While the total sample of AM CVn stars are optical systems, those with the shortest orbital periods have been expected to be observed with LISA, hopefully, will be observed by eLISA, and may be observed both in the optical and X-rays thanks to high mass-transfer rates (see Figure 23*).
. The emission from the donor was treated as the emission of a cooling white dwarf, using approximations to the cooling WD models of Hansen .
The emission from the accretor was treated as the unperturbed cooling luminosity of the white dwarf.37
 and RXJ0806.3+1527  and V407 Vul ) as ROSAT sources and because of the possibility of comparison with the ROSAT all-sky survey. Most AM CVn systems experience a short (106 – 107 yr) “direct impact” stage in the beginning of mass-transfer [281, 518, 464]. Hence, in modeling the X-ray emission from AM CVn systems one has to distinguish two cases: the direct impact and disc accretion.
In the case of direct impact a small area of the accretor’s surface is heated. One may assume that the total accretion luminosity is emitted as black body radiation with a temperature given by75*). The fraction of the surface that is radiating depends on the details of the accretion. It was set to 0.001, consistent with expectations for a ballistic stream  and the observed X-ray emission of V407 Vul, the known direct-impact system .
In the presence of a disc, the X-ray emission was assumed to be coming from a boundary layer with temperature –13 erg s–1 cm–2 were selected. Then, the intrinsic flux in this band, the distance and the estimate of the Galactic hydrogen absorption  can be used to estimate the detectable flux.
Figure 39* presents the resulting model for the sample limited by = 20 mag., which is typical for optically-detected AM CVn stars. In the top panel shown are 220 systems only detectable in X-ray and 330 systems also detectable in the -band. One may distinguish two subpopulations in the top panel: In the shortest period range there are systems with white-dwarf donors with such high that even sources close to the Galactic center are detectable. Spatially, these objects are concentrated in a small area on the sky. At longer periods the X-rays get weaker (and softer) so only the systems close to the Earth can be detected. They are more evenly distributed over the sky. Several of these systems are also detectable in the optical (filled symbols). There are 30 systems that are close enough to the Earth that the donor stars can be seen as well as the discs (filled squares). Above = 600 s, the systems with helium-star donors show up and have a high enough mass transfer rate to be X-ray sources, the closer ones of which are also visible in the optical, as these systems always have a disc. The bottom panel shows the 1230 “conventional” AM CVn systems, detectable only by optical emission, which for most systems emanates only from their accretion disc. Of this population 170 objects closest to the Earth also have a visible donor. The majority of the optically-detectable systems with orbital periods between 1000 and 1500 s are expected to show outbursts due to viscous-thermal disc instability [775, 623, 379], which could enhance the chance of their discovery.
In  the estimates of optical and X-ray emission from AM CVn stars were applied to predict their numbers that can be detected in gravitational waves and the electromagnetic spectrum by LISA. These results were also discussed in the previous version of this review. However, in view of the cancellation of the LISA mission and the different sensitivity of eLISA, new insights into the problem of the resolution of low-frequency GW sources and the recognition that the most promising for detection of new objects with future facilities is the IR-band, these results may be considered as outdated and a new study of the problem is necessary.
To conclude this section, we stress several points that are important for the understanding of the formation and evolution of compact WD binaries.
- The major issue concerning compact WD binaries – DDs, IDDs, UCXBs – is their number. Theoretical predictions strongly depend on assumed parameters and range within an order of magnitude (see references in Section 10). The treatment of common envelopes and the distribution of stars over are, perhaps, the most crucial points. On the other hand, observational estimates suffer from numerous selection effects and resulting incompleteness of the samples. For instance, the local sample of all WD is considered to be 100% complete within 13 pc and only 85% complete within 20 pc [715, 716].
- The space density of the observed AM CVn-stars is by a factor close to 50 lower than the “optimistic” predictions, but the situation can be improved: about half of the known and candidate systems were found within the past decade, predominantly thanks to systematic searches for candidates in the SDSS and PTF data. Hopes for future discoveries may be associated with facilities that will become operative in the next 10 years, like JWST or E-ELT.
- Another key issue in the study of detached and interacting WD binaries is the determination of their distances. In this aspect, the GAIA space probe, which is aimed at constructing a three-dimensional map of our Galaxy via measurements of positions and radial velocities of about one billion stars (see ), appears to be the most helpful.
A failure to discover a significant number of detached and interacting double degenerates or to confirm the current ideas on their structure and evolution would mean that serious drawbacks exist either in the implementation of the known stellar evolution physics and observational statistical data in the population synthesis codes, or in our understanding of the processes occurring in compact binaries, or in the treatment of selection effects. Special attention in theoretical studies has to be paid to the onset of mass-transfer.
Above we presented some of the current ideas on the formation and evolution of compact binaries that can be interesting for general relativity and cosmology and on signals that can be expected from them in the waveband of space-born GW detectors. There is another side of the problem – the analysis of the GW signal, would it be detected. This topic is out of the scope of this brief review. We refer the reader to several papers discussing methods of detecting and subtracting individual binary signals from a data stream with many overlapping signals , of inferring properties of the distribution of white-dwarf binaries , of determining the accuracy of parameter estimation of low-mass binaries [750, 653], and the discussion of the wealth of information that may be provided by eLISA .