Compensated isocurvature perturbations and the cosmic microwave background

Measurements of cosmic microwave background (CMB) anisotropies constrain isocurvature fluctuations between photons and nonrelativistic particles to be subdominant to adiabatic fluctuations. Perturbations in the relative number densities of baryons and dark matter, however, are surprisingly poorly constrained. In fact, baryon-density perturbations of fairly large amplitude may exist if they are compensated by dark-matter perturbations, so that the total density remains unchanged. These compensated isocurvature perturbations (CIPs) leave no imprint on the CMB at observable scales, at linear order. B modes in the CMB polarization are generated at reionization through the modulation of the optical depth by CIPs, but this induced polarization is small. The strongest known constraint $\lesssim 10\%$ to the CIP amplitude comes from galaxy-cluster baryon fractions. Here, it is shown that modulation of the baryon density by CIPs at and before the decoupling of Thomson scattering at $z\sim 1100$ gives rise to CMB effects several orders of magnitude larger than those considered before. Polarization B modes are induced, as are correlations between temperature/polarization spherical-harmonic coefficients of different $lm$. It is shown that the CIP field at the surface of last scatter can be measured with these off-diagonal correlations. The sensitivity of ongoing and future experiments to these fluctuations is estimated. Data from the WMAP, ACT, SPT, and Spider experiments will be sensitive to fluctuations with amplitude $\sim 5–10\%$. The Planck satellite and Polarbear experiment will be sensitive to fluctuations with amplitude $\sim 3\%$. SPTPol, ACTPol, and future space-based polarization methods will probe amplitudes as low as $\sim 0.4\%–0.6\%$. In the cosmic-variance limit, the smallest CIPs that could be detected with the CMB are of amplitude $\sim 0.05\%$.

Figure 1

Dependence of physically relevant scales and Thomson scattering visibility function $g(z)$ on amplitude of a global CIP perturbation $\Delta$. Top left panel shows dependence of the angular sound horizon $l_{\mathrm{s}}$ on $\Delta$. Top right panel shows dependence of the diffusion damping scale $l_{\mathrm{d}}$ on $\Delta$. Bottom panel shows $g(z)$ evaluated for 3 different values of $\Delta$.

Figure 2

CMB B-mode polarization power spectra induced by CIP perturbations at decoupling (black solid line), compared with the effects of CIPs at reionization, for which two contributions are shown: patchy screening (red dotted line), and patchy scattering (blue short-dashed line). The amplitude for the CIP power spectrum is that which saturates the ${\Delta }_{\mathrm{cl}}\lesssim 0.08$ bound from clusters [35].

Figure 3

Predicted noise power spectrum $\delta C_L$ in the reconstructed CIP perturbation field ${\Delta }_{LM}$ in four different ongoing/future CMB anisotropy experiments, as a function of angular scale $L$. We separately plot the noise for distinct pairs of observables: TT is shown as an orange (short-dash–long-dashed), TE as a green (dotted-dashed) line, EE as a magenta (short-dashed) line, BE as a blue (long-dashed) line, and BT as a red (dotted) line. Also shown (black solid line) is the power spectrum $C_L^{\Delta }$, marked signal, for a scale-invariant spectrum of CIPs with the maximum amplitude allowed by galaxy clusters. Each panel shows estimates for a different experiment, as indicated in the figure. The beige (grey) shaded region shows the range of $L$ that is not included in our estimates, due to finite sky coverage effects, as discussed in Sec. 5c.

Figure 4

Predicted noise power spectrum $\delta C_L$ in the reconstructed CIP perturbation field ${\Delta }_{LM}$ in four different ongoing/future CMB anisotropy experiments, as a function of angular scale $L$. Colors (line styles) are as in Fig. 3. We separately plot the noise for distinct pairs of observables: TT is shown as an orange (short-dash–long-dashed), TE as a green (dotted-dashed) line, EE as a magenta (short-dashed) line, BE as a blue (long-dashed) line, and BT as a red (dotted) line. Also shown (black solid line) is the power spectrum $C_L^{\Delta }$ for a scale-invariant spectrum of CIPs with the maximum amplitude allowed by galaxy clusters. Each panel shows estimates for a different experiment, as indicated in the figure. The beige (grey) shaded region shows the range of $L$ that is not included in our estimates, due to finite sky coverage effects, as discussed in Sec. 5c.

