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 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 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 . 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 . The Planck satellite and Polarbear experiment will be sensitive to fluctuations with amplitude . SPTPol, ACTPol, and future space-based polarization methods will probe amplitudes as low as . In the cosmic-variance limit, the smallest CIPs that could be detected with the CMB are of amplitude .
The concordance cosmological model posits a nearly scale-invariant spectrum of primordial density fluctuations with adiabatic initial conditions, for which the ratios of neutrino, photon, baryon, and dark-matter number densities are homogeneous. The simplest inflationary models predict adiabatic fluctuations Guth and Pi (1982); Linde (1982); Bardeen et al. (1983); Hawking (1982); Mukhanov and Chibisov (1982); Starobinsky (1982), and adiabatic fluctuations are consistent with measurements of cosmic microwave background (CMB) temperature/polarization anisotropies Larson et al. (2011) and the clustering of galaxies Padmanabhan et al. (2007); Tegmark et al. (2004).
Isocurvature perturbations are fluctuations in the ratios of number densities of various particle species. They are produced in topological-defect models for structure formation Brandenberger (1994) and in more complicated models of inflation Linde (1984); Linde and Mukhanov (1997); Mukhanov et al. (1992); Axenides et al. (1983); Langlois and Riazuelo (2000); Langlois (1999). CMB temperature anisotropies limit the amplitude of baryon isocurvature perturbations (fluctuations in the baryon-to-photon ratio) Peebles (1999a, b) and CDM isocurvature perturbations (fluctuations in the dark-matter–to–photon ratio) Burns (1997); Seckel and Turner (1985); Hu (1999); Hu et al. (1995) to be of the total perturbation amplitude Enqvist et al. (2000); Enqvist and Kurki-Suonio (2000); Larson et al. (2011); Komatsu et al. (2011); Väliviita and Giannantonio (2009); Zunckel et al. (2011); Bucher et al. (2004); Kawasaki and Sekiguchi (2008); Beltrán et al. (2005); Seljak et al. (2006); Beltrán et al. (2004); Takahashi et al. (2009).
Our intuition thus suggests the matter in the early Universe was very smoothly distributed. It therefore comes as somewhat of a surprise to learn that perturbations in the baryon density can be almost arbitrarily large, as long they are compensated by dark-matter perturbations such that the total nonrelativistic matter density remains unchanged Gordon and Pritchard (2009); Holder et al. (2010). These compensated isocurvature perturbations (CIPs) thus obey
where , , and are fractional energy density perturbations in the dark matter, baryons, and photons, respectively, while and are the homogeneous dark matter and baryon densities.
CIPs induce no curvature perturbation at early times, and they therefore leave the photon density—and thus large-angle CMB fluctuations—unchanged at linear order. CIPs induce baryon motion through baryon-pressure gradients, but these motions occur only at the baryon sound speed which, at the time when Thomson scattering first decouples (, decoupling hereafter), is . The effects of these motions on CMB temperature and polarization anisotropies thus occur only on distances smaller than times the sound horizon at decoupling or CMB multipole moments Gordon and Pritchard (2009); Gordon and Lewis (2003); Lewis and Challinor (2007), scales far smaller than those probed by CMB experiments.
The effect of CIPs on galaxy surveys is also believed to be small Gordon and Pritchard (2009). Big-bang nucleosynthesis (BBN) and galaxy-cluster baryon fractions constrain the CIP perturbation amplitude to be Holder et al. (2010). Measurements of fluctuations in 21-cm radiation from atomic hydrogen during the dark ages may be sensitive to these perturbations Barkana and Loeb (2005); Lewis and Challinor (2007); Gordon and Pritchard (2009); Kawasaki et al. (2011), but these measurements are a long way in the future.
In Ref. Holder et al. (2010), it was shown that, although CIPs produce no observable effect on the CMB at linear order in perturbation theory, they modulate the CMB fluctuations produced by adiabatic perturbations. In particular, it was shown that B modes in the CMB polarization are produced by the angular modulation in the reionization optical depth induced by the CIP.
Here, we consider the additional effects on CMB fluctuations that arise from modulation of the baryon density by CIPs at and before decoupling. CIPs modulate the free-electron density. They thus change the photon diffusion length and thickness of the surface of last scattering (SLS) on different patches of sky. CIPs also change the weight of the baryon-photon plasma and thus the details of the acoustic-peak structure in the CMB power spectrum. Variation in the baryon density from one region on the sky to another thus leads to a modulation of the small-scale power spectrum from one region of sky to another. This induces B modes in the polarization and nontrivial higher-order correlations in the temperature/polarization map analogous to those induced by variations of other cosmological parameters Sigurdson et al. (2003) and those induced by weak gravitational lensing Lewis and Challinor (2006).
