Accessing the homogeneity scale with 21 cm intensity mapping surveys

TL;DR

This work introduces a framework to test the Cosmological Principle by probing the homogeneity scale $R_{ m H}$ with 21 cm intensity mapping, explicitly quantifying how telescope beam smoothing biases the measurement. By modeling beam convolution in configuration space via a Gaussian beam and its damping factor $\\mathcal{B}(k,\mu)$, the authors connect beam width $\sigma$ to a suppressed and redistributed clustering signal, and define $R_{ m H}$ through the correlation dimension $\mathcal{D}_2^{\rm obs}$ with a threshold $2.97$. They derive a redshift-dependent maximum beam width $\sigma_{ m max}(z)$, fit it for different cosmologies, and map accessible versus inaccessible regions in the $\sigma\times z$ plane, applying the results to current and upcoming single-dish 21 cm IM instruments. The study provides a quantitative forecast for instrumental requirements and offers a theoretical basis for future observational work, including cross-correlations with optical surveys and adaptation to other tracers. Overall, it establishes a path to measure a fundamental cosmological test with 21 cm IM, while clearly delineating beam-related limitations to guide instrument design and analysis pipelines.

Abstract

The homogeneity scale, $R_{\rm H}$, offers a fundamental test of the Cosmological Principle, yet it has not yet been measured with 21cm intensity mapping surveys. A key limitation for such a measurement is the telescope beam, which artificially smooths the observed signal. We quantify this effect using the two-point correlation function and the correlation dimension, $\mathcal{D}_2(r)$, to model how beam convolution suppresses intrinsic clustering. For any given redshift $z$, we identify a maximum beam width, $σ_{\rm max}(z)$, beyond which the homogeneity scale cannot be recovered. This limit defines an inaccessible region in the $σ\times z$ parameter space, where $R_{\rm H}$ is erased by beam smoothing. Applying this framework to several current and upcoming radio telescopes, we assess their ability to probe $R_{\rm H}$. Our results provide the first quantitative forecast of the instrumental requirements for measuring the cosmic homogeneity scale with 21cm IM, and establish a theoretical basis for future observational applications.

Figures (4)

• Figure 1: Illustration of the impact of different beam widths, $\sigma$, on the 2pCF at a fixed redshift, $z=0.8$. The black dashed line corresponds to the $\sigma=0$ case (no beam). All cases assume the fiducial $\Lambda$CDM cosmology.
• Figure 2: Illustration of how the correlation dimension, $\mathcal{D}_2$, is impacted with increasing beam width. Black dashed line corresponds to the $\sigma=0$ case. Dotted black line represents $\mathcal{D}_2(r)=2.97$. All cases are calculated considering the fiducial $\Lambda$CDM cosmology, fixing $z=0.8$. The inner plot shows an example of $\mathcal{D}_2$, for $\sigma = 0.5\degr$, defining the four regimes, (a)-(d), delimited by the physical size of the beam, $R_{\rm beam}$, the $r$ value at which $\mathcal{D}_2$ reaches a local minimum, $R_{\rm min}$, and the r value at which $\mathcal{D}_2$ crosses the dashed line, that is, the transition scale to homogeneity, $R_{\rm H}$. See text for details.
• Figure 3: Detectability of the transition scale, $R_{\rm H}$, for varying beam widths ($\sigma$), illustrated for the fixed redshift, $z=0.8$. The black solid (orange dashed) line represents $R^{\Lambda {\rm CDM}}_{\rm H}$ as affected by $\sigma$ considering the fiducial $\Lambda$CDM ($\Lambda$CDM$\nu$) model. The solid dark and dashed light grey lines represent $R_{\rm H}(\sigma_{\rm max}) = R_{\rm min}(\sigma_{\rm max})$, with $R_{\rm min}(\sigma_{\rm max})$ calculated for $\Lambda$CDM and $\Lambda$CDM$\nu$ models, respectively. The blue and green regions represent $R_{\rm H}$ values that could be measured, while the Inaccessible region, in grey, defines the $R_{\rm H}$ scales that would be erased by the beam effect. See text for details.
• Figure 4: Detectability of $R_{\rm H}$ in the $\sigma \times z$ parameter space. The black solid (orange dashed) line represents $\sigma_{\rm max}(z)$ as fitted by equation \ref{['eq:sgima_max']} for the $\Lambda$CDM ($\Lambda$CDM$\nu$) model. The coloured lines show the $\sigma(z) \times z$ for several 21 cm IM instruments. Their redshift (frequency) ranges are summarised in Table \ref{['tab:instruments']} and $\sigma(z)$ values calculated using equation \ref{['eq:beam_width']}.