Three limits to the physical world

Also shown on the diagram is the present density of the Universe, equal to the critical density in the Friedman-Lemaître-Robertson-Walker metric, ρ FLRW ~10 -120 . The corresponding line is labeled "Einstein limit" in reference to Einstein's cosmological constant: the Universe is known to be currently dark energy dominated, dark energy being well described by a cosmological constant Λ contributing a term ρ Λ =Λ/8πG to the energy density of the Universe (ρ Λ ~ρFLRW ).

The third summit of the triangle made by the Heisenberg, Schwarzschild and Einstein limits corresponds to an object having ξ10 60 and η1, namely a mass m Λ such that ρ Λ ~mΛ 4 , i.e. m Λ ~10 -30 or 10 meV, in the range currently accepted for the lightest known fermions, neutrinos. It implies that dark energy -or the cosmological constantsets a lower limit to the mass of composite quantum objects in the range of neutrino masses [2]. Indeed, lines of constant masses are straight lines η=m√ξ with m ranging from the neutrino mass, ~10 -30 , to the mass of the Universe, ~10 60 , with the unit Planck mass (~10 19 GeV) inbetween. Similarly, lines of constant sizes are straight lines η=l/√ξ with l ranging from the unit Planck scale (~10 -33 cm) to the size of the Universe, ~10 60 , with the scale of the “neutrino” wave packet (~10 30 ) in the middle.

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