, the thermalization process, mediated by Compton scattering, double Compton emission and Bremsstrahlung, is so efficient that practically all energy just increases the average CMB temperature.

Around z ≃ 4 × 10⁴, a few per cent of the energy is stored by the residual distortion, and the amplitude of this signal depends strongly on redshift (see Fig. 1). Although small in terms of energy density, the residual distortion reaches ≃10–20 per cent of M(ν) and Y_SZ(ν) at high frequencies, and can even be comparable to M(ν) at ν ≲ 100 GHz. The fraction of energy release to the residual distortion is extremal around z ≃ 3.8 × 10⁴ (see Fig. 2), while the low-frequency amplitude of the residual distortion is largest at z ≃ 6.2 × 10⁴ (see Fig. 1). In Fig. 1, we can also observe a small dependence of the phase of the residual distortion on the redshift of energy release. The redshift-dependent phase shift of the residual distortion provides model-independent information about the time dependence of the energy-release process, while analysis of the superposition between μ- and y-distortion can only be interpreted in a model-dependent way.

Figure 1.

Residual SD at different redshifts. For the construction, we assumed {ν_min, ν_max, Δν_s} = {30, 1000, 1} GHz and diagonal noise covariance.

Open in new tab Download slide

$Energy branching ratios, $\mathcal {J}_{k}(z)$ according to equation (A1) (in the figure the symbol $J\equiv \mathcal {J}$). We multiplied $\mathcal {J}_{R}(z)$ by 10 to make it more visible. For the construction, we assumed {νmin, νmax, Δνs} = {30, 1000, 1} GHz and diagonal noise covariance.$

Figure 2.

Energy branching ratios, |$\mathcal {J}_{k}(z)$| according to equation (A1) (in the figure the symbol |$J\equiv \mathcal {J}$|⁠). We multiplied |$\mathcal {J}_{R}(z)$| by 10 to make it more visible. For the construction, we assumed {ν_min, ν_max, Δν_s} = {30, 1000, 1} GHz and diagonal noise covariance.

Open in new tab Download slide

Fig. 2 also shows that μ-distortion and temperature shift have a significant overlap around z ≃ 10⁵. There |$\mathcal {J}_{\mu }(z)$| exceeds unity, while |$\mathcal {J}_{T}(z)$| is negative. Similarly, for the chosen experimental setting |$\mathcal {J}_{R}(z)$| is negative, ensuring energy conservation. Although below z ≃ 10⁵ photon production becomes very weak and the thermalization of distortions to a temperature shift ceases, the shape of the distortion still projects on to |${\bf {G}}_{\rm T}$|⁠, leading to |$\mathcal {J}_{T}(z)\ne 0$|⁠. When thinking about the different contributions to the total distortion signal these points should be kept in mind.

Another way to define the temperature shift is to integrate the distortion over all frequencies. Scattering terms, to which the μ- and y-distortion are related, conserve photon number density, so that any deviation from zero should be caused by contributions from a temperature shift, related to |${\bf {G}}_{\rm T}(\nu )$|⁠. This approach was used by Chluba (2013b), where by construction |$0<\mathcal {J}_{k}(z)<1$| for k ∈ {T, y, μ, R}. In practice, i.e. with contaminations from foregrounds, this procedure may not be applicable, and simultaneous fitting of different spectral components is expected to work better. We therefore did not further follow this path.

Dependence on experimental settings

It is clear that the decomposition [R(ν, z) and |$\mathcal {J}_k(z)$|] presented above depends on the chosen values for {ν_min, ν_max, Δν_s}. Changing the frequency resolution has a rather small effect, while changing ν_min is more important (see Fig. 3). The differences are therefore mainly driven by the way the distortion projects on to G_T, M and Y_SZ between ν_min and ν_max rather than how precisely the channels are distributed over this interval.

Figure 3.

Residual function at redshift z ≃ 38 000 but for different instrumental settings. The annotated values are {ν_min, ν_max, Δν_s} and we assumed diagonal noise covariance.

Open in new tab Download slide

Also, so far we assumed uniform and uncorrelated noise in the different channels. In this case, the construction of the modes becomes independent of the value of ΔI_c, but more generally one has to include this into the eigenmode analysis. This can be achieved by redefining the scalar product of two frequency vectors, e.g. |${\bf {a}}\cdot {\bf {b}}\equiv \sum _{ij} a_i \,C^{-1}_{ij}\,b_j$|⁠, where C_ij is the full noise covariance matrix. Similarly, signals related to foregrounds can be included when performing the decomposition of the Green's function. These are expected to lead to a degradation of the signal towards both lower and higher frequencies; however, these aspects are beyond the scope of this paper and will be explored in another work.

Energy release and branching ratios

The amplitude of the SD is directly linked to the total energy that was released over the cosmic history. One way, which has been widely applied in the cosmology community, to make this connection is to use the effective μ and y-parameter to characterize the associated distortion, μ ≃ 1.4 Δρ_γ/ρ_γ|_μ and y ≃ (1/4) Δρ_γ/ρ_γ|_y (Zeldovich & Sunyaev 1969; Sunyaev & Zeldovich 1970). The total energy release causing distortions is Δρ_γ/ρ_γ|_dist = Δρ_γ/ρ_γ|_y + Δρ_γ/ρ_γ|_μ, with the partial contributions, Δρ_γ/ρ_γ|_y and Δρ_γ/ρ_γ|_μ, from the y- and μ-era, respectively. In terms of the energy-release history, |$\mathcal {Q}(z^{\prime })={\,\rm d}(Q/\rho _\gamma )/{\,\rm d}\ln z^{\prime }\approx (1+z^{\prime })\,{\,\rm d}(Q/\rho _\gamma )/{\,\rm d}z^{\prime }$|⁠, the effectivey- and μ-parameters can be written as

\begin{eqnarray} y&\approx &\displaystyle\frac{1}{4}\,\int ^{z_{\mu , y}}_0 \mathcal {Q}(z^{\prime }) {\,\rm d}\ln z^{\prime } \nonumber \\ \mu &\approx & 1.4\,\displaystyle\int _{z_{\mu , y}}^\infty \mathcal {J}_{\rm bb}(z^{\prime }) \, \mathcal {Q}(z^{\prime }) {\,\rm d}\ln z^{\prime }, \end{eqnarray}

(5)

where we introduced the SD visibility function, |$\mathcal {J}_{\rm bb}(z)\approx {{\rm e}^{-(z/z_\mu )^{5/2}}}$|⁠, with thermalization redshift z_μ ≃ 2 × 10⁶ (e.g. see Hu & Silk 1993a). The visibility function accounts for efficient thermalization process for redshifts z ≳ z_μ, at which only the average temperature of the CMB is increased and no distortion is created. In equation (5), the transition between the μ- and y-era is modelled as step-function at z_{μ, y} ≃ 5 × 10⁴.

The decomposition, equation (5), into μ- and y-distortion is only rough and has to be refined for the future generation of CMB experiments. Our approach described in this section provides a natural extension. By inserting equation (4) into equation (3) and integrating over all ν we find that the total change of the CMB photon energy density, ρ_γ, caused by energy release is given by

\begin{eqnarray} \left.\displaystyle\frac{\Delta \rho _\gamma }{\rho _\gamma }\right|_{\rm tot} &=& \left.\frac{\Delta \rho _\gamma }{\rho _\gamma }\right|_{T} +\left.\frac{\Delta \rho _\gamma }{\rho _\gamma }\right|_{y} +\left.\frac{\Delta \rho _\gamma }{\rho _\gamma }\right|_{\mu } +\left.\frac{\Delta \rho _\gamma }{\rho _\gamma }\right|_{R} \nonumber \\ &\equiv & 4\Delta _T + 4y + \mu /\alpha + \varepsilon \nonumber \\ \Delta _T&=& \frac{1}{4}\left.\frac{\Delta \rho _\gamma }{\rho _\gamma }\right|_{T} =\frac{1}{4}\int \mathcal {J}_{T}(z^{\prime })\,\mathcal {Q}(z^{\prime }) {\,\rm d}\ln z^{\prime } \nonumber \\ y&=&\frac{1}{4}\left.\frac{\Delta \rho _\gamma }{\rho _\gamma }\right|_{y} =\frac{1}{4}\int \mathcal {J}_{y}(z^{\prime })\,\mathcal {Q}(z^{\prime }) {\,\rm d}\ln z^{\prime } \nonumber \\ \mu &=&\alpha \left.\frac{\Delta \rho _\gamma }{\rho _\gamma }\right|_{\mu } =\alpha \int \mathcal {J}_{\mu }(z^{\prime })\,\mathcal {Q}(z^{\prime }) {\,\rm d}\ln z^{\prime } \nonumber \\ \varepsilon &=&\left.\frac{\Delta \rho _\gamma }{\rho _\gamma }\right|_{R} =\int \,\mathcal {J}_{R}(z^{\prime }) \,\mathcal {Q}(z^{\prime }) {\,\rm d}\ln z^{\prime }, \end{eqnarray}

(6)

with α ≃ 1.401. In addition to Δ_T = ΔT/T₀ (defining a relative temperature shift), y- and μ-parameter, we defined ε to characterize the energy stored in the residual distortion. For a given energy-release history or mechanism, these numbers can be directly computed, but only y, μ and ε can be used to study the energy-release mechanism. The integrals can be carried out as a simple inner product in the discretized redshift vector space, making parameter estimation very efficient.

The expressions, equation (6), for μ and y are very similar to the usual formulae, equation (5). The main difference is that here the origin of the redshift-dependent window functions, |$\mathcal {J}_k(z)$|⁠, becomes apparent, being related to the representation of the different quasi-orthogonal components to the SD. Equations (6) are thus a generalization, introducing visibility functions, or branching ratios |$\mathcal {J}_k$|⁠, for k = μ, y, T and residual distortion, R(ν), respectively. They are, however, dependent on the experimental settings (Section 2.2.1).

PRINCIPAL COMPONENT DECOMPOSITION FOR THE RESIDUAL DISTORTION

In the previous section, we showed that for a given experimental setting the Green's function can be decomposed into quasi-orthogonal parts. The primordial distortion is then fully described by the parameters p = {Δ_T, y, μ} and a residual distortion

\begin{equation} \Delta I^R_{i} \equiv \int R_i(z^{\prime }) \, \mathcal {Q}(z^{\prime }) {\,\rm d}\ln z^{\prime }, \end{equation}

(7)

which can be computed knowing the function R_i(z). To constrain the energy-release history, Δ_T, can be omitted, while interpretation of y and μ only give model-dependent constraints on |$\mathcal {Q}(z^{\prime })$| we discuss this point below. We now ask how much can be learned about the redshift dependence of |$\mathcal {Q}(z^{\prime })$| by analysing |$\Delta I^R_{i}$|⁠. Since the overall signal is only a correction to the main superposition of μ and y-distortion signals, the experimental sensitivity has to be high or the overall energy release ought to be large. By construction, |$\Delta I^R_{i}$| is orthogonal to the space spanned by y and μ-distortion. We can thus perform a simple principal component decomposition for |$\Delta I^R_{i}$| to get a handle on |$\mathcal {Q}(z^{\prime })$|⁠. For this we discretize the energy-release integral, equation (7), as a sum

\begin{eqnarray} \Delta I^R_{i} &\approx \sum _a \hat{R}_{i}(z_a) \, \mathcal {Q}_a, \end{eqnarray}

(8)

where |$\hat{R}_{i}(z_a)=R_{i}(z_a)\,\Delta \ln z$| and |$\mathcal {Q}_a=\mathcal {Q}(z_a)$|⁠. For our computations, we distributed the bins logarithmically between z_min = 10³ and z_max = 5 × 10⁶ with log-spacing Δln z = 2.135 × 10⁻², i.e. 400 grid points, using the mid-point integration rule. While only accurate at the level of ≃0.1 per cent, this approximation is sufficient for deriving the basis functions. When computing the SD from a given energy-release history we still explicitly carry out the full integral, equation (3), using Patterson quadrature rules (Patterson 1968). The Fisher-information matrix for measuring energy-release history, |$\mathcal {Q}_a$|⁠, from the observed residual intensities |$\Delta I_i^R$| is

\begin{eqnarray} \mathcal {F}_{ab}= \frac{1}{\Delta I_{\rm c}^2}\sum _i\, \frac{\mathrm{\partial} \Delta I^R_{i}}{\mathrm{\partial} \mathcal {Q}_a}\, \frac{\mathrm{\partial} \Delta I^R_{i}}{\mathrm{\partial} \mathcal {Q}_b} =\frac{1}{\Delta I_{\rm c}^2}\sum _i\,\hat{R}_{i}(z_a)\,\hat{R}_{i}(z_b), \end{eqnarray}

(9)

where we assumed that the frequency channels, represented by index i, are independent. The eigenvectors of |$\mathcal {F}_{ab}$| determine the principal components, |${\bf {E}}^{(k)}$|⁠, of the problem. The eigenvalues, λ_k, furthermore determine how well one might be able to recover |$\mathcal {Q}(z)$| for a given sensitivity ΔI_c.

The eigenmodes are vectors in discretized-redshift space, which we normalize as |${\bf {E}}^{(k)}\cdot {\bf {E}}^{(l)} =\delta _{kl}$|⁠. The energy-release history, |$\mathcal {Q}(z)$|⁠, and the residual distortion, |$\Delta I^R_{i}$|⁠, can then be written as

\begin{eqnarray} \mathcal {{\bf {Q}}}& \approx &\sum _k {\bf {E}}^{(k)} \,\mu _k, \quad \Delta I^R_{i}\approx \sum _k S^{(k)}_{i} \,\mu _k,\nonumber\\ S^{(k)}_{i}&=&\sum _a \hat{R}_{i}(z_a) \, E^{(k)}_a, \end{eqnarray}

(10)

where μ_k and |$S^{(k)}_{i}$| are the amplitude and distortion signal of the k^th eigenmode, respectively. By construction, the eigenvectors, |${\bf {E}}^{(k)}$|⁠, span an ortho-normal basis, while all |${\bf {S}}^{(k)}$| only define an orthogonal basis (generally |${\bf {S}}^{(k)}\cdot {\bf {S}}^{(l)} \ge \delta _{kl}$|⁠). We furthermore defined the energy-release vector |$\mathcal {{\bf Q}}=(\mathcal {Q}(z_0), \mathcal {Q}_R(z_1),\ldots , \mathcal {Q}(z_{n}))^{T}$| of |$\mathcal {Q}(z)$| in different redshift bins and the mode amplitudes |$\mu _k= {\bf {E}}^{(k)}\cdot \mathcal {{\bf {Q}}}$|⁠. The expected absolute error in the recovered mode amplitudes μ_k is determined by |$\Delta \mu _k=1/\sqrt{\lambda _k}\propto \Delta I_{\rm c}$|⁠. This scaling implies that for a given frequency range and resolution the eigenvalue problem only has to be solved once. This is possible because we assume the same sensitivity in each channel, but generalization is straightforward.

Results for the eigenvectors and eigenvalues

In Fig. 4, we show the first few |${\bf {E}}^{(k)}$| and |${\bf {S}}^{(k)}$| for a PIXIE-like experiment. We defined the signs of the modes such that the mode amplitudes are positive for |$\mathcal {Q}={\rm const}>0$|⁠. The first energy-release mode, |${\bf {E}}^{(1)}$|⁠, has a maximum at z ≃ 5.3 × 10⁴, while higher modes show more variability, extending both towards lower and higher redshift. The corresponding distortion modes, |${\bf {S}}^{(k)}$|⁠, show increasing variability and decreasing overall amplitude with growing k. They capture all corrections to the simple superposition of pure μ- and y-distortion, needed to morph between these two extreme cases.

$First few eigenmodes ${\bf {E}}^{(k)}$ and ${\bf {S}}^{(k)}$ for PIXIE-type settings (νmin = 30 GHz, νmax = 1000 GHz and Δνs = 15 GHz). In the mode construction, we assumed that energy release only occurred at 103 ≤ z ≤ 5 × 106.$

Figure 4.

