Aun Breath is the Ideal of Tachiai.
When Rikishi prepare for Shoubu on the Dohyou they should enter into the state of Aun Breath before the last Shikiri.
By entering into Aun Breath is meant that both Rikishi will synchronize their breathing, inhaling and exhaling at the same time. Figuratively speaking it means to enter Aun at the Tachiai. This way not only their breathing but also their minds will become one.
After getting into Shikiri, both Rikishi will inhale fully, whereby Power enters, then exhale fully, then they will inhale seven to eight tens of a breath at which point the Tachiai will occurr. This is the Aun Breath Ideal of Tachiai. This is reflected in the Haku iki suu iki tomaru iki, one of Sumou Sayings. When a Tachiai occurrs in this way, it is described by Aun no kokyuu de tatsu, one of Sumou Sayings.
This is the ultimate in the Sumou Ideal of Synchronization. It's reflected in the Tachiai ha aun no kokyuu ga risou, one of Sumou Sayings.
Aun Breath is the last step of Synchronization after all the synchronized pre-bout Shoubu moves. It means Synchronization of the delicate feelings and thoughts of both Rikishi.
Aun was a Buddhist concept originally and Aun Breath is mentioned in Koumyou shingon kanju youmon jou.
It leads directly to mind Synchronization whereby Rikishi enter a state of mental unison unaffected by the outside world. For this reason Tachiai occurrs based on synchronized mental agreement of both Rikishi and a Gyouji doesn't start the Shoubu but the Rikishi themselves do.
In Amateur Sumou the Shoubu is started by the Gyouji and loses this most important aspect of mental Synchronization. It is just a physical exercise.