Figure 5

Predicted noise power spectrum $\delta C_L$ in the reconstructed CIP perturbation field ${\Delta }_{LM}$ for the proposed EPIC satellite, as a function of angular scale $L$. Colors (line styles) are as in Fig. 3. We separately plot the noise for distinct pairs of observables: TT is shown as an orange (short-dash–long-dashed), TE as a green (dotted-dashed) line, EE as a magenta (short-dashed) line, BE as a blue (long-dashed) line, and BT as a red (dotted) line. Also shown (black solid line) is the power spectrum $C_L^{\Delta }$ for a scale-invariant spectrum of CIPs with the maximum amplitude allowed by galaxy clusters. The beige (grey) shaded region shows the range of $L$ that is not included in our estimates, due to finite sky coverage effects, as discussed in Sec. 5c.

Figure 6

Predicted noise power spectrum $\delta C_L$ in the reconstructed CIP perturbation field ${\Delta }_{LM}$ for an ideal cosmic-variance limited experiment, as a function of angular scale $L$. Colors (line styles) are as in Fig. 3. We separately plot the noise for distinct pairs of observables: TT is shown as an orange (short-dash–long-dashed), TE as a green (dotted-dashed) line, EE as a magenta (short-dashed) line, BE as a blue (long-dashed) line, and BT as a red (dotted) line. Also shown (black solid line) is the power spectrum $C_L^{\Delta }$ for a scale-invariant spectrum of CIPs with the maximum amplitude allowed by galaxy clusters. The beige (grey) shaded region shows the range of $L$ that is not included in our estimates, due to finite sky coverage effects, as discussed in Sec. 5c.

Figure 7

Combined (predicted) noise power spectrum $\delta C_L$ in the reconstructed CIP perturbation field ${\Delta }_{LM}$ for 9 different CMB experiments, as a function of angular scale $L$. Colors and line styles for each experiment are indicated in the legend (center). Here, noise for the 5 different estimators (TT, TE, EE, BE, and BT) is added in quadrature. Noise curves terminate at L values where modes become inaccessible due to finite sky effects, as discussed in Sec. 5c. Also shown (black solid line) is the power spectrum $C_L^{\Delta }$, marked signal, for a scale-invariant spectrum of CIPs with the maximum amplitude allowed by galaxy clusters. The left panel shows predicted total errors for the indicated experiments. The right panel shows the predicted signal-to-noise ratio that results from these errors, evaluated using Eq. (66) and assuming a scale-invariant spectrum of CIPs [evaluated using Eq. (47)]. The signal-to-noise ratio is plotted as a function of the rms CIP fluctuation ${\Delta }_{\mathrm{cl}}$ on cluster scales. The range of fluctuations ${\Delta }_{\mathrm{cl}}$ excluded by cluster measurements [35] is shown as a beige (grey) band, bounded by a vertical black line with rightward pointing arrows. The black line with upward arrows attached shows the “detection” region, defined by $S/N=3$.

Figure 8

Lowest-order first-derivative power spectra, as defined in Secs. 2, 3. These are necessary to reconstruct ${\Delta }_{LM}$, as described in Sec. 5. We use the numerical methods of Appendix  to obtain these curves using a modified version of the camb [61] code.

Figure 9

Second-order first-derivative power spectra, as defined in Secs. 2, 3. These are necessary to estimate the corrected power spectra $C_l^{{\mathrm{XX}}^{\prime },(2)}$. We use the numerical methods of Appendix  to obtain these curves using a modified version of the camb [61] code.

Figure 10

Second-derivative power spectra, as defined in Secs. 2, 3. These are necessary to estimate the corrected power spectra $C_l^{{\mathrm{XX}}^{\prime },(2)}$. We use the numerical methods of Appendix  to obtain these curves using a modified version of the camb [61] code.