First few eigenmodes |${\bf {E}}^{(k)}$| and |${\bf {S}}^{(k)}$| for PIXIE-type settings (ν_min = 30 GHz, ν_max = 1000 GHz and Δν_s = 15 GHz). In the mode construction, we assumed that energy release only occurred at 10³ ≤ z ≤ 5 × 10⁶.

Open in new tab Download slide

In Table 1, we summarize the projected errors for the first six mode amplitudes. The errors, Δμ_k, increase rapidly with mode number (this is how we order the eigenmodes), meaning that for a fixed amplitude of the distortion signal the information in the higher modes can only be accessed at higher spectral sensitivity.

Table 1.

Open in new tab

Forecasted 1σ errors of the first six eigenmode amplitudes, |${\bf {E}}^{(k)}$|⁠. We also give |$\varepsilon _k=4\sum _i S^{(k)}_i/\sum _i G_{i, T}$|⁠, and the scalar products |${\bf {S}}^{(k)}\cdot {\bf {S}}^{(k)}$| (in units of [10⁻¹⁸ W m⁻² Hz⁻¹ sr⁻¹]²). The fraction of energy release to the residual distortion and its uncertainty are given by ε ≈ ∑_kε_k μ_k and |$\Delta \varepsilon \approx (\sum _{k}\varepsilon _k^2\Delta \mu _k^2)^{1/2}$|⁠, respectively. For the mode construction we used PIXIE-settings ({ν_min, ν_max, Δν_s} = {30, 1000, 15} GHz and channel sensitivity ΔI_c = 5 × 10⁻²⁶ W m⁻² Hz⁻¹ sr⁻¹). The errors roughly scale as |$\Delta \mu _k\propto \Delta I_{\rm c}/\sqrt{\Delta \nu _{\rm s}}$|⁠.

k	Δμ_k	Δμ_k/Δμ₁	ε_k	\|${\bf {S}}^{(k)}\cdot {\bf {S}}^{(k)}$\|
1	1.48 × 10⁻⁷	1	−6.98 × 10⁻³	1.15 × 10⁻¹
2	7.61 × 10⁻⁷	5.14	2.12 × 10⁻³	4.32 × 10⁻³
3	3.61 × 10⁻⁶	24.4	−3.71 × 10⁻⁴	1.92 × 10⁻⁴
4	1.74 × 10⁻⁵	1.18 × 10²	8.29 × 10⁻⁵	8.29 × 10⁻⁶
5	8.52 × 10⁻⁵	5.76 × 10²	−1.55 × 10⁻⁵	3.45 × 10⁻⁷
6	4.24 × 10⁻⁴	2.86 × 10³	2.75 × 10⁻⁶	1.39 × 10⁻⁸

k	Δμ_k	Δμ_k/Δμ₁	ε_k	\|${\bf {S}}^{(k)}\cdot {\bf {S}}^{(k)}$\|
1	1.48 × 10⁻⁷	1	−6.98 × 10⁻³	1.15 × 10⁻¹
2	7.61 × 10⁻⁷	5.14	2.12 × 10⁻³	4.32 × 10⁻³
3	3.61 × 10⁻⁶	24.4	−3.71 × 10⁻⁴	1.92 × 10⁻⁴
4	1.74 × 10⁻⁵	1.18 × 10²	8.29 × 10⁻⁵	8.29 × 10⁻⁶
5	8.52 × 10⁻⁵	5.76 × 10²	−1.55 × 10⁻⁵	3.45 × 10⁻⁷
6	4.24 × 10⁻⁴	2.86 × 10³	2.75 × 10⁻⁶	1.39 × 10⁻⁸

Table 1.

Open in new tab

Forecasted 1σ errors of the first six eigenmode amplitudes, |${\bf {E}}^{(k)}$|⁠. We also give |$\varepsilon _k=4\sum _i S^{(k)}_i/\sum _i G_{i, T}$|⁠, and the scalar products |${\bf {S}}^{(k)}\cdot {\bf {S}}^{(k)}$| (in units of [10⁻¹⁸ W m⁻² Hz⁻¹ sr⁻¹]²). The fraction of energy release to the residual distortion and its uncertainty are given by ε ≈ ∑_kε_k μ_k and |$\Delta \varepsilon \approx (\sum _{k}\varepsilon _k^2\Delta \mu _k^2)^{1/2}$|⁠, respectively. For the mode construction we used PIXIE-settings ({ν_min, ν_max, Δν_s} = {30, 1000, 15} GHz and channel sensitivity ΔI_c = 5 × 10⁻²⁶ W m⁻² Hz⁻¹ sr⁻¹). The errors roughly scale as |$\Delta \mu _k\propto \Delta I_{\rm c}/\sqrt{\Delta \nu _{\rm s}}$|⁠.

k	Δμ_k	Δμ_k/Δμ₁	ε_k	\|${\bf {S}}^{(k)}\cdot {\bf {S}}^{(k)}$\|
1	1.48 × 10⁻⁷	1	−6.98 × 10⁻³	1.15 × 10⁻¹
2	7.61 × 10⁻⁷	5.14	2.12 × 10⁻³	4.32 × 10⁻³
3	3.61 × 10⁻⁶	24.4	−3.71 × 10⁻⁴	1.92 × 10⁻⁴
4	1.74 × 10⁻⁵	1.18 × 10²	8.29 × 10⁻⁵	8.29 × 10⁻⁶
5	8.52 × 10⁻⁵	5.76 × 10²	−1.55 × 10⁻⁵	3.45 × 10⁻⁷
6	4.24 × 10⁻⁴	2.86 × 10³	2.75 × 10⁻⁶	1.39 × 10⁻⁸

k	Δμ_k	Δμ_k/Δμ₁	ε_k	\|${\bf {S}}^{(k)}\cdot {\bf {S}}^{(k)}$\|
1	1.48 × 10⁻⁷	1	−6.98 × 10⁻³	1.15 × 10⁻¹
2	7.61 × 10⁻⁷	5.14	2.12 × 10⁻³	4.32 × 10⁻³
3	3.61 × 10⁻⁶	24.4	−3.71 × 10⁻⁴	1.92 × 10⁻⁴
4	1.74 × 10⁻⁵	1.18 × 10²	8.29 × 10⁻⁵	8.29 × 10⁻⁶
5	8.52 × 10⁻⁵	5.76 × 10²	−1.55 × 10⁻⁵	3.45 × 10⁻⁷
6	4.24 × 10⁻⁴	2.86 × 10³	2.75 × 10⁻⁶	1.39 × 10⁻⁸

Knowing the signal eigenvectors, |${\bf {S}}^{(k)}$|⁠, we can directly relate the mode amplitudes, μ_k, to the fractional energy, ε, stored by the residual distortion. It thus allows us to estimate how much information is contained by the residual distortion. Since integration over frequency can be written as a sum over all frequency bins, with |$\varepsilon _k=4\sum _i S^{(k)}_i/\sum _i G_{i, T}$| we have ε ≈ ∑_kε_k μ_k. The first six ε_k are given in Table 1. The signal modes, |${\bf {S}}^{(1)}$| and |${\bf {S}}^{(2)}$|⁠, contribute most to the energy, while energy release into the higher modes is suppressed by an order of magnitude or more.

Even if individual mode amplitudes cannot be separated, the total energy density contained in the residual distortion might still be detectable. The error of ε can be found using Gaussian error propagation, |$\Delta \varepsilon \approx (\sum _k \varepsilon _k^2 \Delta \mu _k^2)^{1/2}\simeq \lbrace 3.68 \times 10^{-9}, 3.53 \times 10^{-9}, 3.14 \times 10^{-9},2.84 \times 10^{-9}\rbrace$|⁠, where the numbers show, respectively, uncertainties when all modes, all but μ₁, all but μ_k with k ≤ 2 and all but μ_k with k ≤ 3 are included. Another estimator for the residual distortion is the modulus of the residual distortion vector |$|R|^2\approx \sum _k {\bf {S}}^{(k)}\cdot {\bf {S}}^{(k)} \,\mu ^2_k$|⁠. The required scalar product amplitudes are also given in Table 1. Similar to ε, the error of |R|² scales like |$\Delta |R|^2\approx 2 (\sum _k [{\bf {S}}^{(k)}\cdot {\bf {S}}^{(k)} \mu _k]^2 \Delta \mu _k^2)^{1/2}$|⁠. Both ε and |R|² can be used to estimate how much information is left in the residual when the mode hierarchy is truncated at some fixed value k. If the signal-to-noise ratio is larger than unity, more modes should be added.

PARAMETER ESTIMATION USING ENERGY-RELEASE EIGENMODES

In the previous sections, we created a set of orthogonal signal modes that can be constrained by future SD experiments and used to recover part of the energy-release history in a model-independent way. We derived a set of energy-release eigenmodes that describes the residual distortion signal that cannot be expressed as simple superposition of temperature shift, μ- and y-distortion.

As explained above, nothing can be learned from the change in the value of the CMB temperature caused by energy release. Thus, the useful part of the primordial signal is determined by the parameters p_prim = {y, μ, μ_k}. The number of residual modes, μ_k, that can be constrained depends on the typical amplitude of the distortion and instrumental aspects. To the primordial signal, we need to add y_re to describe the late-time y-distortion, and Δ_T to parametrize the uncertainty in the exact value of the CMB monopole. The total distortion signal therefore takes the form

\begin{eqnarray} \Delta I_i&=& \Delta I_i^{T} + \Delta I_i^{y} + \Delta I_i^{\mu } + \Delta I_i^{R} \nonumber \\ \Delta I_i^{T}&=& G_{i, \rm T}\Delta _{T} [1+\Delta _{T}]+Y_{i, \rm SZ}\,\Delta _{T}^2/2 \nonumber \\ \Delta I_i^{y}&=&Y_{i, \rm SZ} \, (y_{\rm re}+y) \nonumber \\ \Delta I_i^{\mu }&=&M_{i} \, \mu , \end{eqnarray}

(11)

where G_{i, T}, Y_{i, SZ} and M_i are the average signals of G_T, Y_SZ and M over the ith channel. The dependence of |$\Delta I_i^{T}$| on Δ_T is quadratic, but since Δ_T ≪ 1, the problem remains quasi-linear, with the second-order term leading to a negligible correction to the covariance matrix, once expanded around the best-fitting value for Δ_T. For estimates one can thus set |$\Delta I_i^{T}\approx G_{i,\rm T}\,\Delta _{T}$| without loss of generality. This defines the parameter set p = {Δ_T, y*, μ, μ_k}, where y* = y_re + y. Note that because of the low-z contribution, it is hard to disentangle the primordial components of Δ_T and y*. The primordial energy release, therefore, is best constrained with μ and the μ_ks.

Errors of Δ_T, y* and μ

As a first step, we estimate the errors on the values of Δ_T, y* and μ assuming PIXIE-like settings. The relevant projections to construct the Fisher matrix, analogous to equation (9), are

\begin{eqnarray} &&{{{\bf G}}_{\rm T}\cdot ({{\bf G}}_{\rm T},{{\bf Y}}_{\rm SZ}, {{\bf M}}) = (2.46 \times 10^{3},1.23 \times 10^{3}, 4.60 \times 10^{2})} \nonumber \\ {{\bf Y}}_{\rm SZ} \cdot ({{\bf Y}}_{\rm SZ}, {{\bf M}}) = (5.37 \times 10^{3}, 5.62 \times 10^{2}) \nonumber \\ {{\bf M}}\cdot {{\bf M}} = 1.23 \times 10^{2}, \end{eqnarray}

(12)

all in units of [10⁻¹⁸ W m⁻² Hz⁻¹ sr⁻¹]². Defining α = ΔI_c/[5 × 10⁻²⁶ W m⁻² Hz⁻¹ sr⁻¹], we expect errors ΔΔ_T ≈ 2.34 × 10⁻⁹ α (or ΔT ≃ 6.4 α nK), Δy* ≈ 1.20 × 10⁻⁹ α and Δμ ≈ 1.37 × 10⁻⁸ α at 1σ level. These numbers are close to the estimates given by Kogut et al. (2011) for the expected 1σ errors on y- and μ-parameter, and show that a huge improvement over COBE/FIRAS (Δy* ≈ 7.5 × 10⁻⁶ and Δμ ≈ 4.5 × 10⁻⁵ at 1σ level) can be expected. Adding the residual distortion eigenmodes to the parameter estimation should not affect these estimates as they are constructed to be orthogonal to the signals from Δ_T, y and μ.

Simple parameter estimation example: proof of concept

To illustrate how the modes can be used to constrain the energy-release history, let us consider |$\mathcal {Q}(z)\equiv 5 \times 10^{-8}$| in the redshift interval 10³ < z < 5 × 10⁶. Using equation (6), this implies a total energy release of Δρ_γ/ρ_γ = 4.26 × 10⁻⁷, with Δρ_γ/ρ_γ|_dist = 4y + μ/α + ε ≈ 4.00 × 10⁻⁷ going into distortions. We also expect y ≃ 4.85 × 10⁻⁸, μ ≃ 2.93 × 10⁻⁷ and Δ_prim ≃ −8.46 × 10⁻⁹ for the primordial distortion. The first three mode amplitudes are μ₁ = 5.14 × 10⁻⁷, μ₂ = 4.34 × 10⁻⁹, and μ₃ = 3.38 × 10⁻⁷, and thus μ₁ should be detectable with a PIXIE-like experiment (see the Δμ_k in Table 1). For illustration, we furthermore assume that the value of the monopole temperature is T₀ = 2.726 K(1 + Δ_f) with Δ_f = 1.2 × 10⁻⁴, and that a low redshift y-distortion with y_re = 4 × 10⁻⁷ is introduced.

We implemented a simple Markov Chain Monte Carlo (MCMC) simulation of this problem using cosmotherm. To compute the primordial distortion signal we used equation (3), i.e. we did not decompose the signal explicitly, but included all contributions to the distortion. We then added a temperature shift with Δ_f = 1.2 × 10⁻⁴ and a y-distortion with y_re = 4 × 10⁻⁷ to the input signal, and analysed it using the model, equation (11), with only μ₁ included. Fig. 5 shows the results of this analysis. All the recovered values and errors agree with the predictions. We can furthermore see that μ₁ does not correlate to any of the standard parameters p_s = {Δ_T, y*, μ}, as ensured by construction. The standard parameters are slightly correlated with each other, since in the analysis we used G_{i, T}, Y_{i, SZ} and M_i which themselves are not orthogonal. Alternatively, one could use the orthogonal basis G_{i, T, ⊥}, Y_{i, SZ} and M_{i, ⊥} (see Appendix A), but since the interpretation of the results is fairly simple we preferred to keep the well-known parametrization. We confirmed that adding more distortion eigenmodes to the estimation problem does not alter any of the constraints on the other parameters. This demonstrates that the eigenmodes constructed above can be directly used for model-independent estimations and compression of the useful information provided by the CMB spectrum.

$Analysis of energy-release history with $\mathcal {Q}(z)=5 \times 10^{-8}$ in the redshift interval 103 < z < 5 × 106 using signal eigenmode, ${{\bf S}}^{(1)}$ (Fig. 4). We assumed {νmin, νmax, Δνs} = {30, 1000, 15} GHz and channel sensitivity ΔIc = 5 × 10−26 W m−2 Hz−1 sr−1. The dashed blue lines and red crosses indicate the expected recovered values. Contours are for 68 per cent and 95 per cent confidence levels. All errors and recovered values agree with the Fisher estimates. We shifted ΔT by Δi = Δf + Δprim with Δf = 1.2 × 10−4 and Δprim ≃ −8.46 × 10−9, where Δprim is the primordial contribution.$

Figure 5.

Analysis of energy-release history with |$\mathcal {Q}(z)=5 \times 10^{-8}$| in the redshift interval 10³ < z < 5 × 10⁶ using signal eigenmode, |${{\bf S}}^{(1)}$| (Fig. 4). We assumed {ν_min, ν_max, Δν_s} = {30, 1000, 15} GHz and channel sensitivity ΔI_c = 5 × 10⁻²⁶ W m⁻² Hz⁻¹ sr⁻¹. The dashed blue lines and red crosses indicate the expected recovered values. Contours are for 68 per cent and 95 per cent confidence levels. All errors and recovered values agree with the Fisher estimates. We shifted Δ_T by Δ_i = Δ_f + Δ_prim with Δ_f = 1.2 × 10⁻⁴ and Δ_prim ≃ −8.46 × 10⁻⁹, where Δ_prim is the primordial contribution.

