A new probe of magnetic fields in the pre–reionization epoch: II. Detectability
In the first paper of this series, we proposed a novel method to probe large–scale intergalactic magnetic fields during the cosmic Dark Ages, using 21–cm tomography. This method relies on the effect of spin alignment of hydrogen atoms in a cosmological setting, and on the effect of magnetic precession of the atoms on the statistics of the 21–cm brightness–temperature fluctuations. In this paper, we forecast the sensitivity of future tomographic surveys to detecting magnetic fields using this method. For this purpose, we develop a minimum–variance estimator formalism to capture the characteristic anisotropy signal using the two–point statistics of the brightness–temperature fluctuations. We find that, depending on the reionization history, and subject to the control of systematics from foreground subtraction, an array of dipole antennas in a compact–grid configuration with a collecting area slightly exceeding one square kilometer can achieve a detection of Gauss comoving (scaled to present–day value) within three years of observation. Using this method, tomographic 21–cm surveys could thus probe ten orders of magnitude below current CMB constraints on primordial magnetic fields, and provide exquisite sensitivity to large–scale magnetic fields in situ at high redshift.
Magnetic fields are ubiquitous in the universe on all observed scales Durrer and Neronov (2013); Vallee (2004); Neronov and Vovk (2010); Wielebinski (2005); Beck (2012). However, the question of origins of the magnetic fields in galaxies and on large scales is as of yet unresolved. Various forms of dynamo mechanism have been proposed to maintain and amplify them Park et al. (2013), but they typically require the presence of seed fields Durrer and Neronov (2013). Such seed fields may be produced during structure formation through the Biermann battery process or similar mechanisms Naoz and Narayan (2013a, b), or may otherwise be relics from the early universe Durrer and Neronov (2013); Widrow et al. (2012); Kobayashi (2014). Observations of large–scale low–strength magnetic fields in the high–redshift intergalactic medium (IGM) could thus probe the origins of present–day magnetic fields and potentially open up an entirely new window into the physics of the early universe.
Many observational probes have been previously proposed and used to search for large–scale magnetic fields locally and at high redshifts (e. g. Yamazaki et al. (2010); Blasi et al. (1999); Tavecchio et al. (2010); Dolag et al. (2011); Wielebinski (2005); Kunze and Komatsu (2014); Kahniashvili et al. (2013); Shiraishi et al. (2014); Tashiro and Sugiyama (2006); Schleicher et al. (2009)). Amongst the most sensitive tracers of cosmological magnetic fields is the cumulative effect of Faraday rotation in the cosmic–microwave–background (CMB) polarization maps, which currently places an upper limit of Gauss (in comoving units) using data from the Planck satellite Planck Collaboration et al. (2015a). In Paper I of this series Venumadhav et al. (2014), we proposed a novel method to detect and measure extremely weak cosmological magnetic fields during the pre–reionization epoch (the cosmic Dark Ages). This method relies on data from upcoming and future 21–cm tomography surveys Madau et al. (1997); Loeb and Zaldarriaga (2004), many of which have pathfinder experiments currently running Greenhill and Bernardi (2012); Bowman et al. (2011); Parsons et al. (2014); Carilli (2008); Vanderlinde and Chime Collaboration (2014); DeBoer and HERA (2015), with the next–stage experiments planned for the coming decade Carilli (2008); DeBoer and HERA (2015).
In Paper I, we calculated the effect of a magnetic field on the observed 21–cm brightness–temperature fluctuations, and in this Paper, we focus on evaluating the sensitivity of future 21–cm experiments to measuring this effect. As we pointed out in Paper I, the 21–cm signal from the cosmic Dark Ages has an intrinsic sensitivity to capturing the effect of the magnetic fields in the IGM that are more than ten orders of magnitude smaller than the current upper limits on primordial magnetic fields from the CMB. In the following, we demonstrate that a square–kilometer array of dipole antennas in a compact grid can reach the sensitivity necessary to detect large–scale magnetic fields that are on the order of Gauss comoving (scaled to present day, assuming adiabatic evolution of the field due to Hubble expansion).
