Searching for dark matter in the CMB: A compact parametrization of energy injection from new physics

High-precision measurements of the temperature and polarization anisotropies of the cosmic microwave background radiation have been employed to set robust constraints on dark matter annihilation during recombination. In this work we improve and generalize these constraints to apply to energy deposition during the recombination era with arbitrary redshift dependence. Our approach also provides more rigorous and model-independent bounds on dark matter annihilation and decay scenarios. We employ principal component analysis to identify a basis of weighting functions for the energy deposition. The coefficients of these weighting functions parameterize any energy deposition model and can be constrained directly by experiment. For generic energy deposition histories that are currently allowed by WMAP 7 data, up to 3 principal component coefficients are measurable by Planck and up to 5 coefficients are measurable by an ideal cosmic variance limited experiment. For WIMP dark matter, our analysis demonstrates that the effect on the CMB is described well by a single (normalization) parameter and a universal redshift dependence for the energy deposition history. We give WMAP 7 constraints on both generic energy deposition histories and the universal WIMP case.

Figure 1

Rate of Hydrogen ionization from energy deposition, relative to the number density of ionized Hydrogen ($n_{\mathrm{ion}}^0$) when there is no energy deposition. The lines shown are the cases of constant $p_{\mathrm{ann}}$ and $p_{\mathrm{dec}}$, corresponding to on-the-spot energy deposition from dark matter annihilation and dark matter decay, respectively.

Figure 2

The effect of the number of included $\ell$’s, and the number of included frequency bands, on the constraint on a constant-$p_{\mathrm{ann}}$ energy deposition history; here we show the value of $p_{\mathrm{ann}}$ corresponding to a $2\sigma$ signal.

Figure 3

The degree of nonlinearity in the computed significance of a sample energy deposition history, for $p_{\mathrm{ann}}$ constant, using WMAP 7 noise parameters. We show the ratio of (1) the $S/N$ estimated by a linear extrapolation from small energy deposition to (2) the true $S/N$ (estimated as in Sec. 2c), as a function of $p_{\mathrm{ann}}$. The solid, dashed and dotted lines indicate the WMAP 7 $2\sigma$ upper limit on $p_{\mathrm{ann}}$, the value of $p_{\mathrm{ann}}$ for which the nonlinearity is 10%, and the value for which the nonlinearity is 1%, respectively. The red dot-dashed line indicates the $2\sigma$ upper limit on $p_{\mathrm{ann}}$ that would be obtained by linearly extrapolating the significance from small energy deposition, which overestimates the significance and hence leads to a too-strong constraint.

Figure 4

The first three principal components for WMAP 7, Planck and a CVL experiment, both before and after marginalization over the cosmological parameters.

Figure 5

The first six principal components for Planck after marginalization, in the case of (left) annihilationlike redshift dependence with linear binning, (center) annihilationlike redshift dependence with log binning, and (right) decaylike redshift dependence with log binning. Note that for decaylike energy deposition histories, the redshift range is extended down to $z=10$ in order to fully capture the effect on the CMB-see Sec. 2. This larger redshift range makes linear binning impractical.

Figure 6

Fractional change to the ionization fraction $x_e$ in the presence of energy deposition, for the first three (marginalized) principal components in Planck. The curve shown is extrapolated from the linear (small energy deposition) regime, with normalization factor ${\epsilon }_{1,2,3}=2\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$.

Figure 7

The mapping of the first three principal components for Planck, after marginalization, into $\delta C_{\ell }$ space. The PCs are multiplied by ${\epsilon }_i(z)=2\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$ for all $i$, to fix the normalization of the $\delta C_{\ell }$’s.

Figure 8

The $\perp$ components of the first three principal components for Planck, after marginalization, mapped into $\delta C_{\ell }$ space. The normalization is the same as for Fig. 7.

Figure 9

Decomposition of $\delta C_{\ell }$ from energy deposition with constant $p_{\mathrm{ann}}(z)$ into parallel ($||$) components which can be absorbed by changes in the cosmological parameters, and perpendicular ($\perp$) components that cannot be absorbed by such changes. The overall effect of the energy deposition is suppression of high-$\ell$ modes, due to the increased optical depth, and enhancement of low-$\ell$ polarization modes, as discussed in 8. The suppression at high $\ell$ is clearly seen in the TT and EE spectra; the effect is also present in the TE spectra, with the peaks of $\delta C_{\ell }^{\mathrm{TE}}$ occurring at the troughs of $C_{\ell }^{\mathrm{TE}}$, and vice versa. The normalization here is $p_{\mathrm{ann}}=2\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$, comparable to the latest limits from $WMAP7+\mathrm{ACT}$ [14]. This decomposition depends on the sensitivity of the experiment; the case shown is WMAP 7 single band.