Open in new tab Download slide

Partial recovery of the energy-release history

The energy-release eigenmodes define an ortho-normal basis to describe the energy-release history over the considered redshift range. In the limit of extremely high sensitivity and very fine spectral coverage (≡ all modes can be measured) a complete reconstruction of the input history would be possible. Since realistically only a finite number of energy-release eigenmodes (two or three really) might be measured, this means that a partial but model-independent reconstruction of the input energy-release history can be derived.

Considering the simple example, |$\mathcal {Q}=5 \times 10^{-8}$|⁠, in Fig. 6 we show the comparison of input history and the corresponding reconstruction if one, three or five modes can be measured. Clearly, the SD signal can only probe energy release around z ≃ 5 × 10⁴, providing the means to obtain a wiggly recovery of the input history. The SD signal created by an energy-release history that is constant, or has the other shapes is virtually indistinguishable from the observational point of view, because the energy release from the oscillatory parts does not leave any significant traces. Still, the trajectories of energy-release histories from different scenarios are directly constrained once the set of μ_k is known. This is one of the interesting model-independent ways of interpreting CMB SD results.

$Partial recovery of the input energy-release history, $\mathcal {Q}= 5 \times 10^{-8}$.$

Figure 6.

Partial recovery of the input energy-release history, |$\mathcal {Q}= 5 \times 10^{-8}$|⁠.

Open in new tab Download slide

Overall picture and how to apply the eigenmodes

We now have all the pieces together to explain how to interpret and use the eigenmode decomposition presented above. Given the distortion data, |$\Delta I^{\rm d}_{i}$| (we assume that foregrounds have been removed perfectly), in different frequency channels we can estimate the spectral model parameters p_m = {Δ_T, y*, μ, μ_k}. Using the signal eigenvectors, |${{\bf S}}^{(k)}$|⁠, we can directly obtain the mode amplitudes by |$\mu _k\approx \sum _i \Delta I^{\rm d}_{i}\,{{\bf S}}^{(k)}_i/|{{\bf S}}^{(k)}|^2$|⁠. Similarly, we can compute y* and μ as simple scalar products of the data vector with |${{\bf Y}}_{\rm SZ}$| and |${{\bf M}}_{\perp }$|⁠. The errors can be deduced using Table 1 and Section 4.1. At this point, we have compressed all the useful information contained by the CMB spectrum into a few numbers, p_m. The number of operations needed to compute the SD from a given ERS also roughly reduces by a factor of η ≃ (m + 2)/N_freq, where m is the included number of eigenmodes and N_freq the number of channels. For PIXIE, this means η^{− 1} ≃ 15-20 times improvement of the performance, when using the signal eigenmodes for parameter estimation.

The μ-parameter provides an integral constraint on the energy release, with redshift-dependent weighting function, |$\mathcal {J}_\mu (z)$| (see Fig. 2). Many energy-release histories can give rise to exactly the same value of μ. Still any specific scenario has to reproduce this number, although an interpretation becomes model dependent at this point. Similarly, the recovered y-parameter can only be interpreted in a model-dependent way. Since only the combination y* = y_re + y can be constrained, the model-dependent step allows us to deduce an estimate for y_re, but otherwise does not help constraining the energy-release history unless y_re is known (precisely) by another method. Conversely, y_re remains uncertain, since a large contribution to y* could be caused by pre-recombination energy release. This compromises our ability to learn about reionization and structure formation by studying the average CMB spectrum.

On the other hand, the recovered eigenmode amplitudes μ_k allow us to constrain the energy-release history, |$\mathcal {Q}(z)$|⁠, in a model-independent way (Section 4.3 and Fig. 6). Since we can only expect the first few modes to be measured, from Fig. 4 it is clear that one is most sensitive to energy release around z ≃ 5 × 10⁴. ERSs with little activity during that epoch will project weakly on to μ_k. Different ERSs are furthermore expected to have specific eigenspectra, μ_k, which in principle allows distinguishing them and constraining their specific model parameters. Computing the eigenspectra as a function of parameters can thus be used to quickly explore degeneracies between models. It is also clear that for ERSs with m parameters, at least m distortion parameters (excluding y) have to be observable. To distinguish between different types of models generally one additional parameter has to be measured and the eigenspectra of the scenarios have to be sufficiently orthogonal with respect to the experimental sensitivity. We find that even for optimistic setting typically no more than the first three eigenmodes plus μ can be measured, so that in the foreseeable future energy-release models with more than four parameters cannot be constrained without providing additional information.

CONSTRAINTS ON DIFFERENT SCENARIOS AND MODEL COMPARISON

The signal decomposition and residual eigenmodes developed in the previous sections provide new insight into the primordial energy-release analysis. This is because we collapse the multifrequency data (order ≃100 numbers) to lower dimensions, with only a handful number of parameters required to describe the distortion signal. In this section, we shall present a few illustrative examples to illustrate how to use the signal eigenmodes in the analysis. In particular, we consider three different classes of early ERSs: dissipation of acoustic modes, particle annihilation and decaying particles. We summarize the parameters and eigenspectra for some examples in Table 2.

Table 2.

Open in new tab

Eigenspectra for different energy-release scenarios. The mode amplitudes were scaled by the variable A, as indicated. An asterisk (*) indicates that the parameter can be detected at more than 1σ with PIXIE-like sensitivity, while a dagger (†) shows that five times the sensitivity is required for a 1σ detection. The last few rows are ρ_k = [μ_k/Δμ_k]/[μ/Δμ], which give a representation that shows how the difficulty of a measurement relative to μ increases. Also, by comparing the numbers between models one can directly estimate how hard it is to distinguish them experimentally.

	Dissipation	Dissipation	Annihilation	Annihilation	Decay	Decay	Decay
Shape	n_S = 1	n_S = 0.96	<σv > =const	<σv > ∝ (1 + z)	z_X = 2 × 10⁴	z_X = 5 × 10⁴	z_X = 10⁵
parameters	n_run = 0	n_run = −0.02	(s-wave)	(p-wave, rel.)	(t_X = 5.8 × 10¹⁰ s)	(t_X = 9.2 × 10⁹ s)	(t_X = 2.3 × 10⁹ s)
A	\|$\frac{A_\zeta }{2.2 \times 10^{-9}}$\|	\|$\frac{A_\zeta }{2.2 \times 10^{-9}}$\|	\|$\frac{f_{\rm ann, s}}{2 \times 10^{-23}\,{\rm eV \, s^{-1}}}$\|	\|$\frac{f_{\rm ann, p}}{10^{-27}\,{\rm eV \, s^{-1}}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|
y/A	4.70 × 10⁻⁹ *	3.52 × 10⁻⁹ *	5.18 × 10⁻¹⁰ †	4.84 × 10⁻¹⁰ †	1.41 × 10⁻⁷ *	8.47 × 10⁻⁸ *	2.96 × 10⁻⁸ *
μ/A	3.11 × 10⁻⁸ *	1.16 × 10⁻⁸ †	3.99 × 10⁻⁹ †	8.35 × 10⁻⁸ *	2.27 × 10⁻⁷ *	7.07 × 10⁻⁷ *	1.01 × 10⁻⁶ *
μ₁/A	5.42 × 10⁻⁸ †	2.95 × 10⁻⁸	6.84 × 10⁻⁹	2.10 × 10⁻⁸	1.59 × 10⁻⁶ *	3.34 × 10⁻⁶ *	2.36 × 10⁻⁶ *
μ₂/A	1.01 × 10⁻⁹	−5.19 × 10⁻⁹	2.61 × 10⁻¹⁰	2.03 × 10⁻⁸	−1.74 × 10⁻⁶ *	−4.95 × 10⁻⁷ †	2.47 × 10⁻⁶ *
μ₃/A	3.53 × 10⁻⁸	1.91 × 10⁻⁸	4.39 × 10⁻⁹	3.04 × 10⁻⁸	1.66 × 10⁻⁶ †	−3.83 × 10⁻⁷	7.1 × 10⁻⁷ †
μ₄/A	2.26 × 10⁻⁹	−5.13 × 10⁻⁹	4.39 × 10⁻¹⁰	3.71 × 10⁻⁸	−1.38 × 10⁻⁶	6.85 × 10⁻⁷	−1.23 × 10⁻⁸
y/A	3.9 σ *	2.9 σ *	0.43 σ †	0.40 σ †	117 σ *	70.6 σ *	24.7 σ *
μ/A	2.3 σ *	0.85 σ †	0.29 σ †	6.1 σ *	16.6 σ *	51.6 σ *	73.8 σ *
ρ₁	0.161 †	0.235	0.159	2.33 × 10⁻²	0.648 *	0.437 *	0.216 *
ρ₂	5.86 × 10⁻⁴	−8.07 × 10⁻³	1.18 × 10⁻³	4.37 × 10⁻³	−0.138 *	−1.26 × 10⁻² †	4.41 × 10⁻² *
ρ₃	4.31 × 10⁻³	6.25 × 10⁻³	4.17 × 10⁻³	1.38 × 10⁻³	2.78 × 10⁻² †	−2.06 × 10⁻³	2.66 × 10⁻³ †
ρ₄	5.72 × 10⁻⁵	−3.49 × 10⁻⁴	8.66 × 10⁻⁵	3.49 × 10⁻⁴	−4.79 × 10⁻³	7.63 × 10⁻⁴	−9.55 × 10⁻⁶

	Dissipation	Dissipation	Annihilation	Annihilation	Decay	Decay	Decay
Shape	n_S = 1	n_S = 0.96	<σv > =const	<σv > ∝ (1 + z)	z_X = 2 × 10⁴	z_X = 5 × 10⁴	z_X = 10⁵
parameters	n_run = 0	n_run = −0.02	(s-wave)	(p-wave, rel.)	(t_X = 5.8 × 10¹⁰ s)	(t_X = 9.2 × 10⁹ s)	(t_X = 2.3 × 10⁹ s)
A	\|$\frac{A_\zeta }{2.2 \times 10^{-9}}$\|	\|$\frac{A_\zeta }{2.2 \times 10^{-9}}$\|	\|$\frac{f_{\rm ann, s}}{2 \times 10^{-23}\,{\rm eV \, s^{-1}}}$\|	\|$\frac{f_{\rm ann, p}}{10^{-27}\,{\rm eV \, s^{-1}}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|
y/A	4.70 × 10⁻⁹ *	3.52 × 10⁻⁹ *	5.18 × 10⁻¹⁰ †	4.84 × 10⁻¹⁰ †	1.41 × 10⁻⁷ *	8.47 × 10⁻⁸ *	2.96 × 10⁻⁸ *
μ/A	3.11 × 10⁻⁸ *	1.16 × 10⁻⁸ †	3.99 × 10⁻⁹ †	8.35 × 10⁻⁸ *	2.27 × 10⁻⁷ *	7.07 × 10⁻⁷ *	1.01 × 10⁻⁶ *
μ₁/A	5.42 × 10⁻⁸ †	2.95 × 10⁻⁸	6.84 × 10⁻⁹	2.10 × 10⁻⁸	1.59 × 10⁻⁶ *	3.34 × 10⁻⁶ *	2.36 × 10⁻⁶ *
μ₂/A	1.01 × 10⁻⁹	−5.19 × 10⁻⁹	2.61 × 10⁻¹⁰	2.03 × 10⁻⁸	−1.74 × 10⁻⁶ *	−4.95 × 10⁻⁷ †	2.47 × 10⁻⁶ *
μ₃/A	3.53 × 10⁻⁸	1.91 × 10⁻⁸	4.39 × 10⁻⁹	3.04 × 10⁻⁸	1.66 × 10⁻⁶ †	−3.83 × 10⁻⁷	7.1 × 10⁻⁷ †
μ₄/A	2.26 × 10⁻⁹	−5.13 × 10⁻⁹	4.39 × 10⁻¹⁰	3.71 × 10⁻⁸	−1.38 × 10⁻⁶	6.85 × 10⁻⁷	−1.23 × 10⁻⁸
y/A	3.9 σ *	2.9 σ *	0.43 σ †	0.40 σ †	117 σ *	70.6 σ *	24.7 σ *
μ/A	2.3 σ *	0.85 σ †	0.29 σ †	6.1 σ *	16.6 σ *	51.6 σ *	73.8 σ *
ρ₁	0.161 †	0.235	0.159	2.33 × 10⁻²	0.648 *	0.437 *	0.216 *
ρ₂	5.86 × 10⁻⁴	−8.07 × 10⁻³	1.18 × 10⁻³	4.37 × 10⁻³	−0.138 *	−1.26 × 10⁻² †	4.41 × 10⁻² *
ρ₃	4.31 × 10⁻³	6.25 × 10⁻³	4.17 × 10⁻³	1.38 × 10⁻³	2.78 × 10⁻² †	−2.06 × 10⁻³	2.66 × 10⁻³ †
ρ₄	5.72 × 10⁻⁵	−3.49 × 10⁻⁴	8.66 × 10⁻⁵	3.49 × 10⁻⁴	−4.79 × 10⁻³	7.63 × 10⁻⁴	−9.55 × 10⁻⁶

Table 2.

Open in new tab

Eigenspectra for different energy-release scenarios. The mode amplitudes were scaled by the variable A, as indicated. An asterisk (*) indicates that the parameter can be detected at more than 1σ with PIXIE-like sensitivity, while a dagger (†) shows that five times the sensitivity is required for a 1σ detection. The last few rows are ρ_k = [μ_k/Δμ_k]/[μ/Δμ], which give a representation that shows how the difficulty of a measurement relative to μ increases. Also, by comparing the numbers between models one can directly estimate how hard it is to distinguish them experimentally.