The rest of this Paper is organized as follows. In §II, we summarize the main results of Paper I. In §III, we define our notation and review the basics of the 21–cm signal and its measurement. In §IV, we derive minimum–variance estimators for uniform and stochastic magnetic fields. In §V, we set up the Fisher formalism necessary to forecast sensitivity of future surveys. In §VI, we present our sensitivity forecasts. In §VII we summarize and discuss the implications of our results. Supporting materials are presented in the appendices.
Ii Summary of the Method
Magnetic moments of hydrogen atoms in the excited (triplet) state of the 21–cm line transition tend to align with the incident quadrupole of the 21–cm radiation from the surrounding medium. This effect of “ground–state alignment” Yan and Lazarian (2008, 2012) arises in a cosmological setting due to velocity–field gradients. In the presence of an external magnetic field, the emitted 21–cm quadrupole is misaligned with the incident quadrupole, due to atomic precession; this is illustrated in Fig. 1. The resulting emission anisotropy can be used to trace magnetic fields at high redshifts.
The main result of Paper I was derivation of the 21–cm brightness–temperature fluctuation111Standard notation, used in other literature and in Paper I of this series, for this quantity is ; however, we use here to simplify our expressions. , including the effects of magnetic precession, as a function of the line–of–sight direction ,
where the magnetic field is along the axis in the rest frame of the emitting atoms (in which the spin–zero spherical harmonics are defined in the usual way); is a density–fluctuation Fourier mode corresponding to the wave vector whose direction is along the unit vector ; , , and parametrize the rates of depolarization of the ground state by optical pumping and atomic collisions, and the rate of magnetic precession (relative to radiative depolarization), respectively (defined in detail in Paper I), and are all functions of redshift ; and are the spin temperature and the CMB temperature at redshift , respectively. Fig. 2 illustrates the effect of the magnetic field on the brightness temperature emission pattern in the frame of the emitting atoms; shown are the quadrupole patterns corresponding to the last term of Eq. (1), for various strengths of the magnetic field. Notice that there is a saturation limit for the field strength—for a strong field, the precession is much faster than the decay of the excited state of the forbidden transition, and the emission pattern asymptotes to the one shown in the bottom panel of Fig. 2. Above this limit, the signal cannot be used to reconstruct the strength of the field. However, in this “saturated regime”, it is still possible to distinguish the presence of a strong magnetic field from the case of no magnetic field, as we discuss in detail in §V.
The effect of quadrupole misalignment arises at second order in optical depth (it is a result of a two–scattering process), and is thus a small correction to the total brightness–temperature fluctuation. However, owing to the long lifetime of the excited state of the forbidden transition (during which even an extremely slow precession can have a large cumulative effect on the direction of the quadrupole, at second order), the misalignment is exquisitely sensitive to magnetic fields in the IGM at redshifts prior to cosmic reionization. As we showed in Paper I, a minuscule magnetic field of Gauss (in comoving units) produces order–one changes in the direction of the quadrupole. This implies that a high–precision measurement of the 21–cm brightness–temperature two–point correlation function intrinsically has that level of sensitivity to magnetic fields prior to the epoch of reionization (when most of the IGM is still neutral). We now proceed to develop a formalism to search for magnetic fields at high redshifts using this effect, and to forecast the sensitivity of future 21–cm experiments.
Before focusing on the estimator formalism (presented in the following Section), we review the basics of 21–cm brightness–temperature fluctuation measurements. In §III.1, we set up our notation and review definitions of quantities describing sensitivity of interferometric radio arrays; in §III.2, we focus on the derivation of the noise power spectrum; and in §III.3, we discuss the effects of the array configuration and its relation to coverage of modes in the plane.