As we show below, the effects of CIPs on CMB fluctuations from decoupling are several orders of magnitude larger than those from reionization, and so the CMB should provide a far more sensitive probe of CIPs than envisioned in Ref. Holder et al. (2010). We therefore follow through and develop the formalism required to look for CIPs with the CMB. To do so, we write down the minimum-variance estimators that can be constructed from a CMB temperature-polarization map for the CIP field as a function of position on the sky. We evaluate the noise with which the CIP field can be reconstructed and estimate the signal-to-noise with which a scale-invariant spectrum of CIPs may be detected with various experiments.
We conclude that data from WMAP, Spider, ACT, and SPT are sensitive to CIP amplitudes of . The Planck satellite The Planck Collaboration (2006) and Polarbear experiment are sensitive to CIP amplitudes as small as . Upcoming ground-based polarization experiments (ACTPol Niemack et al. (2010) and SPTPol Dvorkin and Smith (2009); McMahon et al. (2009)) or a post-Planck CMB-polarization experiment along the lines of the proposed EPIC experiment Bock et al. (2009) could detect fluctuations of . In the cosmic-variance limit, sensitivity to fluctuations of amplitude is possible.
Our principal motivation in studying CIPs is curiosity: can we determine empirically, rather simply assume, that the primordial baryon fraction is homogeneous and traces the dark matter? Still, there may be theoretical motivation as well. For example, curvaton models for inflation may generate CIPs Lyth et al. (2003); Gupta et al. (2004); Gordon and Lewis (2003); Enqvist et al. (2009), with amplitudes approaching the regime detectable by EPIC Gordon and Pritchard (2009). It may also be that recent models Kaplan et al. (2009); Buckley and Randall (2011); Allahverdi et al. (2011); Gu et al. (2011); Heckman and Rey (2011); McDonald (2010) that connect the baryon asymmetry and dark-matter density have implications for CIPs. Additionally, the techniques introduced in this paper could be used to empirically disentangle a CDM isocurvature fluctuation from a baryon isocurvature fluctuation, using CMB data. These modes are usually treated as degenerate in the analysis of CMB observations.
In Ref. Grin et al. (2011), we presented our basic conclusions. Here we present in detail our results, their derivation, and the computational methods used. We calculate the induced temperature anisotropies in Sec. II and the induced polarization anisotropies in Sec. III. In Sec. IV, we compute the expected corrections to CMB power spectra for a scale-invariant spectrum of CIPs and compare the B-mode power spectrum induced by CIPs at decoupling with that induced at reionization. In Sec. V, we construct minimum-variance estimators for CIPs. We then assess in Sec. VI the sensitivity of ongoing and upcoming experiments to CIPs, and we conclude in Sec. VII. Useful relations involving tensor spherical harmonics are presented in Appendix A. Numerical derivatives of transfer functions are discussed in Appendix B. Second-order harmonic expansions for CMB transfer functions are derived in Appendix C. Throughout, we use as our fiducial cosmological parameters those from Ref. Larson et al. (2011).
Ii Perturbed line-of-sight (LOS) formalism: Temperature
Here we review the standard calculation of the temperature-fluctuation power spectrum for primordial adiabatic density perturbations. We then show how this calculation is altered in the presence of CIPs.
ii.1 General line-of-sight solution for temperature
The spherical-harmonic coefficients for the CMB temperature can be written
where is the CMB temperature in direction , and are spherical harmonics. The Fourier transform of the primordial gravitational potential for wave-vector is , while denotes a spherical Bessel function. The conformal time is here an integration variable, and denotes its value today. The function , obtained via the numerical solution of the Boltzmann equations Bertschinger (1995); Seljak and Zaldarriaga (1996); Ma and Bertschinger (1995), encodes how much a real-space primordial-potential perturbation contributes to the temperature anisotropy . It depends on the relation between initial gravitational-potential fluctuations and radiation-density fluctuations at decoupling, as well as the recombination history.
