Gradient-based explorations of the ACT DR4 likelihood surface in ΛCDM. Left: Newton-Raphson (orange) and ADAM (purple) minimisers use gradient information to find the best-fit point of the ACT DR4 data set (light to dark points). We highlight steps in the n_s − Ω_bh² plane, though the minimisation is performed over all ΛCDM parameters and the calibration parameter of the data set. While the Newton-Raphson algorithm directly descends to the best-fit point, the ADAM minimiser follows a more complicated trajectory with some stochasticity to ensure it does not get trapped in local minima. The grey contours in the background show the 1 and 2σ confidence region obtained from the Hessian, which is a by-product of the Newton-Raphson algorithm. Right: comparison of Metropolis-Hastings (MH; blue) and No-U-Turns (orange) chains run using the ACT DR4 data set in ΛCDM. We focus on H₀ samples here, though the MCMC chains explore the full parameter space. The top and middle panels show the H₀ values and acceptance rate ƒ_A (values in faint lines, rolling average over 50 samples in full colour) for an excerpt of 150 samples, respectively. The bottom panel shows the power spectrum P(k) of the H₀ samples of the full-length chain (measured spectra in faint lines, parametric fit (Dunkley et al. 2005) in full colour). The NUTS samples are less correlated by eye than the MH samples, which is confirmed by the flatter power spectrum. Additionally, the acceptance rate of the gradient-based algorithm is substantially higher, with few rejected samples.