	Dissipation	Dissipation	Annihilation	Annihilation	Decay	Decay	Decay
Shape	n_S = 1	n_S = 0.96	<σv > =const	<σv > ∝ (1 + z)	z_X = 2 × 10⁴	z_X = 5 × 10⁴	z_X = 10⁵
parameters	n_run = 0	n_run = −0.02	(s-wave)	(p-wave, rel.)	(t_X = 5.8 × 10¹⁰ s)	(t_X = 9.2 × 10⁹ s)	(t_X = 2.3 × 10⁹ s)
A	\|$\frac{A_\zeta }{2.2 \times 10^{-9}}$\|	\|$\frac{A_\zeta }{2.2 \times 10^{-9}}$\|	\|$\frac{f_{\rm ann, s}}{2 \times 10^{-23}\,{\rm eV \, s^{-1}}}$\|	\|$\frac{f_{\rm ann, p}}{10^{-27}\,{\rm eV \, s^{-1}}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|
y/A	4.70 × 10⁻⁹ *	3.52 × 10⁻⁹ *	5.18 × 10⁻¹⁰ †	4.84 × 10⁻¹⁰ †	1.41 × 10⁻⁷ *	8.47 × 10⁻⁸ *	2.96 × 10⁻⁸ *
μ/A	3.11 × 10⁻⁸ *	1.16 × 10⁻⁸ †	3.99 × 10⁻⁹ †	8.35 × 10⁻⁸ *	2.27 × 10⁻⁷ *	7.07 × 10⁻⁷ *	1.01 × 10⁻⁶ *
μ₁/A	5.42 × 10⁻⁸ †	2.95 × 10⁻⁸	6.84 × 10⁻⁹	2.10 × 10⁻⁸	1.59 × 10⁻⁶ *	3.34 × 10⁻⁶ *	2.36 × 10⁻⁶ *
μ₂/A	1.01 × 10⁻⁹	−5.19 × 10⁻⁹	2.61 × 10⁻¹⁰	2.03 × 10⁻⁸	−1.74 × 10⁻⁶ *	−4.95 × 10⁻⁷ †	2.47 × 10⁻⁶ *
μ₃/A	3.53 × 10⁻⁸	1.91 × 10⁻⁸	4.39 × 10⁻⁹	3.04 × 10⁻⁸	1.66 × 10⁻⁶ †	−3.83 × 10⁻⁷	7.1 × 10⁻⁷ †
μ₄/A	2.26 × 10⁻⁹	−5.13 × 10⁻⁹	4.39 × 10⁻¹⁰	3.71 × 10⁻⁸	−1.38 × 10⁻⁶	6.85 × 10⁻⁷	−1.23 × 10⁻⁸
y/A	3.9 σ *	2.9 σ *	0.43 σ †	0.40 σ †	117 σ *	70.6 σ *	24.7 σ *
μ/A	2.3 σ *	0.85 σ †	0.29 σ †	6.1 σ *	16.6 σ *	51.6 σ *	73.8 σ *
ρ₁	0.161 †	0.235	0.159	2.33 × 10⁻²	0.648 *	0.437 *	0.216 *
ρ₂	5.86 × 10⁻⁴	−8.07 × 10⁻³	1.18 × 10⁻³	4.37 × 10⁻³	−0.138 *	−1.26 × 10⁻² †	4.41 × 10⁻² *
ρ₃	4.31 × 10⁻³	6.25 × 10⁻³	4.17 × 10⁻³	1.38 × 10⁻³	2.78 × 10⁻² †	−2.06 × 10⁻³	2.66 × 10⁻³ †
ρ₄	5.72 × 10⁻⁵	−3.49 × 10⁻⁴	8.66 × 10⁻⁵	3.49 × 10⁻⁴	−4.79 × 10⁻³	7.63 × 10⁻⁴	−9.55 × 10⁻⁶

	Dissipation	Dissipation	Annihilation	Annihilation	Decay	Decay	Decay
Shape	n_S = 1	n_S = 0.96	<σv > =const	<σv > ∝ (1 + z)	z_X = 2 × 10⁴	z_X = 5 × 10⁴	z_X = 10⁵
parameters	n_run = 0	n_run = −0.02	(s-wave)	(p-wave, rel.)	(t_X = 5.8 × 10¹⁰ s)	(t_X = 9.2 × 10⁹ s)	(t_X = 2.3 × 10⁹ s)
A	\|$\frac{A_\zeta }{2.2 \times 10^{-9}}$\|	\|$\frac{A_\zeta }{2.2 \times 10^{-9}}$\|	\|$\frac{f_{\rm ann, s}}{2 \times 10^{-23}\,{\rm eV \, s^{-1}}}$\|	\|$\frac{f_{\rm ann, p}}{10^{-27}\,{\rm eV \, s^{-1}}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|	\|$\frac{f_{\rm X}/z_{\rm X}}{1\,{\rm eV}}$\|
y/A	4.70 × 10⁻⁹ *	3.52 × 10⁻⁹ *	5.18 × 10⁻¹⁰ †	4.84 × 10⁻¹⁰ †	1.41 × 10⁻⁷ *	8.47 × 10⁻⁸ *	2.96 × 10⁻⁸ *
μ/A	3.11 × 10⁻⁸ *	1.16 × 10⁻⁸ †	3.99 × 10⁻⁹ †	8.35 × 10⁻⁸ *	2.27 × 10⁻⁷ *	7.07 × 10⁻⁷ *	1.01 × 10⁻⁶ *
μ₁/A	5.42 × 10⁻⁸ †	2.95 × 10⁻⁸	6.84 × 10⁻⁹	2.10 × 10⁻⁸	1.59 × 10⁻⁶ *	3.34 × 10⁻⁶ *	2.36 × 10⁻⁶ *
μ₂/A	1.01 × 10⁻⁹	−5.19 × 10⁻⁹	2.61 × 10⁻¹⁰	2.03 × 10⁻⁸	−1.74 × 10⁻⁶ *	−4.95 × 10⁻⁷ †	2.47 × 10⁻⁶ *
μ₃/A	3.53 × 10⁻⁸	1.91 × 10⁻⁸	4.39 × 10⁻⁹	3.04 × 10⁻⁸	1.66 × 10⁻⁶ †	−3.83 × 10⁻⁷	7.1 × 10⁻⁷ †
μ₄/A	2.26 × 10⁻⁹	−5.13 × 10⁻⁹	4.39 × 10⁻¹⁰	3.71 × 10⁻⁸	−1.38 × 10⁻⁶	6.85 × 10⁻⁷	−1.23 × 10⁻⁸
y/A	3.9 σ *	2.9 σ *	0.43 σ †	0.40 σ †	117 σ *	70.6 σ *	24.7 σ *
μ/A	2.3 σ *	0.85 σ †	0.29 σ †	6.1 σ *	16.6 σ *	51.6 σ *	73.8 σ *
ρ₁	0.161 †	0.235	0.159	2.33 × 10⁻²	0.648 *	0.437 *	0.216 *
ρ₂	5.86 × 10⁻⁴	−8.07 × 10⁻³	1.18 × 10⁻³	4.37 × 10⁻³	−0.138 *	−1.26 × 10⁻² †	4.41 × 10⁻² *
ρ₃	4.31 × 10⁻³	6.25 × 10⁻³	4.17 × 10⁻³	1.38 × 10⁻³	2.78 × 10⁻² †	−2.06 × 10⁻³	2.66 × 10⁻³ †
ρ₄	5.72 × 10⁻⁵	−3.49 × 10⁻⁴	8.66 × 10⁻⁵	3.49 × 10⁻⁴	−4.79 × 10⁻³	7.63 × 10⁻⁴	−9.55 × 10⁻⁶

In the following, we precede in a step-by-step manner: we first give details about the parametrizations of the different ERSs (Section 5.1). In Section 5.2, we illustrate the general dependence of the distortion signals on the model parameters, while in Section 5.3 we discuss future detection limits for μ and μ_k. We close our analysis in Section 5.4 by providing details about direct model comparisons.

Parametrization of the energy-release scenarios

The two cases for the dissipation of small-scale acoustic modes presented in Table 2 are computed according to Chluba et al. (2012b), using the standard parametrization of the primordial curvature power spectrum, |$\mathcal {P}_\zeta (k)\equiv A_\zeta \,(k/k_0)^{n_{\rm S}-1+\frac{1}{2} n_{\rm run} \ln (k/k_0)}$|⁠, with the pivot-scale k₀ = 0.05 Mpc⁻¹. The associated SD is thus a family of three parameters (A_ζ, n_S, n_run), with heating rate defined by (cf. Chluba et al. 2012b; Chluba & Grin 2013)

\begin{eqnarray} \left.\frac{{\,\rm d}(Q/\rho _\gamma )}{{\,\rm d}z}\right|_{\rm ac} & \approx 2 D^2 \int ^\infty _{k_{\rm cut}} \mathcal {P}_\zeta (k) \, \mathrm{\partial} _z {\rm e}^{-2k^2/k_{\rm D}^2} {\,\rm d}\ln k, \end{eqnarray}

(13)

where k_D(z) is the dissipation scales, k_cut ≃ 1 Mpc⁻¹ denotes the k-space cut-off scale⁷ and D² ≃ 0.81 is the heating efficiency for adiabatic modes (assuming the standard value for the effective number of relativistic neutrino species N_eff = 3.046). The distortion depends on the type of initial conditions (adiabatic versus isocurvature); however, as shown by Chluba & Grin (2013), the differences can be captured by redefining the heating efficiency, the spectral index and its running. Thus, a discussion of the SD caused by adiabatic modes sweeps the whole parameter space. For a scale-invariant power spectrum |${\,\rm d}(Q/\rho _\gamma )/{\,\rm d}z|_{\rm ac} \propto z^{-1}$| so that |$\mathcal {Q}_{\rm ac}\simeq {\rm const}$|⁠.

The two annihilation scenarios given in Table 2 are for s-wave and p-wave annihilation cross-section with redshift dependence 〈σv〉 = const and 〈σv〉 ∝ (1 + z), respectively. The heating rate can be parametrized as (see also Chluba & Sunyaev 2012)

\begin{eqnarray} \left.\displaystyle\frac{{\,\rm d}(Q/\rho _\gamma )}{{\,\rm d}z}\right|_{\rm ann} & \approx f_{\rm ann}\,\displaystyle\frac{N_{\rm H}(z) (1+z)^{2+\lambda }}{H(z)\,\rho _\gamma (z)}, \end{eqnarray}

(14)

where λ = 0 for s-wave and λ = 1 for p-wave annihilation. Furthermore, N_H(z) ≃ 1.9 × 10⁻⁷(1 + z)³ cm⁻³ denotes the number density of hydrogen nuclei, and H(z) ≃ 2.1 × 10⁻²⁰(1 + z)² s⁻¹ is the Hubble rate, assuming radiation domination. Thermally produced dark matter particles are expected to have s-wave annihilation cross-section with possible amplification due to Sommerfeld-enhancement (e.g. see Hannestad & Tram 2011). The p-wave scenario corresponds to a Majorana particle which either is still relativistic after freeze out [e.g. a sterile neutrino with low abundance (Ho & Scherrer 2013)], or shows v⁻¹ ∝ (1 + z)⁻¹ Sommerfeld-enhanced annihilation cross-section (e.g. see Chen & Zhou 2013). For the non-relativistic case the cross-section drops even faster towards lower redshifts, < σv > ≃(1 + z)², causing practically no energy release at late times. The annihilation efficiency, f_ann, parametrizes all the dependences of the energy-release rate on the mass of the particle, its abundance and overall annihilation cross-section. We have |$\mathcal {Q}_{\rm ann, s}\simeq {\rm const}$| and |$\mathcal {Q}_{\rm ann, p}\propto (1+z)$|⁠. Fixing the redshift dependence of the annihilation cross-section (more elaborate scenarios are possible but beyond the scope of this work), the distortion is a one parameter family that only depends on f_ann.

Finally, in Table 2 we consider three decaying particle scenarios. The total energy release in all these cases is Δρ_γ/ρ_γ ≃ 6.4 × 10⁻⁷ and the energy-release rate is parametrized as (cf. Chluba & Sunyaev 2012)

\begin{eqnarray} \left.\displaystyle\frac{{\,\rm d}(Q/\rho _\gamma )}{{\,\rm d}z}\right|_{\rm dec}&\approx \epsilon _{\rm X}\,\displaystyle\frac{N_{\rm H}(z) (1+z_{\rm X}) \Gamma _{\rm X}}{H(z)\rho _\gamma (z)\,(1+z)} \,\exp \left(- \Gamma _{\rm X}\, t\right) \end{eqnarray}

(15)

with ϵ_X = f_X/z_X parametrizing the energy-release efficiency and Γ_X ≃ 2H(z_X) denoting the particle decay rate. The efficiency factor f_X depends on the mass and abundance of the decaying particle and the efficiency of energy transfer to the baryons. In the radiation dominated era one has |$\mathcal {Q}_{\rm dec}\propto \epsilon _{\rm X} \, z^{-3} \, \exp (-[z_{\rm X}/z]^2)$|⁠. The distortion is thus a two parameter family. Well-motivated candidates comprise excited states of dark matter (e.g. Finkbeiner & Weiner 2007; Pospelov & Ritz 2007), or other, dynamically unimportant relic particles (see Kawasaki, Kohri & Moroi 2005; Feng 2010; Kohri & Takahashi 2010; Pospelov & Pradler 2010, for more references).

Shape of the distortion signal

Annihilating particles

We start with the annihilation scenarios, for which the distortion has a fixed shape and only the overall amplitude changes, depending on the annihilation efficiency, f_ann. The residual distortion signals are illustrated in Fig. 7 (upper panel). We scaled the total distortion such that in both cases μ = 10⁻⁸. This emphasizes the differences in the shape of the distortion rather than its overall amplitude. The residual distortion is significantly smaller for the p-wave scenario, showing that most of the energy is released during the μ-era (y, μ₁ < μ, see Table 2). The small difference in the phase and amplitude of the residual distortion relative to μ in principle allows discerning the s- and p-wave cases, however, a detection of μ₁ is required to break the degeneracy. The values of y, μ and μ_k given in Table 2 fully specify the shape of the distortion for s- and p-wave annihilation scenarios and all other cases can be obtained by rescaling the overall amplitude appropriately.

Comparison of μ-distortion with the residual distortion for different ERSs. We rescaled all cases to have μ = 10−8. For the decomposition we used {νmin, νmax, Δνs} = {30, 1000, 1} GHz. The arrows indicate the direction of increasing parameter. For the dissipation scenarios we set nrun = 0. We furthermore rescaled the residual signal by the annotated values to make the distortion more visible.

Figure 7.

Comparison of μ-distortion with the residual distortion for different ERSs. We rescaled all cases to have μ = 10⁻⁸. For the decomposition we used {ν_min, ν_max, Δν_s} = {30, 1000, 1} GHz. The arrows indicate the direction of increasing parameter. For the dissipation scenarios we set n_run = 0. We furthermore rescaled the residual signal by the annotated values to make the distortion more visible.

Open in new tab Download slide

Dissipation of small-scale acoustic modes

The central panel of Fig. 7 illustrates the n_S-dependence of the residual distortion for the dissipation scenario. For different values of n_S, mainly the amplitude of the distortion changes, while the shape and phase of the residual distortion is only mildly affected. For n_S > 1, relative to the scale-invariant case more energy is released at earlier times. This increase the value of μ relative to the μ_ks, implying that the amplitude of the residual distortion decreases. By measuring μ and μ₁ one can thus constrain A_ζ and n_S independently. However, when allowing n_run to vary, also μ₂ (which is significantly harder to access) is required to distinguish these cases. Running again dominantly affects the amplitude of the residual distortion, while changes in the phase of the signal are weaker.

We can represent the dependence of the distortion on the parameters by specifying the amplitude of μ(A_ζ, n_S, n_run) and the ratios μ_k/μ, which are only function of p = {n_S, n_run}. To also rank the variables in terms of the level of difficulty that is met to measure them, we furthermore weight them by their 1σ-errors. This defines the new variable

\begin{eqnarray} \rho _k=\frac{\mu _k/\Delta \mu _k}{\mu /\Delta \mu }, \end{eqnarray}

(16)

and the distortion parameter set p_d = {y, μ, ρ_k}. To give an example, having ρ_k ≡ 1 means the value of μ₁ is as hard to measure as μ, while ρ_k < 1 means it is |$\rho _k^{-1}$| times harder. If μ is observable with significance, s_μ > 1, then ρ_k < 1 only implies non-detections of μ_k if also ρ_ks_k < 1. The real advantage of this variable is that it parametrizes the shape of the energy-release history without depending on the overall amplitude. Its error is simply |$\Delta \rho _k\approx (1+\rho _k^2)^{1/2}\,\Delta \mu /\mu \approx \Delta \mu /\mu$|⁠, where in the last step we assumed |ρ_k| ≪ 1. Especially for model comparisons, this parametrization is very useful (see Section 5.4).

In Fig. 8, we show the dependence of the first three ρ_k on n_S and n_run. We emphasize that in the considered range of parameters for fixed n_run the vector |${{\bf \rho }}=(\rho _1, \rho _2\, \rho _3)$| is uniquely linked to n_S. The different curves have, however, very similar shapes when varying n_run. The equivalent shift in n_S for each ρ_k differs slightly and also depends on n_S, so that sensitivity to p = {A_ζ, n_S, n_run} can be expected. Both ρ₁ and ρ₃ vary rather slowly, while ρ₂ changes sign around n_S ≃ 1. This indicates that if μ, μ₁ and μ₂ are measurable, most sensitive constraints on p = {A_ζ, n_S, n_run} are expected around n_S ≃ 1. However, since in the considered range ρ₂ ≃ 10⁻³–10⁻², it is already clear that pretty high precision for the measurement of μ is needed (see Fig. 9). We can furthermore see that around n_S ≃ 1.2–1.4, the dependence of ρ₂ on n_run is rather weak, and degenerate with n_S. This indicates that high sensitivity is required to discern different cases in this regime.

Figure 8.