ii.2 Temperature anisotropies with homogeneous baryon fraction
In the standard calculation, this transfer function is the same in all directions; i.e., . In this case, Eq. (2) simplifies, via orthogonality of the s, yielding Zaldarriaga and Harari (1995); Seljak and Zaldarriaga (1996)
The temperature power spectrum is then easily obtained using Eq. (3), averaging over realizations of the potential perturbation, and using the identity , where is the Dirac delta function and the primordial-potential power spectrum, and the angle brackets denote an average over realizations of the primordial potential. We then find Ma and Bertschinger (1995)
is the CMB temperature power spectrum, written in terms of a transfer function,
ii.3 Temperature anisotropies with CIPs: Single CIP realization
In the presence of a compensated isocurvature perturbation, the baryon and dark-matter fractions vary from one point in the Universe to another, and so the transfer function now acquires some direction () dependence. The CIP involves small changes,
in the cosmological parameters between different lines of sight . Here, is the value of the CIP in direction at the surface of last scatter (or reionization—we will make these statements more precise below). Note that we define it so that it is the fractional perturbation in the baryon (rather than dark-matter) density associated with the CIP. From Eq. (7), the change in the total density is , and so this is indeed a compensated isocurvature perturbation.
In a general treatment of perturbed recombination/decoupling, one would follow the set of equations for the electron, dark-matter, photon, and neutrino densities, velocities, and the gravitational potential at second order, as in Refs. Senatore et al. (2009a, b); Khatri and Wandelt (2009). In the case of CIPs, however, the CIP amplitude does not evolve for all observationally accessible scales, and we can thus model the effect of CIPs as a modulation in the cosmological parameters and .
We can build some intuition for the effect of CIP perturbations on the CIP by considering a globally constant CMB perturbation . We run the camb Lewis and Challinor (2002) code with a global perturbation of the form in Eq. (7) for a variety of values. We evaluate the angular sound horizon at the surface of last scatter as a function of , using the expressions in Ref. Dodelson (2003). We see in the top left panel of Fig. 1 that, as the plasma is more loaded down with baryons in the presence of a CIP with a positive value, the decrease in sound speed moves the CMB acoustic peaks to smaller angular scales.
CMB temperature anisotropies are suppressed on angular scales due to diffusion damping. Using the expressions in Ref. Zaldarriaga and Harari (1995) and the camb Lewis and Challinor (2002) code, we evaluate and show the results in the top right panel of Fig. 1. We see that, as photons diffuse over smaller distances, as a result of higher local baryon density in the presence of a CIP with positive , the transition to exponential damping of CMB anisotropies occurs at higher .
In the bottom panel of Fig. 1, we show the visibility functions for different values of ; is the optical depth due to Thomson scattering. The peak of the visibility function is the redshift , at which most CMB photons last scatter. In the presence of a positive (negative) CIP, decoupling occurs later (earlier) due to higher (lower) baryon density.
To calculate the effects on the CMB moments , we perturb the line-of-sight solutions, Eqs. (2) and (3). This approach is relatively simple and amenable to rapid computation. The results should be accurate for multipole moments for the CIP, as the baryon fluctuation can be considered as roughly constant in a given direction across the thickness of the surface of last scatter on such scales. We discuss in Sec. IV.1 below how we extrapolate these results to smaller angular scales () with a Limber approximation.
We proceed by Taylor expanding in real space:
where is the value of under the null hypothesis . We expand
in terms of spherical-harmonic coefficients for the angular variation in the CIP at the surface of last scatter. We then apply the expansion in Eq. (8) to linear order in to the line-of-sight expression, Eq. (2), and integrate over angles to obtain the first-order correction,
to in the presence of a CIP, where
and the arrays inside parentheses are Wigner- symbols. Throughout, we use the indices and exclusively for the decomposition of the CIP, while lower-case indices are used for the multipole moments of the CMB observables , , and . Sums over () are always taken over the range (, while sums over () are formally taken over (); in practice, a maximum value is used for numerical evaluation, as discussed in Sec. IV. The monopole corresponds to a global shift in and and we absorb this term into the cosmological parameters themselves.
For a given realization of the CIP field—that is, for a given set of —the covariance between temperature moments is now
for (a generalization that will be useful below), and is the temperature power spectrum in the absence of CIPs. Here,
describes the change in the transfer function, Eq. (6), with .
In deriving these results, we have taken an average over realizations of the primordial-potential power spectrum , but we have restricted our consideration to a given realization of the CIP. In Sec. V, we build the formalism to reconstruct from these off-diagonal temperature correlations as well their generalization to polarization.
ii.4 Temperature anisotropies with CIPs: Average over CIP realizations
We now take an ensemble average over many realizations of both the primordial potential field and the CIP field. This allows us, given a spectrum of CIPs, to calculate the effects of these CIPs on the power spectrum of CMB fluctuations measured on the entire sky.
