Gravitational Lensing of Pregalactic 21 cm Radiation
Low-frequency radio observations of neutral hydrogen during and before the epoch of cosmic reionization will provide hundreds of quasi-independent source planes, each of precisely known redshift, if a resolution of arcminutes or better can be attained. These planes can be used to reconstruct the projected mass distribution of foreground material. A wide-area survey of 21 cm lensing would provide very sensitive constraints on cosmological parameters, in particular on dark energy. These are up to 20 times tighter than the constraints obtainable from comparably sized, very deep surveys of galaxy lensing although the best constraints come from combining data of the two types. Any radio telescope capable of mapping the 21cm brightness temperature with good frequency resolution ( 0.05 MHz) over a band of width 10 MHz should be able to make mass maps of high quality. If the reionization epoch is at very large amounts of cosmological information will be accessible. The planned Square Kilometer Array (SKA) should be capable of mapping the mass with a resolution of a few arcminutes depending on the reionization history of the universe and how successfully foreground sources can be subtracted. The Low-Frequency Array (LOFAR) will be able to measure an accurate matter power spectrum if the same conditions are met.
Max Plank Institut für Astrophysics, Karl-Schwarzchild-Str. 1, 85741 Garching, Germany
The pregalactic 21 cm radiation originates from a period when most of the hydrogen in the universe was neutral before most of the galaxies formed and emitted ionizing radiation. This is often called the dark ages. At the universe cooled to the extent where nearly all of the hydrogen became neutral. We know this from the spectrum of the CMB. Some time between and most of the hydrogen in the universe reionized. We know this from the lack of sufficient Lyman- absorption in high redshift quasar spectra. The 21 cm line comes from the hyperfine splitting of the ground state of hydrogen (coupling between the spin of the electron and proton). It is redshifted to several meters when it reaches us.
There will be many statistically independent regions of 21 cm emission at different redshifts (or frequencies) at a single position on the sky. Gravitational lensing will distort the images of these independent sources in a coherent way. By “stacking” up the 21 cm maps in different frequencies the lensing distortion can be separated from the intrinsic structure in the 21 cm emission [1, 2].
There are several telescopes being planned and built to observe the 21 cm radiation from the epoch of reionization. Among these are the 21 Centimeter Array (21CMA, formerly known as PAST)111http://21cma.bao.ac.cn/, the Mileura Widefield Array (MWA) Low Frequency Demonstrator (LFD)222ttp://www.haystack.mit.edu/ast/arrays/mwa/, the core array of LOFAR (Low Frequency Array)333www.lofar.org and the core of SKA (Square Kilometer Array)444www.skatelescope.org/. Only LOFAR and SKA will have sufficient collecting area to be relevant for gravitational lensing observations as presently conceived.
2 Imaging Dark Matter
When low-frequency radio telescopes become sufficiently powerful to map the signal from high-redshift 21 cm emission/absorption within 10 or more statistically independent bands, the noise in the lensing maps of foreground mass will be limited by the number of frequency bands and not by the noise in the temperature maps themselves. This noise level is called the irreducible level because it depends only on the statistics of the source and the frequency range of the telescope. The irreducible noise level is low enough that a high fidelity map of the foreground mass distribution could be made. At SKA-like resolution objects with virial masses as small as would be clearly visible. The 21 cm emission is intrinsically correlated to a relevant degree for frequency separations below 0.05 MHz on 1 arcmin scales so decreasing the bandwidth below this level produces no improvement in the lensing noise . Simulated surface density maps have been made by ray-tracing through the Millennium simulation .
In the near term the telescopes will not reach the irreducible noise limit because of noise in the brightness temperature measurements. Figure 1 shows the noise in multipole space for a SKA-like telescope and a LOFAR-like telescope after 90 days of observing (approximately 3 seasons). Where the noise per mode is well below the expected signal a high fidelity reconstruction can be made. SKA should be able to image the matter with a resolution of a few arc-minutes. Exceptional regions such as around large galaxy clusters would have higher fidelity down to smaller scales. A LOFAR-like telescope will probably not be able to image typical fluctuations in the mass density. It has been assumed for figure 1 that the universe very rapidly reionized at and that the frequency range of the observations goes down to 100 MHz or for SKA and 114 MHz or for LOFAR. Reionization could increase the signal or decrease it depending on how and when it occurs. See  for details.
Increasing the collecting area of the core of SKA by a factor of two would reduce the noise to close to the irreducible limit. Increasing the resolution of the telescope would reduce the irreducible noise both because of the number of statistically independent redshift slices increases and because the number of independent patches on the sky increases. If a resolution of 6 arcsec (fwhm) could be achieved, every halo more massive than the Milky Way’s () would be clearly visible back to z10! Since almost no halos more massive then would have formed by this would be a complete inventory of all halos above .
21 cm lensing surveys could be cross-correlated with any other survey of foreground objects including galaxy lensing surveys to get additional tomographic information. Images of the mass distribution through nearly the entire depth of the observable universe would be of enormous value for the study of cosmology and galaxy formation, and a very direct test for the existence of dark matter.
3 Measuring Cosmological Parameters
21 cm lensing is also capable of measuring the matter power spectrum and its evolution. Cross-correlating the convergence maps from 21 cm lensing at different redshifts with each other and with galaxy lensing surveys provides information on the evolution of structure formation. Any cosmological parameters that affect the power spectrum and/or its evolution can be probed in this way. Dark energy is of particular interest and 21 cm lensing would provide a unique probe of its behavior at redshifts above as well as substantially constricting the constraints on it below .
For estimating cosmological parameters the requirements on resolution and frequency range are not as demanding as for imaging dark matter, but survey area is of greater importance. At the irreducible noise level the uncertainties in the cosmological parameters are dominated by sample variance if 10s of independent redshift slices are used. Even for SKA, sample variance should be the most important source of error for multipole number ( arcmin / ). This can be seen in figure 1. Where the noise per mode is below the expected signal the sample variance will dominate in the power spectrum measurement. The noise in the power spectrum estimate on these scales can only be reduced by increasing the area of the survey. For comparison, an ambitious all-sky galaxy lensing survey would reach the sample variance limit at an of a few hundred. For the proposed radio interferometers the survey area depends primarily on information storage and processing speed which should improve over time. Table 1 shows some estimates of the constraints that would be possible for a SKA-like survey including modes up to combined with an idealized galaxy lensing survey.
The power spectrum of the projected mass density or convergence could also be measured. Figure 1 shows the forecasted error bars for SKA and LOFAR-like telescopes. Both telescopes should be capable of measuring the power spectrum well and SKA very well.
|galaxies survey||galaxies survey alone|
|with SKA-like 21 cm survey|
|0.008 (0.008)||0.03 (0.02)|
Much will depend on future instrument design and the as yet unknown characteristics of the 21 cm absorption and emission, particularly around the epoch of reionization. If reionization happens unexpectedly early there may not be a large enough range of redshift within the observed frequency range. Despite this, the planned specifications for SKA might enable us to make high fidelity maps of the matter distribution all the way back to and, if enough area can be surveyed, very good statistical information would be possible enabling very high precision measurements of cosmological parameters. Realistic upgrades to the collecting area and array size of the next generation of telescopes would greatly improve their sensitivity to lensing.
Acknowledgements.The author would like to thank the organizers for an excellent conference and for the generous award.
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