Dependence of ρ_k on n_S and n_run. The heavy lines are for n_run = 0, while all other lines are for n_run = {−0.03, −0.02, −0.01, 0.01, 0.02} in each group. For reference, we marked the case n_run = 0.02. We also indicated parts of the curves that are negative.

Open in new tab Download slide

1σ-detection limits for μ, μ1, μ2 and μ3 caused by dissipation of small-scale acoustic modes for PIXIE-like settings. We used the standard parametrization for the power spectrum with amplitude, Aζ, spectral index, nS, and running nrun around pivot scale k0 = 45 Mpc−1. The heavy lines are for nrun = 0, while all other lines are for nrun = {−0.1, 0.1} in each group. For reference, we marked the case nrun = 0.1.

Figure 9.

1σ-detection limits for μ, μ₁, μ₂ and μ₃ caused by dissipation of small-scale acoustic modes for PIXIE-like settings. We used the standard parametrization for the power spectrum with amplitude, A_ζ, spectral index, n_S, and running n_run around pivot scale k₀ = 45 Mpc⁻¹. The heavy lines are for n_run = 0, while all other lines are for n_run = {−0.1, 0.1} in each group. For reference, we marked the case n_run = 0.1.

Open in new tab Download slide

Decaying relic particles

The lower panel of Fig. 7 illustrates the dependence of the distortion signal caused by a decaying particle on its lifetime. Shorter lifetime means most energy is released at earlier times so that the distortion is closer to a pure μ-distortion with a smaller residual distortion. Increasing the lifetime (lowering z_X), the overall amplitude of the residual distortion increases and shows a small phase shift towards higher frequencies. These are the main signatures that allow measuring the particle lifetime, and in principle only μ and μ₁ are needed to achieve this goal.

In Fig. 10, we show the eigenspectra, ρ_k, for decaying particle scenarios as a function of z_X. In the considered range, the vector |${{\bf \rho }}$| is uniquely linked to z_X, e.g. since ρ₁ never shows any degeneracy. One can thus hope to be able to constrain p = {ϵ_X, z_X} using SD measurements. For z_X ≲ 10⁴, all curves become rather flat, so that CMB distortions are less sensitive to the precise lifetime of the particle and a large uncertainty in z_X is expected. Similarly, at high redshift (z ≳ 2 × 10⁵) the amplitude ρ₁ decreases so that sensitivity to the particle lifetime is diminished (see also Chluba 2013a). Around z ≃ 5 × 10⁴, ρ₁ and ρ₂ show the largest variation with z_X and thus the highest sensitivity to the particle lifetime. In particular, ρ₁ and ρ₂ both change sign in this range. This also suggests that finding ρ₂ < 0 provides indication for a decaying particle over a dissipation scenario, giving one possible criterion for model selection (see Section 5.4).

Figure 10.

Dependence of ρ_k on the lifetime of the decaying particle. We indicated parts of the curves that are negative.

Open in new tab Download slide

Detectability of the distortion signal

Annihilating particles

The signal caused by the s-wave annihilation scenario given in Table 2 is undetectable with a PIXIE-like experiment, but could be detected at ≃3σ with PRISM. The distortion signal depends on only one free parameter, f_{ann, s}, for which we chose a value that is close to the upper 1σ bound derived from current CMB anisotropy measurements (Galli et al. 2009; Hütsi, Hektor & Raidal 2009; Slatyer, Padmanabhan & Finkbeiner 2009; Hütsi et al. 2011; Planck Collaboration et al. 2013d). The spectral sensitivity needs to be increased ≃4 times over PIXIE to detect the s-wave μ-distortion signature, while a factor of ≃ 22 improvement is needed to recover the first distortion eigenmode, μ₁.

The considered p-wave scenario illustrates how the eigenspectra change when the redshift scaling of the energy-release rate is modified. Since most of the energy is liberated at early times, the distortion signal is dominated by μ (see Section 5.2). With a PIXIE-like experiment the first distortion eigenmode remains undetectable and even PRISM will not suffice to measure this number values. Again, the distortion is just determined by f_{ann, p}, but since the eigenspectrum differs from the one of the s-wave scenario, by measuring the first eigenmode these are distinguishable. For f_{ann, s} ≃ 2 × 10⁻²³ eV s⁻¹ and f_{ann, p} ≃ 4.8 × 10⁻²⁹ eV s⁻¹, s- and p-wave scenarios both give rise to μ ≃ 4 × 10⁻⁹. In this case, μ_{1, p} ≃ 9.7 × 10^{− 10} for the p-wave, and μ_{1, s} ≃ 6.4 × 10^{− 9} for the s-wave case. Thus, by increasing the sensitivity ≃ 22 times over PIXIE, the s- and p-wave scenarios in principle become distinguishable (μ₁ from the s-wave scenario would be detected at 1σ, while for a p-wave case μ₁ should be consistent with zero). These findings are in good agreement with those of Chluba (2013a), where an MCMC analysis was used.

A PIXIE-type experiment could place independent 1σ-limits of f_{ann, s} ≲ 6.9 × 10⁻²³ eV s⁻¹ and f_{ann, p} ≲ 1.6 × 10⁻²⁸ eV s⁻¹ on the annihilation efficiency, with practically all the information coming from μ itself (see also Chluba 2013a). Using the parametrization according to the recent Planck papers (Planck Collaboration et al. 2013d), this implies⁸p_{ann, s} < 9.2 × 10⁻⁶ m³ kg⁻¹ s⁻² (95 per cent c.l.), which is several times weaker than the current CMB anisotropy limit obtained with Planck (p_ann < 3.1 × 10⁻⁶ m³ kg⁻¹ s⁻²). Uncertainties in the modelling of the energy-deposition rates indicate that this limit is in fact slightly weaker, but still once the full polarization data from Planck is included, one does expect an improvement of this bound to p_ann < 1.7 × 10⁻⁷ m³ kg⁻¹ s⁻² (Galli et al. 2013). Thus, only an increase of the spectral sensitivity by a factor of ≃ 50 over PIXIE could make future CMB distortion measurements a competitive probe for annihilating dark matter particles, although one should emphasize that SD would still give an independent measurement, suffering from very different systematics. In the future, PRISM might allow direct detection of a dark matter annihilation signature, if p_{ann, s} ≳ 4.6 × 10⁻⁷ m³ kg⁻¹ s⁻².

Dissipation of small-scale acoustic modes

From measurements of the CMB anisotropies at large scales we have A_ζ ≃ 2.2 × 10⁻⁹, n_S ≃ 0.96 and n_run ≃ −0.02 (Planck Collaboration et al. 2013d). Using these values and extrapolating all the way to wavenumber k ≃ few × 10⁴ Mpc^{− 1}, we obtain the distortion parameters given in Table 2. For comparison, we also show the case with no running and scale-invariant power spectrum. For these two dissipation scenarios the y-parameter will contribute at a few σ-level to y* = y_re + y for a PIXIE-like experiment, while no information can be extracted from the residual distortion (none of the μ_k can be detected). For a scale-invariant power spectrum also a non-vanishing μ-parameter could be found (≃2.3σ) with a PIXIE-like experiment (see also Chluba et al. 2012b). For PRISM, a more than 20 σ detection of μ for a scale-invariant power spectrum should be feasible, while for A_ζ ≃ 2.2 × 10⁻⁹, n_S ≃ 0.96 and n_run ≃ 0 we expect a ≃ 17 σ detection of μ.

Since the y-parameter is degenerate with y_re, only μ can be used to place constraints in these cases, however, the degeneracy among model parameters is very large. For example, the small difference in the value of μ for the two considered cases can be compensated by adjusting A_ζ at small scales. Increasing the sensitivity 10 times over PIXIE will allow an additional detection of the first eigenmode (≃3.7σ and ≃2.0σ for the two dissipation scenarios given Table 2, respectively). In this case, the parameter degeneracies (A_ζ, n_S, and n_run) can be partially broken (two numbers, μ and μ₁, are used to limit three variables). Improvement by another factor of 10 allows marginal detections of the second mode amplitude, but to truly constrain the shape of the small-scale power spectrum (assuming the standard parametrization) using SD data alone an overall factor ≳ 200 over PIXIE will be necessary, making this application of SDs rather futuristic (see also Chluba 2013a).

These simple estimates indicate that SD alone only provide competitive constraints on A_ζ, n_S and n_run for much higher spectral sensitivity; however, SD data can help to slightly improve the constraint on n_run when combined with future CMB anisotropy measurements (see Powell 2012; Khatri & Sunyaev 2013, for similar discussion). This is simply because both A_ζ and n_S can be tightly constrained with the CMB anisotropy measurement, while the long lever arm added with SD measurements improves the sensitivity to running of the power spectrum. We illustrate this in Fig. 11 for PRISM and current constraints from Planck, Wilkinson Microwave Anisotropy Probe (WMAP) (e.g. Komatsu et al. 2011) and high ℓ data from ACT (e.g. Dunkley et al. 2011) and SPT (e.g. Keisler et al. 2011). For the standard power spectrum, SD data add little with respect to A_ζ and n_S, but do improve the constraint on n_run for n_run > −0.02. However, similar improvements can also be expected from future small-scale (Stage IV) CMB measurements (Abazajian et al. 2013). At PIXIE's sensitivity, we do not find any significant improvement of power spectrum constraints derived from CMB anisotropy measurements when adding the SD data.

Figure 11.

Forecasted constraints on A_ζ, n_S and n_run. The case labelled Planck+WP+highL uses the published covariance matrix of Planck with inclusion of WMAP polarization data and the high ℓ data from ACT and SPT. The case labelled PRISM is based on estimates given in André et al. (2013) for the PRISM imager and spectrometer part. The upper panel shows the 2D contours and marginalized distributions for A_ζ, n_S and n_run, while the lower panel illustrates the expected improvement (decrease) in the measurement uncertainty of the PRISM imager over Planck (horizontal lines) and the additional gain when adding the PRISM SD data. Note that the PRISM spectrometer is about one order of magnitude more sensitive that PIXIE.

Open in new tab Download slide

Dissipation of small-scale acoustic modes: generalization

The above statements assume that the three-parameter Ansatz for the primordial curvature power spectrum holds for more than six to seven decades in scales. Strictly speaking, the exact shape and amplitude of the small-scale power spectrum are unknown and a large range of viable early-universe models (e.g. Salopek, Bond & Bardeen 1989; Starobinskij 1992; Ivanov, Naselsky & Novikov 1994; Randall, Soljačić & Guth 1996; Stewart 1997; Copeland et al. 1998; Starobinsky 1998; Chung et al. 2000; Hunt & Sarkar 2007; Joy, Sahni & Starobinsky 2008; Barnaby et al. 2009; Barnaby 2010; Ben-Dayan & Brustein 2010; Achúcarro et al. 2011; Céspedes, Atal & Palma 2012) producing enhanced small-scale power exist (see, Chluba et al. 2012a, for more examples and simple SD constraints). Observationally, the amplitude of the primordial small-scale power spectrum is limited to A_ζ ≲ 10⁻⁷–10⁻⁶ at wavenumber 3 Mpc^{− 1} ≲ k ≲ few × 10⁴ Mpc^{− 1} (the range that is of most interest for CMB distortions) using ultracompact mini-haloes (Bringmann, Scott & Akrami 2012; Scott, Bringmann & Akrami 2012). Although slightly model-independent, this still leaves a lot of room for non-standard dissipation scenarios, with enhanced small-scale power.

To study how well the small-scale power spectrum might be constrained by future SD measurements, it is convenient to consider the shape and amplitude of the curvature power spectrum at 3 Mpc^{− 1} ≲ k ≲ few × 10⁴ Mpc^{− 1} independent of the large-scale power spectrum. We therefore change the question as follows: by shifting the pivot scale to k₀ = 45 Mpc⁻¹ (corresponding to heating around z_diss ≃ 4.5 × 10⁵[k/10³ Mpc⁻¹]^2/3 ≃ 5.7 × 10⁴) and using the standard parametrization for the power spectrum, how large does the power spectrum amplitude, A_ζ(k₀ = 45 Mpc⁻¹), have to be to obtain a 1σ-detection of μ, μ₁, μ₂ and μ₃, respectively? The results of this exercise are shown in Fig. 9 for PIXIE settings. Around n_S ≃ 1, a detection of μ is possible for A_ζ ≳ 10⁻⁹, while A_ζ ≳ 6 × 10⁻⁹ is necessary to also have a detection of μ₁. In this case, two of the three model parameters can in principle be constrained independently. To also access information from μ₂ and μ₃ one furthermore needs A_ζ ≳ 10⁻⁷. In this case, we could expect to break the degeneracy between all three parameters with a PIXIE-type experiment.

The detection limits depend both on the value of n_S and n_run. For n_run < 0, in total less energy is released so that larger A_ζ is required for a detection. For n_S > 1, more power is found at k > 45 Mpc⁻¹, so that more energy is released in the μ-era. Consequently, the μ-distortion can be detected for lower A_ζ. Similarly, when increasing n_S, less energy is released around z ≃ 5 × 10⁴, so that the value of μ₁ decreases. Thus, larger A_ζ is required to warrant a detection of μ₁.

The above statements can be phrased in another way. Assuming A_ζ ≃ 10⁻⁹ and n_S ≃ 1, at least a factor of 5 improvement over PIXIE sensitivity is needed to allow constraining combinations of two power spectrum parameters. To determine all p = {A_ζ, n_S, n_run} independently an overall factor of ≳ 200 improvement over PIXIE sensitivity is required, although in this (very conservative) case the corresponding constraints would still not be competitive with those obtained using large-scale CMB anisotropy measurements.

We can also ask the question of how well the power spectrum parameters can be constrained for different cases. If only μ is available, then the corresponding constraints on small-scale power spectrum parameters remain rather weak, but could still be used to limit the parameters space (e.g. Chluba et al. 2012a,b). If μ and μ₁ can be accessed, we can limit the overall amplitude of the power spectrum for given pairs of n_S and n_run. This can be seen from the upper panel of Fig. 12, where we illustrate the possible parameter space of μ, ρ₁ ∝ μ₁/μ and ρ₂ ∝ μ₂/μ in some range of n_S and n_run. For the considered sensitivity and fiducial value of A_ζ, the errors on μ and ρ₁ are very small, but since A_ζ can be adjusted without affecting ρ₁, the measurement is not independent of n_S and n_run.

Parameter range of μ, μ1 and μ2 for dissipation scenarios. We assumed PIXIE settings with five times its sensitivity, and a power spectrum amplitude Aζ(k0 = 45 Mpc−1) = 5 × 10−8 (i.e. A ≡ Aζ/5 × 10−8). The heavy solid black lines are for nrun = 0, while the thin solid brown lines indicate nS = const. The other light lines are for nrun = {−0.2, −0.1, 0.1, 0.2}. The open symbols mark nS in steps ΔnS = 0.1. The blue symbols with error bars (tiny in the upper panel) are for nS = {0.5, 1, 1.5, 1.8} and nrun = 0. They illustrate how the error scales in different regions of the parameter space. Measurements in the μ-ρ1 plane can be used to fix the overall amplitude of the small-scale power spectrum for a given pair nS and nrun, but no independent constraint on nS and nrun can be deduced. The constraints on ρ1 and ρ2 allow us to partially break the remaining degeneracy.

Figure 12.

Parameter range of μ, μ₁ and μ₂ for dissipation scenarios. We assumed PIXIE settings with five times its sensitivity, and a power spectrum amplitude A_ζ(k₀ = 45 Mpc⁻¹) = 5 × 10⁻⁸ (i.e. A ≡ A_ζ/5 × 10⁻⁸). The heavy solid black lines are for n_run = 0, while the thin solid brown lines indicate n_S = const. The other light lines are for n_run = {−0.2, −0.1, 0.1, 0.2}. The open symbols mark n_S in steps Δn_S = 0.1. The blue symbols with error bars (tiny in the upper panel) are for n_S = {0.5, 1, 1.5, 1.8} and n_run = 0. They illustrate how the error scales in different regions of the parameter space. Measurements in the μ-ρ₁ plane can be used to fix the overall amplitude of the small-scale power spectrum for a given pair n_S and n_run, but no independent constraint on n_S and n_run can be deduced. The constraints on ρ₁ and ρ₂ allow us to partially break the remaining degeneracy.