Figure 10

(a) The sensitivity for Planck (single-band), after marginalization, for various models subject to constraints from WMAP 7 single-band at $2\sigma$. The left figure assumes annihilationlike energy deposition and the right figure assumes decaylike energy deposition. The top panels show: (1) assuming $p(z)\propto e_i(z)$ for each PC, (2) the generic case where all PC coefficients have equal magnitudes $|{\epsilon }_i|=\epsilon =2/\sqrt{\sum _i{\lambda }_i^{WMAP}}$, (3) constant $p(z)$, and (4) taking $p(z)\propto f(z)$, with $f(z)$ from the models in 10. For the left figure, the hatched region indicates the range of results from changing the ionization history calculator and including or neglecting the effects of helium and Lyman-$\alpha$ photons, described in Sec. 1c, 1d. The bottom panels show some sample $z_{\tau }$ models for asymmetric annihilating dark matter (left) and decaying species (right), as discussed in Sec. 4b (the labels describe the initial particle mass, and the SM final state for annihilation or decay), and an extreme case where $p(z)=0$ for $200<z<900$ and constant outside that range. (b) Same as (a), but for a CVL experiment. (c) The models in (a) and (b) for annihilationlike (left) and decaylike (right).

Figure 11

For the $i$th PC, the contribution to the bias to cosmological parameters in WMAP 7 (left panel) and Planck (right panel), relative to the error bars forecast from the Fisher matrix. The normalization is that of the generic case (see discussion in Sec. 4b or Fig. 10), where each PC coefficient has the same absolute value and the overall normalization is the maximum allowed by WMAP 7 at $2\sigma$. The total bias for the parameter $\theta$ is $\sum _i\delta {\theta }_i$.

Figure 12

The universal $e_{\mathrm{WIMP}}$ curve (solid black line), normalized as discussed in the text. Note that the principal component analysis is only performed in the redshift range between the vertical red solid lines; outside these lines, we still plot the linear combination of the input energy deposition histories that has been identified as the first principal component, to serve as a canonical energy deposition history for WIMP annihilation, but the $C_{\ell }$’s are not sensitive to the details of this energy deposition. In the PCA region we also plot the second (dashed) and third (dotted) principal components, with arbitrary normalization.

Figure 13

Fractional change to the ionization fraction $x_e$ in the presence of energy deposition, for the universal $e_{\mathrm{WIMP}}$ curve (red, solid). We also write the $e_{\mathrm{WIMP}}(z)$ curve as a linear combination of the Planck principal components, and show (in black) the effect on the ionization history of the first three principal components individually (weighted by their contribution to $e_{\mathrm{WIMP}}$), and their (weighted) sum. The curves shown are extrapolated from the linear (small energy deposition) regime, with normalization factor $\epsilon =2\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$ for the $e_{\mathrm{WIMP}}$ curve.

Figure 14

Constraints from the seven-year WMAP data (red), and from simulated data for Planck (blue) and a cosmic variance limited experiment (green). The plot shows marginalized one-dimensional distributions and two-dimensional 68% and 95% limits. The mock data for Planck and the CVL experiment assumed no dark matter annihilation. Three Principal Components were used in each run to model the energy deposition from dark matter annihilation. The units of the PC coefficients here are in ${\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}$, with $1\times {10}^{-6}{\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}=1.8\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$.

Figure 15

Bias on the cosmological parameters using models with $\Lambda \mathrm{CDM}+n$ principal components for Planck (left panel) and CVL (right panel) simulated data. Here, the bias is defined as $({\theta }_i-{\theta }_i^{fid})/\sigma ({\theta }_i)$, where ${\theta }_i$ is the value of the mean value of the parameter from CosmoMC, ${\theta }_i^{fid}$ is the WMAP 7 marginalized value used as fiducial for the mock data, and $\sigma ({\theta }_i)$ is $1\mathrm{\text{−}}\sigma$ error bound from the runs. Both sets of mock data assumed a constant-$p_{\mathrm{ann}}(z)$ energy deposition history with $p_{\mathrm{ann}}=1\times {10}^{-6}{\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}=1.8\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$. Note that this plot should not be directly compared to Fig. 11 (which shows the bias per unit-normalized PC, rather than the total remaining bias after a certain number of PCs have been reconstructed from the fit).

Figure 16

Constraints from simulated data for Planck on $\Lambda \mathrm{CDM}$ parameters $+$ 0 principal components (green), $\Lambda \mathrm{CDM}+1$ PCs (magenta), $\Lambda \mathrm{CDM}+3$ PCs (blue), $\Lambda \mathrm{CDM}+5$ PCs (red). The plot shows marginalized one-dimensional distributions and two-dimensional 68% and 95% limits. The mock data for Planck assumed for the solid lines includes energy deposition with constant $p_{\mathrm{ann}}=1\times {10}^{-6}{\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}=1.8\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$. The grey area shows the case of a mock data with no energy injection and a model $\Lambda \mathrm{CDM}+0$ principal components. Only the cosmological parameters are shown.