We denote the ensemble average of a spatially varying field over realizations of the CIP field by . We denote the ensemble average over realizations of both the CIP field and the primordial potential by . From Eqs. (13) and (14), we see that . For an isotropic random field, , so we must thus go to second order in to compute the effects of CIPs on the CMB power spectrum. We thus obtain, to second order in , the temperature power spectrum,
where is the unperturbed moment in Eq. (3), is given by Eq. (10), and is the unperturbed power spectrum given by Eq. (5). The superscript denotes the term arising when expanding to order . We evaluate this term using the second-derivative term in Eq. (LABEL:harmonic). We take an expectation value over CIPs and primordial-potential realizations. We then use Eqs. (2), (11), and (17), identities of Wigner- symbols Varshalovich et al. (1988), and Appendix C to obtain
The CIP power spectrum and total variance obey
while the CMB derivative power spectra are given by
Iii Perturbed line-of-sight formalism: Polarization
We now generalize the analysis above to the CMB polarization. In addition to inducing off-diagonal correlations in the polarization spherical-harmonic coefficients, CIPs will induce B modes. We begin by reviewing the LOS solution for polarization under the null hypothesis of no CIPs. We then compute the effects of CIPs, both for a single realization of the CIPs, and then for an average over realizations of a spectrum of CIPs.
iii.1 Polarization anisotropies with homogeneous baryon fraction
where and are the E- and B mode (“grad” and “curl”, respectively) tensor spherical harmonics, as defined in Appendix A. The right-most indices after the comma are tensor indices. In terms of the Stokes polarization parameters and , the polarization tensor is Kamionkowski et al. (1997); Zaldarriaga and Seljak (1997)
where is the polar angle of the LOS with respect to some origin.
Under the null hypothesis, the polarization pattern at the surface of last scatter is a pure E mode, with multipole moments given by
where , and is the E-mode transfer function, obtainable numerically from Boltzmann codes. The polarization covariance and TE covariance are derived analogously to the results for temperature, yielding Seljak and Zaldarriaga (1996)
and Seljak and Zaldarriaga (1996)
iii.2 Polarization anisotropies with CIPs: single CIP realization
We now generalize the analysis to include the effects of CIPs. In the presence of a CIP field , the real-space polarization tensor may be Taylor expanded as
Just as in the case of temperature, when considering a single realization, we need only consider the first-order terms in Eq. (29). We then utilize Eqs. (24), (26), and the first-derivative piece of the usual expansion for [see Eq. (8)] to obtain an expansion for the polarization tensor in the presence of CIPs:
We may now pick off the induced E- and B-mode multipole moments and at order , using the appropriate integral over a tensor spherical harmonic:
We now evaluate the induced correlations between different temperature/polarization moments. At first order in , . The remaining covariances are
iii.3 Polarization anisotropies with CIPs: Average over CIP realizations
We now extend the ensemble average to multiple realizations of the CIP field. We do this in order to compare the polarization power spectrum induced by CIPs at the surface of last scatter with that induced at reionization. For temperature, the average over realizations of both the CIP and primordial-potential perturbations is given by Eq. (17). Extending this average to , we obtain the XX power spectra, averaged over the entire sky, to second order in :
where is the power spectrum computed with no CIP contribution.
We evaluate Eq. (41) with Eqs. (A12) and (A13) and Wigner- relations to simplify the resulting integrals and sums. Superscript indices and indicate the order of the derivative used to derive the indicated term, as in Sec. II.4. The resulting nonzero power spectra are
The CIP field is a scalar and cannot statistically change the parity of polarization perturbations. This requires that and vanish when averaging over CIP realizations. Algebraically, this is enforced by the vanishing of the relevant Wigner- symbols, as occurs with optical-depth fluctuations at reionization Dvorkin et al. (2009); Dvorkin and Smith (2009) and with gravitational-potential perturbations in weak lensing Hu (2000a); Okamoto and Hu (2003). Indeed, the geometric (Wigner-) symbols obtained are the same as for those effects. CIPs give rise to different dependences for the functions , however, through the dependence on the derivative power spectra , allowing them to be disentangled observationally from gravitational-potential fluctuations along the LOS or optical-depth fluctuations at reionization.
Iv Numerical Results for B- Mode Power Spectra
We now apply the formulas derived in Secs. II.4 and III.3 to compute the power spectra for B modes induced by CIPs at decoupling. We first discuss the form of the angular CIP power spectrum . We then present numerical results for B modes induced at decoupling. For comparison, we then reproduce the calculations of Ref. Holder et al. (2010) of the B modes induced at reionization.
iv.1 Power spectrum of compensated perturbations
Three-dimensional CIP power spectrum
To proceed further, we must make an ansatz for the spectrum of CIPs. Motivated by the curvaton model (which produces a nearly scale-invariant spectrum of CIPs) Lyth et al. (2003); Gordon and Lewis (2003); Gordon and Pritchard (2009), we assume a scale-invariant spectrum for the three-dimensional CIP field ; that is,
where is the Fourier transform of , and is a dimensionless CIP amplitude.