Open in new tab Download slide

If in addition μ₂ can be constrained, then the degeneracy can be broken. For PIXIE-settings and n_S ≃ 0.96, this is only conceivable if the amplitude of the small-scale power spectrum is A_ζ ≳ 10^{− 7}-10^{− 6} (see Fig. 9). As the lower panel of Fig. 12 indicates, the relative dependence on n_run seems rather similar in all parts of parameter space: although the absolute distance between the lines varies relative to the error bars they seem rather constant. To show this more explicitly, from μ, μ₁ and μ₂ we compute the expected 1σ-errors on A_ζ(k₀ = 45 Mpc⁻¹), n_S, and n_run around the fiducial value using the Fisher information matrix, |$\mathcal {F}_{ij}=\Delta \mu ^{-2}\,\mathrm{\partial} _{p_i} \mu \,\mathrm{\partial} _{p_j} \mu +\sum _k \Delta \mu _k^{-2}\,\mathrm{\partial} _{p_i} \mu _k\mathrm{\partial} _{p_j} \mu _k$|⁠, with p ≡ {A_ζ, n_S, n_run}. Fig. 13 shows the corresponding forecasts assuming PIXIE-setting but with five times its sensitivity. If only p ≡ {A_ζ, n_S} are estimated for fixed n_run, the errors of A_ζ and n_S are only a few per cent. When also trying to constrain n_run, we see that the uncertainties in the values of A_ζ and n_S increase by about one order of magnitude, with an absolute error Δn_run ≃ 0.07 rather independent of n_S.

Expected uncertainties of Aζ(k0 = 45 Mpc−1), nS and nrun using measurements of μ, μ1 and μ2. We assumed five times the sensitivity of PIXIE and Aζ = 5 × 10−8 as reference value (other cases can be estimated by simple rescaling). For the upper panel, we also varied nrun as indicated, while in the lower panel it was fixed to nrun = 0.

Figure 13.

Expected uncertainties of A_ζ(k₀ = 45 Mpc⁻¹), n_S and n_run using measurements of μ, μ₁ and μ₂. We assumed five times the sensitivity of PIXIE and A_ζ = 5 × 10⁻⁸ as reference value (other cases can be estimated by simple rescaling). For the upper panel, we also varied n_run as indicated, while in the lower panel it was fixed to n_run = 0.

Open in new tab Download slide

Little information is added when also μ₃ can be measured (we find small differences in the constraints for small n_S when n_run is varied), although for model-comparison μ₃ could become important. Also, for power spectra that result in μ₂ ≃ 0, the detection limit of μ₃ is much lower (see Fig. 9), so that the combination of μ₂ consistent with zero but μ₃ > 0 provides a useful confirmation of the dissipation scenario.

We can also use the results of Fig. 13 to estimate the expected uncertainties for other cases. Adjusting the spectral sensitivity by a factor f = ΔI_c/[10⁻²⁶ W m⁻² Hz⁻¹ sr⁻¹], all curves can be rescaled by this factor to obtain the new estimates for the errors. Similarly, if A_ζ(k₀ = 45 Mpc⁻¹) differs by f_ζ = A_ζ/5 × 10⁻⁸, we have to rescale the error estimates by |$f^{-1}_\zeta$|⁠. We checked the predicted uncertainties for some representative cases using the MCMC method of Chluba (2013a), finding excellent agreement. Overall, our analysis shows that CMB SD measurement provide a unique probe of the small-scale power spectrum, which can be utilized to directly constraint inflationary models. Especially, if the small-scale power spectrum is close to scale invariant with small running, very robust constraints can be expected from PIXIE and PRISM, if A_ζ(k₀ = 45 Mpc^{− 1}) ≃ 10^{− 8}-10^{− 7}.

Decaying relic particles

The distortion signals for the three decaying particle scenarios presented in Table 2 will all be detectable with a PIXIE-like experiment. More generally, Fig. 14 shows the 1σ-detection limits for μ, μ₁, μ₂ and μ₃, as a function of the particle lifetime. CMB SDs are sensitive to decaying particles with ϵ_X = f_X/z_X as low as ≃ 10⁻² eV for particle lifetimes 10⁷ s ≲ t_X ≲ 10¹⁰ s. For PRISM the detection limit will be as low as ϵ_X ≃ 10⁻³ eV in this range. To directly constrain t_X, at least a measurement of μ₁ is needed. At PIXIE sensitivity this means that the lifetime of particles with 2 × 10⁹ s ≲ t_X ≲ 6 × 10¹⁰ s for ϵ_X ≳ 0.1 eV and 3 × 10⁸ s ≲ t_X ≲ 10¹² s for ϵ_X ≳ 1 eV will be directly measurable. Most of this parameter space is completely unconstrained [see upper limit from measurements of the primordial ³He/D abundance ratio⁹ (from fig. 42 of Kawasaki et al. 2005) in Fig. 14]. Higher sensitivity will allow cutting deeper into the parameter space and widen the range over which the particle lifetime can be directly constrained.

Detectability of μ, μ1, μ2 and μ3. The upper panel shows the limits for ϵX = fX/zX, while the lower panel uses the standard yield variable, EvisYX (cp., Kawasaki et al. 2005). For a given particle lifetime, we compute the required value of ϵX for which a 1σ-detection of the corresponding variable is possible with PIXIE. The violet shaded area is excluded by measurements of the primordial 3He/D abundance ratio (1σ-level, adapted from fig. 42 of Kawasaki et al. 2005).

Figure 14.

Detectability of μ, μ₁, μ₂ and μ₃. The upper panel shows the limits for ϵ_X = f_X/z_X, while the lower panel uses the standard yield variable, E_visY_X (cp., Kawasaki et al. 2005). For a given particle lifetime, we compute the required value of ϵ_X for which a 1σ-detection of the corresponding variable is possible with PIXIE. The violet shaded area is excluded by measurements of the primordial ³He/D abundance ratio (1σ-level, adapted from fig. 42 of Kawasaki et al. 2005).

Open in new tab Download slide

To illustrate this aspect even further, we can again study the μ-ρ_k-parameter space covered by decaying particles. The projections into the μ-ρ₁ and ρ₁-ρ₂-plane are shown in Fig. 15 for decay efficiency ϵ_X = 1 eV and PIXIE settings. Varying ϵ_X would move the μ-ρ₁ trajectory left or right, as indicated in the upper panel of Fig. 15. Furthermore, all error bars of ρ_k would have to be rescaled by f = [ϵ_X/1 eV]⁻¹ under this transformation. Measuring μ and ρ₁ is in principle sufficient for independent determination of ϵ_X and the particle lifetime, t_X ≈ [4.9 × 10⁹/(1 + z_X)]² s, with most sensitivity around z_X ≃ 5 × 10⁴ − 10⁵ or t_X ≃ 2 × 10⁹ − 10¹⁰ s for the shown scenario. For shorter lifetime, the SD signal is very close to a pure μ-distortion, with little information in the residual (ρ₁ and ρ₂ are both very small and also show very little variation with redshift). Similarly, for longer lifetimes the particle signature is close to a y-distortion. In both cases, the sensitivity to the lifetime is very weak and only an overall integral constraint can be derived, with large degeneracy between ϵ_X and z_X (see discussion in Chluba 2013a).

Parameter range of μ, μ1 and μ2 for decaying particle scenarios. We assumed PIXIE settings and sensitivity, and ϵX = fX/zX = 1 eV (i.e. A ≡ ϵX/1 eV). The blue symbols with error bars are for zX as labelled. Measurements in the μ-ρ1 plane can be used to constrain zX with the most sensitive range around zX ≃ 5 × 104-105. The ρ1-ρ2 plane can be used to further improve this measurement, but also for model comparison.

Figure 15.

Parameter range of μ, μ₁ and μ₂ for decaying particle scenarios. We assumed PIXIE settings and sensitivity, and ϵ_X = f_X/z_X = 1 eV (i.e. A ≡ ϵ_X/1 eV). The blue symbols with error bars are for z_X as labelled. Measurements in the μ-ρ₁ plane can be used to constrain z_X with the most sensitive range around z_X ≃ 5 × 10⁴-10⁵. The ρ₁-ρ₂ plane can be used to further improve this measurement, but also for model comparison.

Open in new tab Download slide

We can again estimate the expected 1σ-errors on ϵ_X and z_X around the fiducial value using the Fisher information matrix, |$\mathcal {F}_{ij}=\Delta \mu ^{-2}\,\mathrm{\partial} _{p_i} \mu \,\mathrm{\partial} _{p_j} \mu +\sum _k \Delta \mu _k^{-2}\,\mathrm{\partial} _{p_i} \mu _k\mathrm{\partial} _{p_j} \mu _k$|⁠, with parameters p ≡ {ϵ_X, z_X}. In Fig. 16, we show the corresponding Fisher-forecasts assuming PIXIE-setting but with five times its sensitivity. We included information from μ, μ₁ and μ₂, because adding μ₃ did not change the forecast significantly. For 1.7 × 10⁴ ≲ z_X ≲ 3.5 × 10⁵ (2 × 10⁹ s ≲ t_X ≲ 8.3 × 10¹⁰ s), the particle lifetime can be constrained to better than ≃20 per cent and ϵ_X can be measured with uncertainty ≲10 per cent. These findings are in good agreement with those of Chluba (2013a), where direct MCMC simulations were performed. CMB SD are thus a powerful probe of early-universe particle physics, providing tight constraints that are independent and complementary to those derived from light element abundances (e.g. Kawasaki et al. 2005; Kohri & Takahashi 2010; Pospelov & Pradler 2010).

Relative error for determination of ϵX = fX/zX and zX using measurements of μ to μ2. We assumed five times the sensitivity of PIXIE and ϵX = 1 eV as reference value (other cases can be estimated by simple rescaling). The corresponding error in the particle lifetime is ΔtX/tX ≃ 2ΔzX/zX.

Figure 16.

Relative error for determination of ϵ_X = f_X/z_X and z_X using measurements of μ to μ₂. We assumed five times the sensitivity of PIXIE and ϵ_X = 1 eV as reference value (other cases can be estimated by simple rescaling). The corresponding error in the particle lifetime is Δt_X/t_X ≃ 2Δz_X/z_X.

Open in new tab Download slide

We emphasize that the CMB spectrum can be utilized to directly probe the particle lifetime, a measurement that cannot be obtained by other means. CMB SDs furthermore provide a calorimetric constraint, which is sensitive to the total heat that is generated in the decay process. For very light relic particles (mass smaller than a few MeV), measurements of light element abundances will not allow placing constraints, while the CMB spectrum should still be sensitive, assuming that the particle is abundant enough.

We also mention, that the Fisher estimates become crude, once the error reaches much more than ≃15–20 per cent. In this case, the likelihood becomes non-Gaussian and non-linear dependences are important. We also find that the solutions can be multimodal, with regions of low probability far away from the fiducial value. This means that MCMC sampling has to be performed in several steps, using wide priors to find regions of interest, followed by re-simulations around different maximum likelihood points. In this case, we refer to the methods developed in Chluba (2013a).

Comparing models using distortion eigenmodes

In the previous section, we presented parameter estimation cases using eigenmodes in a model-by-model basis. Furthermore, since each model has rather unique predictions for the observable μ_ks, the eigenmode analysis opens a new possibility of distinguishing different ERSs. In this section, we shall illustrate this point by some solid examples. First, let us assume that the time dependence of the energy release is fixed. In that case, the shape of eigenspectrum does not change and only the overall amplitude is free, and we can directly use the examples given in Table 2. If only μ can be constrained then different models cannot be distinguished unless some other constraint can be invoked. For example, finding μ ≃ 10⁻⁷ is unlikely to be caused by s-wave annihilation, which is bound to much smaller annihilation efficiencies by CMB anisotropy measurements. It could, however, be caused by a decaying relic particle or the dissipation of small-scale perturbations.

Once some of the μ_k, which directly probe the time dependence of the energy-release history, can be determined with signal-to-noise S/N > 1, one can in principle distinguish between different scenarios. For instance, from Table 2 the dissipation scenario with (n_S, n_run) = (1, 0) has a ρ-vector ρ_diss ≈ (0.161, 5.86 × 10⁻⁴, 4.31 × 10⁻³), while for the s-wave annihilation scenarios we find ρ_{ann, s} ≈ (0.159, 1.18 × 10⁻³, 4.17 × 10⁻³). Comparing the entries of these vectors indicates that the two cases are quasi-degenerate. The small differences stem from the late-time behaviour of |$\mathcal {Q}(z)$| at z ≲ 10⁴, but very high precision is indeed needed to discern them. In addition, by slightly adjusting n_S to ≃ 1.01 one can align these two ρ-vectors nearly perfectly. For the p-wave scenario, we find ρ_{ann, p} ≈ (2.33 × 10⁻², 4.37 × 10⁻³, 1.38 × 10⁻³), which clearly is different from the s-wave annihilation and scale-invariant dissipation scenarios. However, adjusting n_S ≃ 1.67 practically aligns ρ_diss with ρ_{ann, p}. This is expected since for n_run = 0 one has |$\mathcal {Q}_{\rm ac}\propto (1+z)^{3(n_{\rm S}-1)/2}$| (e.g. Chluba et al. 2012b), which becomes |$\mathcal {Q}_{\rm ac}\propto (1+z)$| for n_S ≃ 5/3. Similarly, we find ρ_diss ≈ (0.235, −8.07 × 10⁻³, 6.25 × 10⁻³) for (n_S, n_run) = (0.96, −0.02), which is distinguishable from the s- and p-wave annihilation scenario, if ρ₁ could be measured with ≃ 10 per cent precision. However, as soon as the dissipation model parameters are varied, degeneracies reappear, unless even higher experimental precision is achieved. Therefore, s-wave annihilation scenarios and quasi-scale-invariant dissipation scenarios with small running are observationally hard to distinguish using CMB SD data. Similarly, p-wave and dissipation scenarios with n_S ≃ 1.67 and small running are degenerate. For values of n_S ≠ {1, 1.67}, the degeneracies with annihilation scenarios are less severe and at high spectral sensitivity they in principle can be discerned.

For comparison of dissipation and decaying particle scenarios, let us consider the general case with all model parameters varying. As mentioned above, if only μ is measurable, no distinction can be made, unless priors are used (e.g. we assume that the primordial small-scale power spectrum is determined by extrapolation from large scales but find μ ≃ 10⁻⁷, which cannot be explained in this case). Assuming that μ and μ₁ are measurable, by comparing the upper panels of Figs 12 and 15, it is evident that due to freedom in the overall amplitude (A_ζ and ε_X can be re-scaled), dissipation and decaying particle scenarios again cannot be distinguish in a model-independent way (moving the curves left and right one can make then coincide).

The situation changes when also μ₂ can be measured. Fig. 17 shows the trajectories of dissipation and decaying particle scenarios in the ρ₁-ρ₂-plane for two spectral sensitivities. Assuming that the small-scale power spectrum is quasi-scale-invariant with small running a PIXIE-type experiment will already be able to directly distinguish this from a decaying particle scenario. Allowing large negative running does increase the degeneracy and higher spectral sensitivity is needed to discern these cases. The lower panel illustrates the improvement for five times the sensitivity of PIXIE. Clearly, measurements of μ₁ and μ₂ allow discerning dissipation and decaying particle scenarios over a wide range of the parameter space, with degeneracies appearing for large negative running (n_run ≪ −0.6) and in the limit of large and small spectral index.

Model-comparison for dissipation and decaying particle scenarios in the ρ1-ρ2 plane. We assumed Aζ(k0 = 45 Mpc−1) = 5 × 10−8 and ϵX = 1 eV. The upper panel is for PIXIE sensitivity, while the lower is for five times higher sensitivity.

Figure 17.

Model-comparison for dissipation and decaying particle scenarios in the ρ₁-ρ₂ plane. We assumed A_ζ(k₀ = 45 Mpc⁻¹) = 5 × 10⁻⁸ and ϵ_X = 1 eV. The upper panel is for PIXIE sensitivity, while the lower is for five times higher sensitivity.

