Supersonic baryon-CDM velocities and CMB B-mode polarization
It has recently been shown that supersonic relative velocities between dark matter and baryonic matter can have a significant effect on formation of the first structures in the universe. If this effect is still non-negligible during the epoch of hydrogen reionization, it generates large-scale anisotropy in the free electron density, which gives rise to a CMB B-mode. We compute the B-mode power spectrum and find a characteristic shape with acoustic peaks at . The amplitude of this signal is a free parameter which is related to the dependence of the ionization fraction on the relative baryon-CDM velocity during the epoch of reionization. However, we find that the B-mode signal is undetectably small for currently favored reionization models in which hydrogen is reionized promptly at , although constraints on this signal by future experiments may help constrain models in which partial reionization occurs at higher redshift, e.g. by accretion onto primordial black holes.
It was recently realized Tseliakhovich:2010bj () that supersonic relative velocities between baryons and cold dark matter can have a significant effect on formation of the first structures in the universe. The physics can be described intuitively as follows.
The baryon-CDM relative velocity is coherent on scales smaller than a few comoving Mpc (Fig. 1). Let us imagine dividing the universe into subregions of this size, with a constant value of in each subregion. As shown in Tseliakhovich:2010bj (), the baryon-CDM flow will suppress growth of perturbations whose comoving wavenumber is larger than the freestreaming scale . Equivalently, a mode is suppressed if its wavelength is less than the distance that the baryon-CDM flow travels in a Hubble time.
Perturbations whose wavenumber is larger than the Jeans scale , where is the baryon sound speed, are additionally suppressed by the usual Jeans mechanism (pressure support). For large , Jeans suppression dominates (by an extra factor of in the equations of motion) over the suppression. Therefore, the suppression is important on a window of scales given by , illustrated in Fig. 2. The size of this window is equal to , the Mach number of the flow, so the existence of a range of scales where the suppression dominates is equivalent to a supersonic flow. For , a typical Mach number is , so there exists a range of scales where suppression is an order unity effect.
The first stars in the universe are predicted to form at redshift , in dark matter halos with typical mass . The abundance of such halos depends sensitively on the amplitude of perturbations on a broad range of scales near characteristic wavenumber Mpc. This range of scales is broad enough that there is some overlap with the range where suppression is important, and so the abundance of the first stars is sensitive to the local value of . For larger halo masses, the overlap of scales decreases rapidly, and suppression is less important.
If we spatially average over a large region of the universe, then the overall effect of the baryon-CDM flow is to delay the formation of the first bound structures. What is more distinctive is that the effect is inhomogeneous: the halo abundance will depend on the local value of . Since the power spectrum of the field has a different shape than the usual contributions to the halo power spectrum (i.e. a Poisson term and a term proportional to the matter power spectrum), this makes the effect qualitatively new, e.g. it was shown in Yoo:2011tq () that the effect can shift the apparent scale of the baryon acoustic peak in the halo two-point function.
At redshift , the abundance of halos (or stars) is significantly modulated by the local value of . What is less clear (and a critical question for the phenomenology) is whether the effect “survives” to lower redshift: if we inspect the universe at , can we still tell the difference between a low- and a high- region? The gravitational suppression mechanism described above suggests that the abundance of the largest halos is relatively unaffected by : the scales which are suppressed by are smaller than the scales which are relevant for spherical collapse of these halos (Fig. 2). However, the number (or total luminosity) of luminous objects such as galaxies could depend on , due to a variety of possible feedback effects. An example of positive feedback would be metal enrichment of the intergalactic medium: in a large- region, formation of the first stars is suppressed, leading to lower IGM metallicity, which further suppresses star formation at later times since it will be harder for gas to cool via atomic lines. In the opposite direction, an example of negative feedback would be heating of the IGM by starlight: in a large- region with fewer stars at , the IGM will be colder than average, which makes it easier for clouds of gas to overcome pressure support and collapse gravitationally, leading to increased star formation at later times.
It is also interesting to take an agnostic approach to the question of whether the effect survives to low redshift (i.e. treating the size of the effect as a free parameter), and study the potential impact on various cosmological observables. For example, a recent paper Yoo:2011tq () considered the possibility that the effect survives long enough to affect galaxy abundance at redshifts . Here, we will study the analogous possibility that the effect survives long enough to affect the hydrogen ionization fraction during the epoch of reionization , and compute the CMB B-mode which is generated. In particular, we would like to understand whether large-scale B-modes sourced by the effect can significantly contaminate the gravity wave signal at , since this will be a primary scientific target for polarization experiments in the near future.
Ii B-mode calculation
We assume that the mean ionization fraction at redshift depends weakly on the local value of . By rotation invariance, can only depend on ; it is convenient to change variables by defining the field
which is normalized to have zero mean, and is (to an excellent approximation) independent of redshift as implied by the notation.
We will assume that the effect is small at , so that the ionization fraction can be linearized in :
where we have introduced a free parameter which parameterizes the size of the effect at redshift . Intuitively, is the mean fractional change in produced by a fluctuation in the local value of . The CMB optical depth in direction is given by
where is the comoving electron density at redshift zero, is the Thomson scattering cross section, and is the comoving distance to redshift in the assumed flat cosmology. In the model (2), the angular power spectrum of the field is given (in the Limber approximation) by
Note that we are only considering new contributions to due to variations in the ionization fraction due to the effect; the full power spectrum contains additional terms (e.g. a “one-bubble” term arising from Poisson statistics of individual HII regions) not considered here Mortonson:2006re ().
