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Baryonic power spectrum P(k)

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Power spectrum of the Lyα forest

Since the spatial distribution of the baryons in the Intergalactic Medium is determined by the gravitational amplification of density fluctuations of the dark matter, it is natural to examine whether the matter density fluctuations could be measured through fluctuations in the distribution of baryons. The Cosmic Microwave Background contains a clear imprint of the matter fluctuations when still in the linear regime of gravitational amplification. The baryons in the IGM represent the quasi-linear regime intermediate between the CMB fluctuations and the highly non-linear fluctuations leading to galaxy formation. It may therefore be hoped that the baryons may be used to trace the evolution of the baryon fluctuations at intermediate redshifts and scales. Such measurements would in principle provide independent constraints on the primordial power spectrum P(k) and the geometry of the Universe, and may even provide evidence for novel physics should distortions in the power spectrum from the behaviour expected at the intermediate length scales probed by the baryons be discovered.


Several uncertainties, however, complicate any attempt to determine the primordial power spectrum using the baryons:

1. Only the 1D power spectrum is measured
Only 1D power spectra may be directly measured, not the full 3D power spectrum. This is because currently the only means of inferring the baryonic density fluctuations is through measurements of the Lyα forest. In the future, this difficuly may be circumvented if the 3D distribution of the baryons may be measured through radio tomography using the 21cm line. Until then, efforts must be limited to the fluctuations in the measured flux levels in the spectra of QSOs.
2. The 1D power spectrum is difficult to predict
The 1D matter power spectrum is a convolution over all scales of the underlying 3D matter power spectrum. Since the 3D matter spectrum is non-linear at small scales, predictions for the 1D matter spectrum require high resolution N-body computations. As a result, computations of the 1D matter power spectrum are poorly converged. This effect, however, is much ameliorated by the coherence in the flux measurements resulting from the finite widths of the Lyα absorption features. As a result, the power spectrum of the QSO flux smooths over the most strongly non-linear scale. This more readily allows a converged flux power spectrum on large scales.
3. The QSO flux power spectrum is difficult to predict
The scale of smoothing of the underlying matter fluctuations in the QSO flux fluctuations depends on the dispersion velocity of the baryons and the temperature of the IGM, which is currently unknown. A reliable computation of the IGM temperature requires reionisation simulations . The unknown process by which the Universe was reionised precludes a definitive temperature prediction. Detailed numerical computations are also required to determine the turbulent velocity field broadening the absorption lines. Such computations are expensive, as they require solving the fluid equations along with the gravity equations for the dark matter. In lieu of this, attempts have been made to artificially smooth the matter density fluctuations to mimic these effects in N-body simulations for the dark matter, and dispense with solutions of the fluid equations. The resulting estimates for the smoothed flux power spectrum , however, depend on the smoothing scheme, which is inevitably in part ad hoc.
4. The flux power spectrum is difficult to measure
Uncertainties in the varying continuum of the QSO through the Lyα forest result in uncertainties in the flux power spectrum. The limited number of modes probed is another limitation. While a large number of modes are available at large wavevectors, these are the ones most affected by the smoothing uncertainties. Fewer modes, and consequently larger errors, are available at the smaller wavevectors, hindering precise flux P(k) comparisons between models and observations.
Continued measurement efforts
None of these uncertainties are insurmountable in principle. Efforts are being made by various groups to develop the flux power spectrum into a precise tool for inferring the underlying matter power spectrum. These efforts are particularly interesting since they offer the best promise for measuring the power spectrum on intermediate length scales, amenable neither to Cosmic Microwave Background experiments, which probe the larges scales, nor galaxy surveys, which probe the smallest.
For further information...
A review of recent efforts to infer the matter power specturm from measurements of the Lyα forest in the QSO spectra of the Sloan Digital Sky Survey is found here.
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