Open in new tab Download slide

We mention, however, that when allowing more complex shapes of the small-scale power spectrum, e.g. with bumps caused by particle production during inflation (Chung et al. 2000; Barnaby & Huang 2009; Barnaby 2010), closer resemblance of the energy-release history with the one of a decaying particle can be achieved. In this case, a distinction of the two scenarios will be more challenging. Also, a combination of decaying particle and dissipation scenarios could be possible but would be hard to distinguish from the single scenarios. Nevertheless, CMB SD measurements provide a unique way to study different ERSs allowing direct model-comparisons and distinction in certain situations.

CONCLUSIONS

In this work, we derived a decomposition of the CMB SD signal into temperature shift, y, μ and residual distortion. The residual distortion was defined to be orthogonal to the temperature shift, y- and μ-distortion, taking experimental settings into account. Using this decomposition, we can explicitly show how much energy, at a given instance, is transferred to the various components of the CMB spectrum (Fig. 2). The y-distortion part of the CMB spectrum cannot be used in a model-independent way to learn about the primordial energy-release (occurring at z ≳ 10³), since it is degenerate with y-distortions introduced at later times, by reionization and the formation of structures. The μ-distortion component only provides a measure for the overall (integrated) energy release at |$z\simeq \rm few \times 10^{4}{\rm -}\rm few \times 10^{6}$|⁠, which can again only be interpreted in a model-dependent way. Adding the information in the residual distortion allows us to directly constraint the time-dependence of the energy-release history, and thus provides a way to discern different scenarios (see also, Chluba & Sunyaev 2012; Chluba 2013a).

We took a step forward towards the analysis of future CMB SD data. The information contained in the residual distortion can be compressed into a few numbers. This compression is achieved by performing a principal component analysis to determine residual-distortion and energy-release eigenmodes (see Section 3). It introduces a new set of distortion parameters, μ_k, which parametrize the shape of the residual distortion. We demonstrated that the eigenmodes can be used to simplify the analysis of future distortion data, providing a model-independent way to extract all useful information from the average CMB spectrum. Using this method, we discussed annihilating and decaying particle scenarios, as well as energy release caused by the dissipation of small-scale acoustic modes (corresponding to wave numbers 1 Mpc^{− 1} ≲ k ≲ few × 10⁴ Mpc^{− 1}) for different experimental sensitivities. We showed that future CMB SD measurements will allow direct detection of s-wave annihilation signals if the annihilation efficiency is p_{ann, s} ≳ 4.6 × 10⁻⁷ m³ kg⁻¹ s⁻² using PRISM. Detection limits for dissipation and decaying particle scenarios are shown in Figs 9 and 14, respectively.

CMB SD measurement provide a unique probe of the primordial small-scale power spectrum, which can be utilized to directly constraint inflationary models. Especially, if the small-scale power spectrum is close to scale-invariant with small running, very robust constraints can be expected from PIXIE and PRISM, if the amplitude of curvature perturbations is A_ζ(k₀ = 45 Mpc^{− 1}) ≃ 10^{− 8}-10^{− 7}. These conclusions are in good agreement with those of Chluba (2013a) where a MCMC analysis was used.

For decaying particle models with 2 × 10⁹ s ≲ t_X ≲ 8.3 × 10¹⁰ s and total energy release Δρ_γ/ρ_γ ≃ 6.4 × 10⁻⁷, the particle lifetime can be constrained to better than ≃ 20 per cent and ϵ_X could be measured with uncertainty ≲10 per cent using a PIXIE-type experiment with five times its sensitivity (see Fig. 15 for details). These findings are in good agreement with those of Chluba (2013a), where direct MCMC simulations were performed. CMB SD are thus a powerful probe of early-universe particle physics, providing tight limits that are independent and complementary to those derived from light element abundances (e.g. Kawasaki et al. 2005; Kohri & Takahashi 2010; Pospelov & Pradler 2010) and the CMB anisotropies (Chen & Kamionkowski 2004; Zhang et al. 2007; Giesen et al. 2012).

Finally, we demonstrated how the eigenmode decomposition of the residual distortion can be used for direct model comparison. The dissipation caused by a quasi-scale-invariant power spectrum gives rise to a distortion signature that is degenerate with an s-wave annihilation scenario (see also, Chluba & Sunyaev 2012; Chluba 2013a). However, a combination of future CMB anisotropy constraints with CMB SD measurements might provide the means to disentangle these cases. In particular, a detection of an annihilating particle signature with CMB anisotropy measurements could be independently confirmed using CMB SDs. Furthermore, decaying particle scenarios have distortion eigenspectra that are distinct from the one caused by the dissipation of small-scale acoustic modes, if the power spectrum is neither too blue nor too red and does not show too much negative running (see Fig. 17). This again demonstrates the potential of future CMB distortion measurements and we look forward to extending our method to include more realistic instrumental effects and foregrounds. The principal component decomposition can furthermore be used to determine the optimal experimental settings for the detection of different SDs signatures, another application we plan for the future.

The authors specially thank Yacine Ali-Haïmoud and the referee for insightful comments and suggestions. JC furthermore thanks Kazunori Kohri and Josef Pradler for useful discussions about particle physics scenarios. The authors also thank Rishi Khatri and Rashid Sunyaev for comments on the manuscript, and Silvia Galli for providing simulated covariance matrices for PRISM. Use of the GPC supercomputer at the SciNet HPC Consortium is acknowledged. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund – Research Excellence and the University of Toronto. This work was supported by DoE SC-0008108 and NASA NNX12AE86G.

1

Available at www.Chluba.de/CosmoTherm.

2

This name was coined by Chluba & Sunyaev (2012), but the original derivation (accounting for the effect of Bremsstrahlung) was given in Sunyaev & Zeldovich (1970). Danese & de Zotti (1982) also included the effect of double Compton emission (Lightman 1981; Thorne 1981; Chluba, Sazonov & Sunyaev 2007), and recent improvements to the shape of |$\mathcal {J}_{\rm bb}(z)$| were given by Khatri & Sunyaev (2012a), using semi-analytic approximations.

3

Excursion of the modulating (dihedral) mirror of the Fourier transform spectrometer around the zero-point.

4

At all times, the distortion has to be small compared to the CMB blackbody, since otherwise non-linear effects become important and the Green's function approach is inapplicable.

5

An alternative method is described in Khatri & Sunyaev (2012b).

6

Henceforth, we shall denote vectors with bold font.

7

The exact value of k_cut does not matter much, since it only affects the primordial y-distortion contribution, which we do explicitly not use constrain the underlying ERS.

8

f_{ann, s} ≡ 1.5 × 10⁻¹⁷ eV kg m⁻³ p_{ann, s}

9

In the particle physics community the abundance yield, Y_X = N_X/S, and deposited particle energy, E_vis [GeV], are commonly used. Here, N_X is the particle number density at t ≪ t_X and |$S=\frac{4}{3}\,\frac{\rho }{kT}\simeq 7\,N_\gamma \simeq 2.9 \times 10^{3}\,(1+z)^3\,{\rm cm^{-3}}$| denotes the total entropy density. We thus find ϵ_X ≡ (E_vis Y_X) 10⁹S/[N_H (1 + z_X)] ≃ 1.5 × 10¹⁹(E_vis Y_X)/(1 + z_X).

REFERENCES

Abazajian

K. N.

et al. ,

2013

preprint (arXiv:1309.5381)

Achúcarro

A.

Gong

J.-O.

Hardeman

S.

Palma

G. A.

Patil

S. P.

,

J. Cosmol. Astropart. Phys.

,

2011

, vol.

1

pg.

30

Crossref

Search ADS

André

P.

et al. ,

2013

preprint (arXiv:1310.1554)

Barnaby

N.

,

Phys. Rev. D

,

2010

, vol.

82

pg.

106009

Crossref

Search ADS

Barnaby

N.

Huang

Z.

,

Phys. Rev. D

,

2009

, vol.

80

pg.

126018

Crossref

Search ADS

Barnaby

N.

Huang

Z.

Kofman

L.

Pogosyan

D.

,

Phys. Rev. D

,

2009

, vol.

80

pg.

043501

Crossref

Search ADS

Barrow

J. D.

Coles

P.

,

MNRAS

,

1991

, vol.

248

pg.

52

Crossref

Search ADS

Ben-Dayan

I.

Brustein

R.

,

J. Cosmol. Astropart. Phys.

,

2010

, vol.

9

pg.

7

Crossref

Search ADS

Biagetti

M.

Perrier

H.

Riotto

A.

Desjacques

V.

,

Phys. Rev. D

,

2013

, vol.

87

pg.

063521

Crossref

Search ADS

Brax

P.

Burrage

C.

Davis

A.-C.

Gubitosi

G.

,

J. Cosmol. Astropart. Phys.

,

2013

, vol.

11

pg.

1

Crossref

Search ADS

Bringmann

T.

Scott

P.

Akrami

Y.

,

Phys. Rev. D

,

2012

, vol.

85

pg.

125027

Crossref

Search ADS

Bull

P.

Kamionkowski

M.

,

Phys. Rev. D

,

2013

, vol.

87

pg.

081301

Crossref

Search ADS

Burigana

C.

Danese

L.

de Zotti

G.

,

A&A

,

1991

, vol.

246

pg.

49

Carr

B. J.

Kohri

K.

Sendouda

Y.

Yokoyama

J.

,

Phys. Rev. D

,

2010

, vol.

81

pg.

104019

Crossref

Search ADS

Cen

R.

Ostriker

J. P.

,

ApJ

,

1999

, vol.

514

pg.

1

Crossref

Search ADS

Céspedes

S.

Atal

V.

Palma

G. A.

,

J. Cosmol. Astropart. Phys.

,

2012

, vol.

5

pg.

8

Crossref

Search ADS

Chen

X.

Kamionkowski

M.

,

Phys. Rev. D

,

2004

, vol.

70

pg.

043502

Crossref

Search ADS

Chen

J.

Zhou

Y.-F.

,

J. Cosmol. Astropart. Phys.

,

2013

, vol.

4

pg.

17

Crossref

Search ADS

Chluba

J.

,

MNRAS

,

2013a

, vol.

436

pg.

2232

Crossref

Search ADS

Chluba

J.

,

MNRAS

,

2013b

, vol.

434

pg.

352

Crossref

Search ADS

Chluba

J.

Grin

D.

,

MNRAS

,

2013

, vol.

434

pg.

1619

Crossref

Search ADS

Chluba

J.

Sunyaev

R. A.

,

A&A

,

2004

, vol.

424

pg.

389

Crossref

Search ADS

Chluba

J.

Sunyaev

R. A.

,

MNRAS

,

2012

, vol.

419

pg.

1294

Crossref

Search ADS

Chluba

J.

Sazonov

S. Y.

Sunyaev

R. A.

,

A&A

,

2007

, vol.

468

pg.

785

Crossref

Search ADS

Chluba

J.

Khatri

R.

Sunyaev

R. A.

,

MNRAS

,

2012a

, vol.

425

pg.

1129

Crossref

Search ADS

Chluba

J.

Erickcek

A. L.

Ben-Dayan

I.

,

ApJ

,

2012b

, vol.

758

pg.

76

Crossref

Search ADS

Chung

D. J. H.

Kolb

E. W.

Riotto

A.

Tkachev

I. I.

,

Phys. Rev. D

,

2000

, vol.

62

pg.

043508

Crossref

Search ADS

Copeland

E. J.

Liddle

A. R.

Lidsey

J. E.

Wands

D.

,

Phys. Rev. D

,

1998

, vol.

58

pg.

063508

Crossref

Search ADS

Daly

R. A.

,

ApJ

,

1991

, vol.

371

pg.

14

Crossref

Search ADS

Danese

L.

de Zotti

G.

,

Nuovo Cimento Rivista Ser.

,

1977

, vol.

7

pg.

277

Crossref

Search ADS

Danese

L.

de Zotti

G.

,

A&A

,

1982

, vol.

107

pg.

39

Dent

J. B.

Easson

D. A.

Tashiro

H.

,

Phys. Rev. D

,

2012

, vol.

86

pg.

023514

Crossref

Search ADS

Dunkley

J.

et al. ,

ApJ

,

2011

, vol.

739

pg.

52

Crossref

Search ADS

Farhang

M.

Bond

J. R.

Chluba

J.

,

ApJ

,

2012

, vol.

752

pg.

88

Crossref

Search ADS

Feng

J. L.

,

ARA&A

,

2010

, vol.

48

pg.

495

Crossref

Search ADS

Finkbeiner

D. P.

Weiner

N.

,

Phys. Rev. D

,

2007

, vol.

76

pg.

083519

Crossref

Search ADS

Finkbeiner

D. P.

Galli

S.

Lin

T.

Slatyer

T. R.

,

Phys. Rev. D

,

2012

, vol.

85

pg.

043522

Crossref

Search ADS

Fixsen

D. J.

,

ApJ

,

2009

, vol.

707

pg.

916

Crossref

Search ADS

Fixsen

D. J.

Mather

J. C.

,

ApJ

,

2002

, vol.

581

pg.

817

Crossref

Search ADS

Fixsen

D. J.

Cheng

E. S.

Gales

J. M.

Mather

J. C.

Shafer

R. A.

Wright

E. L.

,

ApJ

,

1996

, vol.

473

pg.

576

Crossref

Search ADS

Galli

S.

Iocco

F.

Bertone

G.

Melchiorri

A.

,

Phys. Rev. D

,

2009

, vol.

80

pg.

023505

Crossref

Search ADS

Galli

S.

Slatyer

T. R.

Valdes

M.

Iocco

F.

,

Phys. Rev. D

,

2013

, vol.

88

pg.

063502

Crossref

Search ADS

Ganc

J.

Komatsu

E.

,

Phys. Rev. D

,

2012

, vol.

86

pg.

023518

Crossref

Search ADS

Giesen

G.

Lesgourgues

J.

Audren

B.

Ali-Haïmoud

Y.

,

J. Cosmol. Astropart. Phys.

,

2012

, vol.

12

pg.

8

Crossref

Search ADS

Hannestad

S.

Tram

T.

,

J. Cosmol. Astropart. Phys.

,

2011

, vol.

1

pg.

16

Crossref

Search ADS

Ho

C. M.

Scherrer

R. J.

,

Phys. Rev. D

,

2013

, vol.

87

pg.

065016

Crossref

Search ADS

Hu

W.

Silk

J.

,

Phys. Rev. D

,

1993a

, vol.

48

pg.

485

Crossref

Search ADS

Hu

W.

Silk

J.

,

Phys. Rev. Lett.

,

1993b

, vol.

70

pg.

2661

Crossref

Search ADS

Hu

W.

Sugiyama

N.

,

ApJ

,

1994

, vol.

436

pg.

456

Crossref

Search ADS

Hu

W.

Scott

D.

Silk

J.

,

ApJ

,

1994a

, vol.

430

pg.

L5

Crossref

Search ADS

Hu

W.

Scott

D.

Silk

J.

,

Phys. Rev. D

,

1994b

, vol.

49

pg.

648

Crossref

Search ADS

Hunt

P.

Sarkar

S.

,

Phys. Rev. D

,

2007

, vol.

76

pg.

123504

Crossref

Search ADS

Hütsi

G.

Hektor

A.

Raidal

M.

,

A&A

,

2009

, vol.

505

pg.

999

Crossref

Search ADS

Hütsi

G.

Chluba

J.

Hektor

A.

Raidal

M.

,

A&A

,

2011

, vol.

535

pg.

A26

Crossref

Search ADS

Illarionov

A. F.

Sunyaev

R. A.

,

Sov. Astron.

,

1975a

, vol.

18

pg.

413

Illarionov

A. F.

Sunyaev

R. A.

,

Sov. Astron.

,

1975b

, vol.

18

pg.

691

Ivanov

P.

Naselsky

P.

Novikov

I.

,

Phys. Rev. D

,

1994

, vol.

50

pg.

7173

Crossref

Search ADS

Jedamzik

K.

Katalinić

V.

