There are some achieves that prove it's flat. In August 2011 the Supernova Legacy Survey team published results showing that Ωk = –0.002 ± 0.006. In February of 2012 the Hubble Space Telescope Cluster Supernova Survey team determined that Ωk = 0.002 ± 0.005. A combination of these two results confirms that the universe possesses a flat geometry to within three places of the decimal (much less than one percent).
