Curvature

The locations of the peaks in the CMB power spectra are highly sensitive to the curvature of the early universe, which affects the angular scale of the sound horizon at recombination. The parameter \(\Omega_{k}\) quantifies the energy density associated with curvature. In a flat universe, \(\Omega_{k}\) = 0, the angular scale of the sound horizon is approximately 1 degree, which aligns with current CMB observations. If the universe is open, \( \Omega_{k} > 0 \), it exhibits a saddle-shaped geometry where light rays diverge. This divergence causes the angular size of CMB fluctuations to appear smaller, shifting the acoustic peaks to smaller angular scales, hence larger multipoles. Conversely, a closed universe, \(\Omega_{k}\) < 0, resembles a spherical geometry where light rays converge. This convergence makes fluctuations appear larger and shifts the peaks of the acoustic oscillations to larger angular scales, hence smaller multipoles. We show this explictly in the figure below, where we can clearly observe how varying \(\Omega_{k}\) shifts the first peak while maintaining the overall shape of the spectra.

Curvature also impacts the low-\(\ell\) spectra through its effect on the late Integrated Sachs–Wolfe effect (see dark energy page); a phenomenon where photons gain or lose energy as they pass through decaying gravitational potentials, leading to an enhancement in temperature fluctuations. Introducing curvature, while keeping the other parameters fixed, will change the expansion history and consequently the evolution of the gravitational potentials. In an open universe, the faster rate of expansion leads to gravitational potentials decaying quicker, enhancing the late ISW effect, and thus icreasing the power at larger scales. In a closed universe, the slower rate of expansion, results in more stable potentials, reducing the late ISW effect and decreasing power at larger scales.

In the panels below, the parameters \(\omega_{m}\), \(\Omega_{\Lambda}\), \(\omega_{b}\), \(\tau\), \(10^{9}A_{s}\), and \(n_{s}\) are held fixed while \(\Omega_{k}\) and \(h\) are varied.