Figure 17

Constraints from simulated data for Planck on $\Lambda \mathrm{CDM}+1$ PCs (magenta), $\Lambda \mathrm{CDM}+3$ PCs (blue), $\Lambda \mathrm{CDM}+5$ PCs (red). The plot shows marginalized one-dimensional distributions and two-dimensional 68% and 95% limits. The mock data for Planck assumed for the solid lines includes energy deposition with constant $p_{\mathrm{ann}}=1\times {10}^{-6}{\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}=1.8\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$, with the PC coefficients shown in units of ${\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}$. Only the principal components and $n_s$ are shown.

Figure 18

Constraints from simulated data for a cosmic variance limited experiment on $\Lambda \mathrm{CDM}+0$ principal components (green), $\Lambda \mathrm{CDM}+1$ PCs (magenta), $\Lambda \mathrm{CDM}+3$ PCs (blue), $\Lambda \mathrm{CDM}+5$ PCs (red), $\Lambda \mathrm{CDM}+7$ PCs (black). The plot shows marginalized one-dimensional distributions and two-dimensional 68% and 95% limits. The mock data for the CVL experiment assumed for the solid lines includes an energy deposition history with constant $p_{\mathrm{ann}}=1\times {10}^{-6}{\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}=1.8\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$. The grey area shows the case of a mock data with no energy injection and a model $\Lambda \mathrm{CDM}+0$ principal components. Only the cosmological parameters are shown.

Figure 19

Constraints from simulated data for a cosmic variance limited experiment on $\Lambda \mathrm{CDM}+1$ PCs (magenta), $\Lambda \mathrm{CDM}+3$ PCs (blue), $\Lambda \mathrm{CDM}+5$ PCs (red), $\Lambda \mathrm{CDM}+7$ PCs (black). The plot shows marginalized one-dimensional distributions and two-dimensional 68% and 95% limits. The mock data for the CVL experiment assumed for the solid lines includes an energy deposition history with constant $p_{\mathrm{ann}}=1\times {10}^{-6}{\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}=1.8\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$, with the PC coefficients shown in units of ${\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}$. Only the principal components and $n_s$ are shown.

Figure 20

Error bars for the cosmological parameters and coefficients of the principal components, in mock Planck data, simulated assuming a constant-$p_{\mathrm{ann}}(z)$ energy deposition history. The points marked “large $p_{\mathrm{ann}}$” have $p_{\mathrm{ann}}=1\times {10}^{-6}{\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}=1.8\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$, while for the points marked “small $p_{\mathrm{ann}}$” the value is a factor of 10 lower. In the left panel the fit to the data is performed using only the standard six cosmological parameters, whereas in the center panel and right panel three PCs are also included in the fit. For ease of comparison, the green triangles in the center panel reproduce the black asterisks in the left panel. For the PCs, the dimensionless uncertainties plotted here should be multiplied by $p_{\mathrm{ann}}$ to obtain the uncertainties on the ${\epsilon }_i$ coefficients. The uncertainties on the cosmological parameters have been divided by the central values of the respective parameters.

Figure 21

Biases to the cosmological parameters in mock Planck data, simulated assuming a constant-$p_{\mathrm{ann}}(z)$ energy deposition history with $p_{\mathrm{ann}}=1\times {10}^{-6}{\mathrm{m}}^3/\mathrm{s}/\mathrm{kg}=1.8\times {10}^{-27}{\mathrm{cm}}^3/\mathrm{s}/\mathrm{GeV}$. The fit to the data is performed using only the standard six cosmological parameters.

Figure 22

An example of the dependence of the principal components on the number of bins and the width of the Gaussians, for the first, third and sixth principal components in Planck, after marginalization over the cosmological parameters (the later principal components are shown as they are more sensitive to small changes in the binning). Black lines indicate the case with 100 linearly-spaced bins and $\sigma =\mathrm{bin}$ width/2; in the left (right) panel, red and blue dots indicate, respectively, the case with 15 (50) linearly-spaced bins and $\sigma =\mathrm{bin}$ width/2, and the case with 15 (50) linearly-spaced bins and $\sigma =\mathrm{bin}$ width/4.

Figure 23

The dependence of a selection of PCs on the range of $\ell$ included in the analysis, for WMAP 7 (left) and Planck (right).

Figure 24

The first three principal components for Planck, after marginalization, computed using RECFAST 1.5 and CosmoRec. In the baseline case (as in CosmoRec), ionization of helium is included but injection of Lyman-$\alpha$ photons is not. We also show the effects of including a contribution to Lyman-$\alpha$ photons, and neglecting helium. The effect of helium ionization on the PCs is negligible because it is approximately a redshift-independent effect.