The redshifted 21–cm signal can be represented with specific intensity at a location in physical space or in Fourier space . In sky coordinates (centered on an emitting patch of the sky), these functions become and , respectively. Here, vector (in the units of comoving Mpc) is a Fourier dual of (comoving Mpc), and likewise, (rad), (rad), and (Hz) are duals of the coordinates (rad), (rad), and (seconds), respectively. Notice that and represent the angular extent of the patch in the sky, while represents its extent in frequency space. The two sets of coordinates are related through linear transformations in the following way
where MHz is the frequency corresponding to the 21–cm line in the rest frame of the emitting atoms; is the Hubble parameter; and is the comoving distance to redshift which marks the middle of the observed data cube where and intervals are evaluated. Note that , for . The convention we use for the Fourier transform is
where Fourier–space functions are denoted with tilde. Similarly,
where the proportionality factor contains the transformation Jacobian . Finally, the relationship between the specific intensity in the –plane and the visibility function is given by the Fourier transform of the frequency coordinate,
Here, is the bandwidth of the observed data cube, centered on (see also Appendix A).
iii.2 Power spectra and noise
In this Section, we derive the noise power spectrum for the brightness–temperature fluctuation measurement. We start by defining a brightness–temperature power spectrum as
where is Dirac delta function. The observable quantity of the interferometric arrays is the visibility function—a complex Gaussian variable with a zero mean and the following variance (see detailed derivation in Appendix A)
where is the sky temperature (which, in principle, includes both the foreground signal from the Galaxy, and the instrument noise, where we assume the latter to be subdominant in the following); is the total time a single baseline observes element in the plane; is the collecting area of a single dish; is the Boltzmann constant; is the bandwidth of a single observation centered on ; and the last in this expression denotes the Kronecker delta.
where we used the standard definition
for the noise power per mode, per baseline.
In the last step, we wish to get from Eq. (11) to the expression for the noise power spectrum that corresponds to observation with all available baselines. To do that, we need to incorporate information about the array configuration and its coverage of the plane. In other words, we need to divide the expression in Eq. (11) by the number density of baselines that observe a given mode at a given time (for a discussion of the coverage, see the following Section). The final result for the noise power spectrum per mode in intensity units is
and in temperature units
iii.3 The UV coverage
The total number density of baselines that can observe mode is related to the (unitless) number density of baselines per element as
where represents an element in the plane. The number density integrates to the total number of baselines ,
where is the number of antennas in the array, and the integration is done on one half of the plane222This is because the visibility has the following property , and only a half of the plane contains independent samples.. We assume that the array consists of many antennas, so that time dependence of is negligible; if this is not the case, time average of this quantity should be computed to account for Earth’s rotation.
In this work, we focus on a specific array configuration that is of particular interest to cosmology—a compact grid of dipole antennas, with a total collecting area of , and a maximum baseline length333Note that for a square with area tiled in dipoles, there is a very small number of baselines longer than , but we neglect this for simplicity. of . In this setup, the beam solid angle is 1 sr, the effective area of a single dipole is , and the effective number of antennas is . For such a configuration, the number density of baselines entering the calculation of the noise power spectrum reads
The relation between and is
where the subscript denotes components perpendicular to the line–of–sight direction , which, in this case, is along the axis. From this, the corresponding number of baselines observing a given is
As a last note, when computing numerical results in §VI, we substitute the –averaged version of the above quantity (averaged between and only, due to the four–fold symmetry of the experimental setup of a square of dipoles) when computing the noise power, in order to account for the rotation of the baselines with respect to the modes in the sky. This average number density reads
assuming a given mode is observable by the array, such that its value is between and , where and are the maximum and minimum baseline lengths, respectively. If this condition is not satisfied, .
Iv Quadratic estimator formalism
We now derive an unbiased minimum–variance quadratic estimator for a magnetic field present in the IGM prior to the epoch of reionization. This formalism is applicable to tomographic data from 21–cm surveys, and is similar to that used in CMB lensing analyses Okamoto and Hu (2003), for example. We assume that the magnetic field only evolves adiabatically, due to Hubble expansion,
where is its present–day value (the value of the field in comoving units). The corresponding estimator is denoted with a hat sign, .
We start by noting that the observed brightness–temperature fluctuations contain contributions from the noise fluctuation (from the instrumental noise plus Galactic foreground emission444Note that this term adds variance to the visibilities due to foregrounds, but we assume the bias in the visibilities is removed via foreground cleaning.) and the signal ,
where can get contribution from both the magnetic–field effects and the (null–case) cosmological 21–cm signal, . The signal temperature fluctuation is proportional to the density fluctuation , with transfer function as the proportionality factor,
where is a unit vector in the direction of . Note that we use the subscript “0” to denote when the transfer function , the temperature fluctuation , their derivatives, or the power spectrum , are evaluated at . Furthermore, we omit explicit dependence of on redshift and on cosmological parameters, and consider it implied. Finally, note that is a function of the direction vector , while the power spectrum is a function of the magnitude , in an isotropic universe. The expression for the transfer function is obtained from Eq. (1),
for a reference frame where the magnetic field is along the –axis. For simplicity of the expressions, we adopt the following notation
where for adiabatic evolution of the magnetic field.
