A large number of extremely low-mass helium white dwarfs (ELM WDs) have been discovered in recent years. The majority of them are found in close binary systems suggesting they are formed either through a common-envelope phase or via stable mass transfer in a low-mass X-ray binary (LMXB) or a cataclysmic variable (CV) system. Here, we investigate the formation of these objects through the LMXB channel with emphasis on the proto-WD evolution in environments with different metallicities. We study for the first time the combined effects of rotational mixing and element diffusion (e.g. gravitational settling, thermal and chemical diffusion) on the evolution of proto-WDs and on the cooling properties of the resulting WDs. We present state-of-the-art binary stellar evolution models computed with MESA for metallicities of Z = 0.02, 0.01, 0.001 and 0.0002, producing WDs with masses between ∼0.16−0.45 M . Our results confirm that element diffusion plays a significant role in the evolution of proto-WDs that experience hydrogen shell flashes. The occurrence of these flashes produces a clear dichotomy in the cooling timescales of ELM WDs, which has important consequences e.g. for the age determination of binary millisecond pulsars. In addition, we confirm that the threshold mass at which this dichotomy occurs depends on metallicity. Rotational mixing is found to counteract the effect of gravitational settling in the surface layers of young, bloated ELM proto-WDs and therefore plays a key role in determining their surface chemical abundances, i.e. the observed presence of metals in their atmospheres. We predict that these proto-WDs have helium-rich envelopes through a significant part of their lifetime. This is of great importance as helium is a crucial ingredient in the driving of the κ-mechanism suggested for the newly observed ELM proto-WD pulsators. However, we find that the number of hydrogen shell flashes and, as a result, the hydrogen envelope mass at the beginning of the cooling track, are not influenced significantly by rotational mixing. In addition to being dependent on proto-WD mass and metallicity, the hydrogen envelope mass of the newly formed proto-WDs depends on whether or not the donor star experiences a temporary contraction when the H-burning shell crosses the hydrogen discontinuity left behind by the convective envelope. The hydrogen envelope at detachment, although small compared to the total mass of the WD, contains enough angular momentum such that the spin frequency of the resulting WD on the cooling track is well above the orbital frequency.

Common-envelope (CE) evolution in massive binary systems is thought to be one of the most promising channels for the formation of compact binary mergers. In the case of merging binary black holes (BBHs), the essential CE phase takes place at a stage when the first BH is already formed and the companion star expands as a supergiant. We aim to decipher the kinds of BH binaries with supergiant companions that could potentially evolve through and survive a CE phase. To this end, we compute envelope binding energies from detailed massive stellar models at different evolutionary stages and metallicities. We make multiple physically extreme choices of assumptions that favor easier CE ejection as well as account for recent advancements in mass-transfer stability criteria. We find that even with the most optimistic assumptions, a successful CE ejection in BH binaries is only possible if the donor is a massive convective-envelope giant, namely a red supergiant (RSG). The same is true for neutron-star binaries with massive companions. In other words, pre-CE progenitors of BBH mergers are BH binaries with RSG companions. We find that because of its influence on the radial expansion of massive giants, metallicity has an indirect but a very strong effect on the chemical profile, density structure, and the binding energies of RSG envelopes. Our results suggest that merger rates from population-synthesis models could be severely overestimated, especially at low metallicity. Additionally, the lack of observed RSGs with luminosities above  log(L/L⊙) ≈ 5.6 − 5.8, corresponding to stars with M ≳ 40 M⊙, puts into question the viability of the CE channel for the formation of the most massive BBH mergers. Either such RSGs elude detection due to very short lifetimes, or they do not exist and the CE channel can only produce BBH systems with total mass ≲50 M⊙. Finally, we discuss an alternative CE scenario in which a partial envelope ejection is followed by a phase of possibly long and stable mass transfer.

Context. A large number of low-mass ( < ∼ 0.20 M ) helium white dwarfs (He WDs) have recently been discovered. The majority of these are orbiting another WD or a millisecond pulsar (MSP) in a close binary system; a few examples are found to show pulsations or to have a main-sequence star companion. There appear to be discrepancies between the current theoretical modelling of such lowmass He WDs and a number of key observed cases, indicating that their formation scenario yet remains to be fully understood. Aims. Here we investigate the formation of detached proto-He WDs in close-orbit low-mass X-ray binaries (LMXBs). Our prime focus is to examine the thermal evolution and the contraction phase towards the WD cooling track and investigate how this evolution depends on the WD mass. Our calculations are then compared to the most recent observational data. Methods. Numerical calculations with a detailed stellar evolution code were used to trace the mass-transfer phase in a large number of close-orbit LMXBs with different initial values of donor star mass, neutron star mass, orbital period, and strength of magnetic braking. Subsequently, we followed the evolution of the detached low-mass proto-He WDs, including stages with residual shell hydrogen burning and vigorous flashes caused by unstable CNO burning. Results. We find that the time between Roche-lobe detachment until the low-mass proto-He WD reaches the WD cooling track is typically Δt proto = 0.5−2 Gyr, depending systematically on the WD mass and therefore on its luminosity. The lowest WD mass for developing shell flashes is ∼0.21 M for progenitor stars of mass M 2 ≤ 1.5 M (and ∼0.18 M for M 2 = 1.6 M ). Conclusions. The long timescale of low-mass proto-He WD evolution can explain a number of recent observations, including some MSP systems hosting He WD companions with very low surface gravities and high effective temperatures. We find no evidence for Δt proto to depend on the occurrence of flashes and thus question the suggested dichotomy in thermal evolution of proto-WDs.

Context. Millisecond pulsars (MSPs) are generally believed to be old neutron stars (NSs) that have been spun up to high rotation rates via accretion of matter from a companion star in a low-mass X-ray binary (LMXB). This scenario has been strongly supported by various pieces of observational evidence. However, many details of this recycling scenario remain to be understood. Aims. Here we investigate binary evolution in close LMXBs to study the formation of radio MSPs with low-mass helium white dwarf companions (He WDs) in tight binaries with orbital periods P orb 2−9 h. In particular, we examine i) if the observed systems can be reproduced by theoretical modelling using standard prescriptions of orbital angular momentum losses (i.e. with respect to the nature and the strength of magnetic braking), ii) if our computations of the Roche-lobe detachments can match the observed orbital periods, and iii) if the correlation between WD mass and orbital period (M WD , P orb ) is valid for systems with P orb < 2 days. Methods. Numerical calculations with a detailed stellar evolution code were used to trace the mass-transfer phase in ∼400 close LMXB systems with different initial values of donor star mass, NS mass, orbital period, and the so-called γ-index of magnetic braking. Subsequently, we followed the orbital and the interior evolution of the detached low-mass (proto) He WDs, including stages with residual shell hydrogen burning. Results. We find that severe fine-tuning is necessary to reproduce the observed MSPs in tight binaries with He WD companions of mass <0.20 M , which suggests that something needs to be modified or is missing in the standard input physics of LMXB modelling. Results from previous independent studies support this conclusion. We demonstrate that the theoretically calculated (M WD , P orb )-relation is in general also valid for systems with P orb < 2 days, although with a large scatter in He WD masses between 0.15−0.20 M . The results of the thermal evolution of the (proto) He WDs are reported in a follow-up paper (Paper II).