  Gravitational Waves (GWs) were observed for the first time in 2015, one
century after Einstein predicted their existence. There is now growing interest
to extend the detection bandwidth to low frequency. The scientific potential of
multi-frequency GW astronomy is enormous as it would enable to obtain a more
complete picture of cosmic events and mechanisms. This is a unique and entirely
new opportunity for the future of astronomy, the success of which depends upon
the decisions being made on existing and new infrastructures. The prospect of
combining observations from the future space-based instrument LISA together
with third generation ground based detectors will open the way towards
multi-band GW astronomy, but will leave the infrasound (0.1 Hz to 10 Hz) band
uncovered. GW detectors based on matter wave interferometry promise to fill
such a sensitivity gap. We propose the European Laboratory for Gravitation and
Atom-interferometric Research (ELGAR), an underground infrastructure based on
the latest progress in atomic physics, to study space-time and gravitation with
the primary goal of detecting GWs in the infrasound band. ELGAR will directly
inherit from large research facilities now being built in Europe for the study
of large scale atom interferometry and will drive new pan-European synergies
from top research centers developing quantum sensors. ELGAR will measure GW
radiation in the infrasound band with a peak strain sensitivity of $4.1 \times
10^{-22}/\sqrt{\text{Hz}}$ at 1.7 Hz. The antenna will have an impact on
diverse fundamental and applied research fields beyond GW astronomy, including
gravitation, general relativity, and geology.