The signal power spectrum in the absence of a magnetic field (null case) is given by
The total measured null–case power spectrum is
In §IV.1, we first consider the case of a field uniform in the entire survey volume; this case is described by a single parameter, . In §IV.2, we move on to the case of a stochastic magnetic field, with a given power spectrum (where is the wavevector of a given mode of the field); in this case, the relevant parameter is the amplitude of this power spectrum, . In both cases, we assume that there is a valid separation of scales: density–field modes in consideration must have much smaller wavelengths than the coherence scale of the magnetic field (or a given mode wavelength for the case of a stochastic magnetic field), and both length scales must fit within the size of the survey.
iv.1 Uniform field
We now derive an estimator for a comoving uniform magnetic field. We adopt the linear–theory approach and start with
where is a small expansion parameter. The observable two–point correlation function in Fourier space is then
where we use the reality of and , assume that the signal and the noise are uncorrelated, and keep only terms linear in . Since we observe only one universe, a proxy for the ensemble average in Eq. (30) is measurement of the product . Thus, an estimate of from a single temperature mode is
where we use the following properties of the Dirac delta function (defined on a finite volume of the survey)
which is related to the Kronecker delta as
The estimator of Eq. (31) is unbiased (with a zero mean), . The covariance of estimators derived from all measured temperature modes involves temperature–field four–point correlation function with three Wick contractions, whose numerator reads
where every ensemble average yielded one factor of volume . Using the final expression in the above Equation, we get
Estimators from all –modes can be combined with inverse–variance weighting as
Expanding the above expression, we get the minimum–variance quadratic estimator for obtained from all temperature–fluctuation modes observed at a given redshift,
Its variance is given by
where the sums are unrestricted. Note that ; this follows from the reality condition on the temperature field, , and from the isotropy of space in the null–assumption case, . Thus, in order to avoid double counting of modes, a factor of appears at the right–hand side of Eq. (38).
Finally, the total sensitivity of a survey covering a range of redshifts is given by integrating the above Equation as
where we transitioned from a sum over modes to an integral, using . The integral is performed over the (comoving) volume of the survey of angular size (at a given redshift, given in steradians), such that the volume element reads
iv.2 Stochastic field
We now examine the case where both the magnitude and the direction of the magnetic field are stochastic random variables, with spatial variation. Note that in this Section we do not assume a particular model for their power spectra, but we do assume a separation of scales, in the sense that we are only concerned with the modes of the magnetic field that correspond to scales much larger than those corresponding to the density and temperature modes used for estimating the field, . We use to denote a component of the magnetic field along one of the three Cartesian–system axes, and to denote position vector in physical space, as before, and start with
where the subscripts and superscripts have the same meaning as before. In Fourier space, we now get
where the last step uses the convolution theorem. The observable two–point correlation function in Fourier space then becomes
where we used the reality condition . In analogy to the procedure of §IV.1, we estimate from pair of modes that satisfy as
where we only focus on terms (). The variance of this estimator (under the null assumption) can be evaluated using the above expression. Furthermore, the full estimator for from all available temperature modes is obtained by combining individual estimates with inverse–variance weights, and with appropriate normalization, in complete analogy to the uniform–field case. For the purpose of forecasting sensitivities, we are interested in the variance of the minimum–variance estimator, or equivalently, the noise power spectrum , given by
with the restriction . The factor of in the denominator corrects for double counting mode pairs, since , and the sum is unconstrained. If we only consider diagonal terms , then the left–hand side of the above Equation becomes equal to . The explicit expression for the noise power spectrum is then
Finally, transitioning from a sum to the integral (like in §V.1), we get the following expression for the noise power spectrum of one of the components of the magnetic field in the plane of the sky,