Olinto

A. V.

,

Phys. Rev. Lett.

,

2000

, vol.

85

pg.

700

Crossref

Search ADS

PubMed

Joy

M.

Sahni

V.

Starobinsky

A. A.

,

Phys. Rev. D

,

2008

, vol.

77

pg.

023514

Crossref

Search ADS

Kawasaki

M.

Kohri

K.

Moroi

T.

,

Phys. Rev. D

,

2005

, vol.

71

pg.

083502

Crossref

Search ADS

Keisler

R.

et al. ,

ApJ

,

2011

, vol.

743

pg.

28

Crossref

Search ADS

Khatri

R.

Sunyaev

R. A.

,

J. Cosmol. Astropart. Phys.

,

2012a

, vol.

6

pg.

38

Crossref

Search ADS

Khatri

R.

Sunyaev

R. A.

,

J. Cosmol. Astropart. Phys.

,

2012b

, vol.

9

pg.

16

Crossref

Search ADS

Khatri

R.

Sunyaev

R. A.

,

J. Cosmol. Astropart. Phys.

,

2013

, vol.

6

pg.

26

Crossref

Search ADS

Kogut

A.

et al. ,

New Astron. Rev.

,

2006

, vol.

50

pg.

925

Crossref

Search ADS

Kogut

A.

et al. ,

J. Cosmol. Astropart. Phys.

,

2011

, vol.

7

pg.

25

Crossref

Search ADS

Kohri

K.

Takahashi

T.

,

Phys. Lett. B

,

2010

, vol.

682

pg.

337

Crossref

Search ADS

Komatsu

E.

et al. ,

ApJS

,

2011

, vol.

192

pg.

18

Crossref

Search ADS

Lightman

A. P.

,

ApJ

,

1981

, vol.

244

pg.

392

Crossref

Search ADS

Lochan

K.

Das

S.

Bassi

A.

,

Phys. Rev. D

,

2012

, vol.

86

pg.

065016

Crossref

Search ADS

Mather

J. C.

et al. ,

ApJ

,

1994

, vol.

420

pg.

439

Crossref

Search ADS

McDonald

P.

Scherrer

R. J.

Walker

T. P.

,

Phys. Rev. D

,

2001

, vol.

63

pg.

023001

Crossref

Search ADS

Miniati

F.

Ryu

D.

Kang

H.

Jones

T. W.

Cen

R.

Ostriker

J. P.

,

ApJ

,

2000

, vol.

542

pg.

608

Crossref

Search ADS

Mortonson

M. J.

Hu

W.

,

ApJ

,

2008

, vol.

672

pg.

737

Crossref

Search ADS

Oh

S. P.

Cooray

A.

Kamionkowski

M.

,

MNRAS

,

2003

, vol.

342

pg.

L20

Crossref

Search ADS

Ostriker

J. P.

Thompson

C.

,

ApJ

,

1987

, vol.

323

pg.

L97

Crossref

Search ADS

Pajer

E.

Zaldarriaga

M.

,

Phys. Rev. Lett.

,

2012

, vol.

109

pg.

021302

Crossref

Search ADS

PubMed

Patterson

T. N. L.

,

Math. Comput.

,

1968

, vol.

22

pg.

847

Crossref

Search ADS

Planck Collaboration

, et al. ,

2013a

preprint (arXiv:1303.5073)

Planck Collaboration

, et al. ,

2013b

preprint (arXiv:1303.5072)

Planck Collaboration

, et al. ,

2013c

preprint (arXiv:1303.5074)

Planck Collaboration

, et al. ,

2013d

preprint (arXiv:1303.5076)

Planck Collaboration

, et al. ,

2013e

preprint (arXiv:1303.5078)

Planck Collaboration

, et al. ,

2013f

preprint (arXiv:1303.5081)

Pospelov

M.

Pradler

J.

,

Ann. Rev. Nucl. Part. Sci.

,

2010

, vol.

60

pg.

539

Crossref

Search ADS

Pospelov

M.

Ritz

A.

,

Phys. Lett. B

,

2007

, vol.

651

pg.

208

Crossref

Search ADS

Powell

B. A.

,

2012

preprint (arXiv:1209.2024)

PRISM Collaboration

, et al. ,

2013

preprint (arXiv:1306.2259)

Randall

L.

Soljačić

M.

Guth

A. H.

,

Nucl. Phys. B

,

1996

, vol.

472

pg.

377

Crossref

Search ADS

Refregier

A.

Komatsu

E.

Spergel

D. N.

Pen

U.-L.

,

Phys. Rev. D

,

2000

, vol.

61

pg.

123001

Crossref

Search ADS

Salopek

D. S.

Bond

J. R.

Bardeen

J. M.

,

Phys. Rev. D

,

1989

, vol.

40

pg.

1753

Crossref

Search ADS

Scott

P.

Bringmann

T.

Akrami

Y.

,

J. Phys.: Conf. Ser.

,

2012

, vol.

375

pg.

032012

Crossref

Search ADS

Seiffert

M.

et al. ,

ApJ

,

2011

, vol.

734

pg.

6

Crossref

Search ADS

Shaw

J. R.

Sigurdson

K.

Pen

U.-L.

Stebbins

A.

Sitwell

M.

,

2013

preprint (arXiv:1302.0327)

Slatyer

T. R.

Padmanabhan

N.

Finkbeiner

D. P.

,

Phys. Rev. D

,

2009

, vol.

80

pg.

043526

Crossref

Search ADS

Starobinskij

A. A.

,

Sov. J. Exp. Theor. Phys. Lett.

,

1992

, vol.

55

pg.

489

Starobinsky

A. A.

,

Gravit. Cosmol.

,

1998

, vol.

4

pg.

88

Stewart

E. D.

,

Phys. Rev. D

,

1997

, vol.

56

pg.

2019

Crossref

Search ADS

Sunyaev

R. A.

Khatri

R.

,

Int. J Mod. Phys. D

,

2013

, vol.

22

pg.

30014

Crossref

Search ADS

Sunyaev

R. A.

Zeldovich

Y. B.

,

Ap&SS

,

1970

, vol.

7

pg.

20

Sunyaev

R. A.

Zeldovich

Y. B.

,

A&A

,

1972

, vol.

20

pg.

189

Tashiro

H.

Sabancilar

E.

Vachaspati

T.

,

Phys. Rev. D

,

2012

, vol.

85

pg.

103522

Crossref

Search ADS

Tashiro

H.

Sabancilar

E.

Vachaspati

T.

,

J. Cosmol. Astropart. Phys.

,

2013

, vol.

8

pg.

35

Crossref

Search ADS

Thorne

K. S.

,

MNRAS

,

1981

, vol.

194

pg.

439

Crossref

Search ADS

Zannoni

M.

Tartari

A.

Gervasi

M.

Boella

G.

Sironi

G.

De Lucia

A.

Passerini

A.

Cavaliere

F.

,

ApJ

,

2008

, vol.

688

pg.

12

Crossref

Search ADS

Zeldovich

Y. B.

Sunyaev

R. A.

,

Ap&SS

,

1969

, vol.

4

pg.

301

Crossref

Search ADS

Zeldovich

Y. B.

Illarionov

A. F.

Syunyaev

R. A.

,

Sov. J. Exp. Theor. Phys.

,

1972

, vol.

35

pg.

643

Zhang

P.

Pen

U.-L.

Trac

H.

,

MNRAS

,

2004

, vol.

355

pg.

451

Crossref

Search ADS

Zhang

L.

Chen

X.

Kamionkowski

M.

Si

Z.

Zheng

Z.

,

Phys. Rev. D

,

2007

, vol.

76

pg.

061301

Crossref

Search ADS

APPENDIX A: ORTHOGONAL BASIS

To define the residual distortion, |${{\bf R}}(z)$|⁠, that is perpendicular to the space spanned by |${{\bf G}}_{\rm T}$|⁠, |${{\bf Y}}_{\rm SZ}$| and |${{\bf M}}$|⁠, we simply follow the Gram–Schmidt orthogonalization procedure. Aligning one axis with |${{\bf Y}}_{\rm SZ}$|⁠, this space is given by the orthonormal basis |${{\bf e}}_{y}= {{\bf Y}}_{\rm SZ}/ |{{\bf Y}}_{\rm SZ}|$|⁠, |${{\bf e}}_{\mu }={{\bf M}}_\perp /|{{\bf M}}_\perp |$| and |${{\bf e}}_{\rm T}={{\bf G}}_{\rm T, \perp }/|{{\bf G}}_{\rm T, \perp }|$|⁠, where |${{\bf M}}_\perp = {{\bf M}}-M_y\,{{\bf e}}_{y}$| and |${{\bf G}}_{\rm T, \perp } ={{\bf G}}_{\rm T}-G_y \,{{\bf e}}_y-G_\mu \,{{\bf e}}_{\mu }$|⁠. With |${{\bf a}}\cdot {{\bf b}}=\sum _i a_i\,b_i$|⁠, the required projections are |$M_y={{\bf e}}_{y}\cdot {{\bf M}}$|⁠, |$G_y={{\bf e}}_{y}\cdot {{\bf G}}_{\rm T}$| and |$G_\mu ={{\bf e}}_{\mu }\cdot {{\bf G}}_{\rm T}$|⁠. Assuming PIXIE-type settings ({ν_min, ν_max, Δν_s} = {30, 1000, 15} GHz), we find |$\lbrace |{{\bf Y}}_{\rm SZ}|, |{{\bf M}}_\perp |, |{{\bf G}}_{\rm T, \perp }|\rbrace \simeq \lbrace 73.3, 7.99, 21.4\rbrace \,10^{-18}\,{\rm W\,m^{-2}\,Hz^{-1}\,sr^{-1}}$| and {M_y, G_y, G_μ} ≃ {7.66, 16.8, 41.5} 10⁻¹⁸ W m⁻² Hz⁻¹ sr⁻¹. From equation (4), we furthermore obtain

\begin{eqnarray} {{\bf R}}(z)&=&{{\bf G}}_{\rm th}(z) - {{\bf G}}_{\rm T}\,\mathcal {J}_{T}(z)/4 - {{\bf Y}}_{\rm SZ} \,\mathcal {J}_{y}(z)/4 - \alpha \,{{\bf M}} \,\mathcal {J}_{\mu }(z) \nonumber \\ \mathcal {J}_{T}(z) &=& 4 \,{{\bf e}}_{\rm T}\cdot {{\bf G}}_{\rm th}(z)/|{{\bf G}}_{\rm T, \perp }| \nonumber \\ \mathcal {J}_{\mu }(z) &=& \alpha ^{-1}\,[{{\bf e}}_{\mu }\cdot {{\bf G}}_{\rm th}(z)-G_\mu \,\mathcal {J}_{T}(z)/4]/|{{\bf M}}_{\perp }| \nonumber \\ \mathcal {J}_{y}(z) &=& 4\,[{{\bf e}}_y\cdot {{\bf G}}_{\rm th}(z)-\alpha \,M_y\,\mathcal {J}_{\mu }(z) -G_y\,\mathcal {J}_{T}(z)/4]/|{{\bf Y}}_{\rm SZ}| \nonumber \\ \mathcal {J}_{R}(z) &=& 1-\mathcal {J}_{T}(z)-\mathcal {J}_{y}(z) - \mathcal {J}_{\mu }(z), \end{eqnarray}

(A1)

where we also introduced |$\mathcal {J}_{R}(z)$|⁠, which determines the amount of energy found in the residual distortion. All |$\mathcal {J}_{k}(z)$| are illustrated in Fig. 2.

Download all slides

Month:	Total Views:
November 2016	1
December 2016	2
January 2017	2
February 2017	9
March 2017	1
April 2017	2
June 2017	4
July 2017	2
August 2017	6
September 2017	3
October 2017	1
December 2017	7
January 2018	12
February 2018	19
March 2018	15
April 2018	15
May 2018	20
June 2018	13
July 2018	22
August 2018	20
September 2018	7
October 2018	10
November 2018	20
December 2018	14
January 2019	9
February 2019	14
March 2019	10
April 2019	22
May 2019	17
June 2019	16
July 2019	13
August 2019	12
September 2019	17
October 2019	7
November 2019	12
December 2019	9
January 2020	8
February 2020	6
March 2020	8
April 2020	8
May 2020	2
June 2020	6
July 2020	9
August 2020	4
September 2020	8
October 2020	11
November 2020	7
December 2020	7
January 2021	8
February 2021	4
March 2021	13
April 2021	6
May 2021	11
June 2021	1
July 2021	10
August 2021	1
September 2021	7
October 2021	8
November 2021	16
December 2021	14
January 2022	6
February 2022	10
March 2022	14
April 2022	11
May 2022	14
June 2022	17
July 2022	22
August 2022	22
September 2022	27
October 2022	5
November 2022	40
December 2022	1
January 2023	27
February 2023	2
March 2023	9
April 2023	11
May 2023	1
June 2023	4
August 2023	7
September 2023	16
October 2023	4
November 2023	9
December 2023	17
January 2024	7
February 2024	14
March 2024	5
April 2024	22
May 2024	10
June 2024	7
July 2024	22
August 2024	2
September 2024	8
October 2024	11
November 2024	7
December 2024	3
January 2025	1
February 2025	10
March 2025	14
April 2025	24
May 2025	4

Article Contents

Teasing bits of information out of the CMB energy spectrum

Abstract

INTRODUCTION

QUASI-ORTHOGONAL DECOMPOSITION OF THE THERMALIZATION GREEN'S FUNCTION

Instrumental aspects

Defining the residual distortion

Dependence on experimental settings

Energy release and branching ratios

PRINCIPAL COMPONENT DECOMPOSITION FOR THE RESIDUAL DISTORTION

Results for the eigenvectors and eigenvalues

PARAMETER ESTIMATION USING ENERGY-RELEASE EIGENMODES

Errors of Δ_T, y* and μ

Simple parameter estimation example: proof of concept

Partial recovery of the energy-release history

Overall picture and how to apply the eigenmodes

CONSTRAINTS ON DIFFERENT SCENARIOS AND MODEL COMPARISON

Parametrization of the energy-release scenarios

Shape of the distortion signal

Annihilating particles

Dissipation of small-scale acoustic modes

Decaying relic particles

Detectability of the distortion signal

Annihilating particles

Dissipation of small-scale acoustic modes

Dissipation of small-scale acoustic modes: generalization

Decaying relic particles

Comparing models using distortion eigenmodes

CONCLUSIONS

REFERENCES

APPENDIX A: ORTHOGONAL BASIS

Citations

Views

Altmetric

Email alerts

Astrophysics Data System

Citing articles via

Latest

Most Read

Most Cited

Article Contents

Teasing bits of information out of the CMB energy spectrum

Abstract

INTRODUCTION

QUASI-ORTHOGONAL DECOMPOSITION OF THE THERMALIZATION GREEN'S FUNCTION

Instrumental aspects

Defining the residual distortion

Dependence on experimental settings

Energy release and branching ratios

PRINCIPAL COMPONENT DECOMPOSITION FOR THE RESIDUAL DISTORTION

Results for the eigenvectors and eigenvalues

PARAMETER ESTIMATION USING ENERGY-RELEASE EIGENMODES

Errors of ΔT, y* and μ

Simple parameter estimation example: proof of concept

Partial recovery of the energy-release history

Overall picture and how to apply the eigenmodes

CONSTRAINTS ON DIFFERENT SCENARIOS AND MODEL COMPARISON

Parametrization of the energy-release scenarios

Shape of the distortion signal

Annihilating particles

Dissipation of small-scale acoustic modes

Decaying relic particles

Detectability of the distortion signal

Annihilating particles

Dissipation of small-scale acoustic modes

Dissipation of small-scale acoustic modes: generalization

Decaying relic particles

Comparing models using distortion eigenmodes

CONCLUSIONS

REFERENCES

APPENDIX A: ORTHOGONAL BASIS

Citations

Views

Altmetric

Email alerts

Astrophysics Data System

Citing articles via

Latest

Most Read

Most Cited

This Feature Is Available To Subscribers Only

Errors of Δ_T, y* and μ