As discussed in the introduction, the strongest constraint to comes from cluster baryon fractions. This constraint tells us that the variance,
in the baryon–to–dark-matter ratio on Mpc scales is . The integral has a formal logarithmic divergence at low which is cut off, however, by the volume occupied by the clusters surveyed. Taking this to be the horizon, , we find . Since the cosmological baryon fraction determines primordial abundances via BBN, there is an additional constraint from astrophysical measurements of these abundances Holder et al. (2010). However, this constraint is less stringent than the one from cluster gas fractions.
Angular CIP power spectrum
When the 3-dimensional field is projected onto a narrow spherical surface, the resulting angular power spectrum for will be for mulipole moments , where and are the conformal time at last scatter and today, respectively, and is the rms conformal-time width of the surface of last scattering (SLS). At smaller angular scales (larger ), the angular variation in is suppressed by the finite width Silk (1968) of the scattering surface. Using the Limber approximation, the angular power spectrum for can be approximated by for . The exact analytic expression we use is obtained from the Limber approximation, approximating the visibility function as a Gaussian. It is
where is a confluent hypergeometric function. We use and for decoupling. We use and for reionization. These values are obtained by directly fitting to the visibility function output by the camb code Lewis and Challinor (2002). Of course, Eq. (47) is an approximation, and the precise shape of the transition from near depends on the interference of Fourier modes of with those of , averaged over the SLS. This issue warrants future study, but the asymptotic behavior at low and high is correct (as shown for an analogous computation in Ref. Pogosian et al. (2011)). Moreover, as we shall see in Sec. VI, most signal-to-noise in CIP reconstruction comes either from or , and so the main conclusions of this work should not be affected.
iv.2 Numerical result for B modes from CIPs at decoupling
Using the Limber approximation with values and appropriate for decoupling, Eqs. (42)–(44) can be used to obtain predictions for the B modes induced by CIPs at decoupling. The results are shown in Fig. 2 for , the largest CIP amplitude consistent with the galaxy-cluster constraint. Appendix B details the calculation of the requisite derivative power spectra. We use a maximum value of .
In Ref. Holder et al. (2010), it was noted that spatial inhomogeneities in the baryon density give rise to angular variations in the optical depth for rescattering of CMB photons at reionization. It was also noted that these inhomogeneities would give rise to B modes primarily at large angular scales by patchy rescattering of CMB photons and at smaller angular scales through patchy screening of the primary CMB polarization from the decoupling epoch. These calculations build upon calculations in Refs. Hu (2000b); Baumann et al. (2003); Doré et al. (2007); Dvorkin et al. (2009); Dvorkin and Smith (2009) where optical-depth fluctuations were postulated to arise from inhomogeneities in the free-electron fraction due to inhomogeneous reionization.
where is the mean optical depth, and is here the angular CIP power spectrum for reionization; i.e., obtained with and . These values are obtained by fitting a Gaussian visibility function to the reionization model of Ref. Lewis (2008).
The contributions of patchy scattering are
where is the rms temperature quadrupole at reionization.
Figure 2 shows the B modes induced by patchy scattering and screening at reionization again using . We see that at all but the largest scales, the decoupling-induced B modes are larger (by up to orders of magnitude) than those induced at reionization.
We thus conclude that the effects of CIPs on CMB fluctuations would be much larger than found in Ref. Holder et al. (2010), particularly at the small scales most important for detection and reconstruction (to be discussed below) of CIPs from the CMB. We thus now move on to show how spatial fluctuations in the baryon–to–dark-matter ratio can be measured with CMB maps.
V Measurement of CIPs with the CMB
In this section, we show how the CIP field can be measured with off-diagonal CMB correlations, building upon analogous prior work on measurement of cosmic-shear fields Seljak (1996); Zaldarriaga and Seljak (1998); Zaldarriaga (2000); Seljak and Zaldarriaga (1999); Hu (2000a); Okamoto and Hu (2003), departures from statistical isotropy Hajian and Souradeep (2003, 2005); Pullen and Kamionkowski (2007), and cosmic birefringence Kamionkowski (2009); Gluscevic et al. (2009); Yadav et al. (2009); Caldwell et al. (2011). Having concluded that the decoupling signal is much bigger than that from reionization, we consider detection/measurement of CIPs at the surface of last scatter.
In Sec. V.1, we construct a minimum-variance quadratic estimator