Linear CMB polarization is generated by Thomson scattering of CMB photons by free electrons. If reionization is spatially homogeneous, then the combination of rotation and parity invariance implies that only E-mode polarization is generated. The CMB B-mode power spectrum generated by inhomogeneous reionization is a sum of scattering and screening terms Hu:1999vq (); Dvorkin:2008tf (); Dvorkin:2009ah (); these correspond respectively to generation of new linear polarization via Thomson scattering by free electrons, and screening of the primary E-mode by inhomogeneities in the optical depth.
where is the effective optical depth to reionization and K is the RMS temperature quadrupole during matter domination.
For calculating the B-mode power spectrum at the 10% level, we can make the approximation that the comoving distance is constant over the range of redshifts which contribute to the integral in (4). We can then factor as the product of a parameter which depends on reionization history and the bias parameter , and a template shape in which depends only on cosmological parameters:
To complete the calculation of the B-mode power spectrum, we need to compute the power spectrum . The two-point correlation function of the vector field is given by:
where are spatial indices and is the relative velocity power spectrum, which can be computed using CAMB Lewis:1999bs (). From the definition (1) of , the power spectrum is a four-point function in which can be computed straightforwardly using (11) and Wick’s theorem. The result can be presented either in Fourier space,
or equivalently in position space as
where the correlation functions , are defined by:
In Fig. 3, we show the B-mode power spectrum for a fiducial reionization model defined as follows. The reionization history is given by the CAMB parameterization, with reionization width , central redshift , and total optical depth . To get a sense for the order of magnitude of , if is constant in , then . In Dalal:2010yt (), the value of at is predicted to be 0.01 (this value is consistent with the first attempts to include the effect in simulations Maio:2010qi (); Stacy:2010gg (); Greif:2011iv (); Naoz:2011if ()), so we will take as our fiducial value.
For this fiducial reionization model, we find that the total B-mode power spectrum generated by baryon-CDM relative velocities is small (roughly 0.14 K-arcmin on large scales). The scattering power spectrum has acoustic peaks at values of which are multiples of , where Mpc is the angular diameter distance to reionization and Mpc is the baryon acoustic scale.
There is one additional effect which has not been included in our B-mode calculation: high- damping due to freestreaming. Our model (2) allows the mean ionization fraction at position to depend on the velocity at the same point ; in reality it will also depend on nearby values of since the mean free path of reionizing photons is nonzero. The qualitative effect will be to suppress the power spectra on scales , where Mpc is the characteristic size of HII regions during reionization, but a precise calculation of the damping tail will depend on model-dependent details.
Iii Observational prospects and discussion
To forecast detectability of the CMB B-modes produced by the effect, we consider several observational scenarios. The error on is given by
where is the lensing B-mode power spectrum and is the noise power spectrum of the experiment.
For a lensing-limited experiment (i.e. ) with and , we find . This forecast effectively treats the lensing B-mode as an extra source of noise and could potentially be improved by incorporating delensing techniques (e.g. Knox:2002pe (); Kesden:2002ku (); Hirata:2003ka ()) which statistically separate the lensing B-mode from other components. Using the forecasting methodology from Smith:2010gu (), we find that the statistical error can be improved to , , or , assuming a Gaussian beam with and polarization noise level = 3, 1, or 0.25 K-arcmin respectively. In all cases, the statistical error is larger than the value , indicating that no statistically significant detection of the signal is possible in our fiducial reionization model.
The basic reason that the B-mode is small in our fiducial model is that we have assumed that reionization is dominated by stars at , and that the abundance of these sources depends weakly on the local value of (i.e. is small at these redshifts). A larger B-mode may be generated if can become large. For example, if positive feedback mechanisms (such as metal enrichment of the IGM) help the effect survive to , then may be enhanced, leading to a larger signal. Models of reionization in which high redshifts contribute non-negligible optical depth should also lead to increased . For example, scenarios where X-ray emission from primordial black holes ionize the universe by a few percent at redshift have been considered e.g. in Ricotti:2007au (). Although a quantitative study is beyond the scope of this paper, the bias parameter could be as large as order unity in these models, since the Hoyle-Bondi rate for accretion of gas onto a black hole is proportional to , and and are of the same order of magnitude (so that statistical fluctuations in lead to order-unity variations in the accretion rate).
In conclusion, although optical depth anisotropy sourced by baryon-CDM relative velocities generates a CMB B-mode in principle, we find that the amplitude is likely to be unobservably small, if currently favored reionization models are correct. On the other hand, no new data analysis is required to fit for the amplitude of the B-mode (one simply fits a multiple of a fixed template shape to the estimated power spectrum), so estimating from future B-mode measurements may be a useful test of the assumptions underlying these models. In any case, since the acoustic features at are on different scales than the ones which are relevant for gravity waves ( and ), the B-mode power spectrum calculated here cannot be a contaminant for the gravity wave signal, even in a reionization model where its amplitude is large.
Acknowledgements. We thank Renyue Cen, Dan Grin, Jerry Ostriker, and David Spergel for useful discussions. SF was supported by the Martin Schwarzschild Fund in Astronomy at Princeton University. KMS was supported by a Lyman Spitzer fellowship in the Department of Astrophysical Sciences at Princeton University. CD was supported by the Kavli Institute for Cosmological Physics (KICP) at the University of Chicago through grants NSF PHY-0114422 and NSF PHY-0551142 and an endowment from the Kavli Foundation and its founder Fred Kavli. CD is additionally supported by the Institute for Advanced Study through the NSF grant AST-0807444 and the Raymond and Beverly Sackler Funds.
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