The linear power spectrum of observed source number counts (1105.5292v2)

Abstract: We relate the observable number of sources per solid angle and redshift to the underlying proper source density and velocity, background evolution and line-of-sight potentials. We give an exact result in the case of linearized perturbations assuming general relativity. This consistently includes contributions of the source density perturbations and redshift distortions, magnification, radial displacement, and various additional linear terms that are small on sub-horizon scales. In addition we calculate the effect on observed luminosities, and hence the result for sources observed as a function of flux, including magnification bias and radial-displacement effects. We give the corresponding linear result for a magnitude-limited survey at low redshift, and discuss the angular power spectrum of the total count distribution. We also calculate the cross-correlation with the CMB polarization and temperature including Doppler source terms, magnification, redshift distortions and other velocity effects for the sources, and discuss why the contribution of redshift distortions is generally small. Finally we relate the result for source number counts to that for the brightness of line radiation, for example 21-cm radiation, from the sources.

• The paper presents a unified framework using linear perturbation theory to connect observed source counts with underlying density, velocity, and gravitational influences.
• It provides analytical and numerical results, including CAMB sources code for computing angular power spectra and cross-correlations with CMB signals.
• The research advances cosmological data analysis by integrating redshift distortions, magnification bias, and radial displacement effects in observable counts.

An Expert Analysis of "The Linear Power Spectrum of Observed Source Number Counts"

This paper, authored by Anthony Challinor and Antony Lewis, explores the intricacies of relating observable astronomical measurements, specifically the number of sources per solid angle and redshift, to the underlying physical quantities such as proper source density and velocity, as well as the line-of-sight potentials. This paper is anchored in the principles of linearized general relativity, aiming to provide a comprehensive theoretical framework to bridge the gap between theoretical predictions in cosmology and actual observations.

The authors methodically construct a framework that accounts for several contributing factors that affect source counts observed in redshift surveys: source density perturbations, redshift distortions, gravitational lensing (magnification), radial displacements, and other linear contributions. These elements are treated cohesively within the context of linear perturbation theory. The work stands as an advancement in consistently including these effects together, ensuring that minor terms aren't overlooked in the analysis of sub-horizon scales.

Key Contributions and Results

1. Theoretical Framework: The paper presents exact results for source number counts using linear perturbation theory and general relativity. The framework encapsulates contributions from a variety of potential perturbations and source velocities which are crucial to interpreting observed data accurately.
2. Analytical Calculations: For magnitude-limited surveys at low redshift, the paper offers linear results considering magnification bias and radial-displacement effects. This enables the calculation of angular power spectra for the total count distribution and cross-correlations with CMB polarization and temperature.
3. Practical Implications: The computations and derived relationships play a pivotal role in improving our understanding of large-scale structure formation and enhancing the interpretation of cosmological data. The integration of complex factors like redshift distortions and magnification bias into observational prediction models is particularly noteworthy.
4. Numerical Results and Code Availability: An important aspect of the paper is the provision of a numerical code (CAMB sources), which is publicly accessible. This code facilitates the computation of source count angular power spectra and their cross-correlation with CMB, incorporating various velocity and post-Newtonian effects.
5. Cross-Correlation Analysis with CMB: The paper extends its analysis to include the cross-correlation of source counts with the Cosmic Microwave Background (CMB). It emphasizes why redshift distortions have a minimal contribution to this correlation, adding depth to the theoretical exploration of cosmological signal interrelations.

Implications for Cosmology and Future Directions

This research facilitates a robust linking of theoretical models to empirical data through its self-consistent treatment of source count perturbations. The outcomes are instrumental in setting accurate constraints on cosmological parameters and the large-scale structure of the universe. They also provide insight into additional observational phenomena such as the ISW effect and its relation to galaxy surveys.

Future research avenues may incorporate further elucidation of non-linear effects, potentially incorporating more complex interactions like those found in alternative gravity theories or advanced models of structure formation beyond the linear regime. Understanding these dynamics can refine predictions for upcoming cosmological surveys with more sensitive instruments.

This paper is a significant step in unifying observational astronomy with cosmological theory, enhancing our capacity to extract meaningful insights about the universe's evolution from the data available from current and future astronomical surveys.