I have addressed this problem by using expanding wave solution of Newton's inverse square law (modified). A Schrodinger like wave equation is developed from Newton's inverse square law for two body particle anti-particle system and Hubble parameter is equated to Planck frequency at Planck time which is of order 10^43. As the universe undergoes accelerated expansion, this value of Hubble parameter rolls down to the present value which is of order 10^{-18}.

Article Periodic quantum gravity and cosmology

Thanks Vikram, for your answer. Best wishes

Recently, via a different approach, I have shown that application of quantum mechanics to a wavefunction representing observable universe, can solve the dark energy problem in a simple way, and can do so without any new physics. Of course as I am sure you realize, this does not prove that it is the solution to the dark energy problem(!), just that it might be. It comes about as a result of an interesting coincidence I found between the present day value of the Hubble parameter (i.e. Hubble's 'constant'), the Planck Length and the mass of the observable universe, i.e. a link between the smallest and largest structures of the universe. What I showed in a recent paper presented at the 2012 AIP congress (and an accompanying refereed paper from this site:

http://www.aip.org.au/info/sites/default/files/Congress2012/01._ERNEST_AIP2012.pdf, and a preprint at http://arxiv.org/ftp/arxiv/papers/1202/1202.5731.pdf - both attached here) was that one can derive the Hubble constant by examining the rate a which a standard quantum wavepacket expands. The coincidence is that, if a wavepacket with a mass equal to that of the observable universe had a size of the Planck Length at the Planck Time, then simple quantum theory predicts that the present day rate of expansion of that packet is almost the same as the Hubble Constant in GR. I find this a remarkable coincidence. (FYI, You're probably well aware of the quantum physics anyway but just in case its not your field, any quantum wavepacket expands at a rate which depends on the mass of the packet and it's initial size: the greater the mass, the less the packet expansion rate, and the smaller the initial size, the greater the packet expansion rate.)

Furthermore, using the mass of the observable universe as the sole fitting parameter one can obtain the correct observed switch-over time from deceleration to acceleration, and the present day rate of acceleration. It is interesting also that ultimately the GR expansion (an expansion of space-time dependent on the average universal density) gets less with time, while the  QM expansion (which is an expansion of the universal quantum wavepacket within that space time) gets greater with time. The QM wavepacket behavior ultimately gives a constant rate of expansion but when the two effects are similar in magnitude, as they have been over the latter part of cosmic history, one observes that the distant galaxy clusters are receding faster now than they were in the past, i.e. we observe an acceleration over cosmic time. One can also show that the quantum wavepacket expansion process does not violate the light speed limit (as wavepacket expansion 'piggybacks' GR expansion) and also does not interfere with structure formation within the packet, i.e. does not affect fragmentation of packet substructure into clusters of galaxies etc., although looking for effects from adjacent 'Planck boxes' (which could be manifested at the very distant regions of the universe) may be a way to look for evidence of the hypothesis. Still it's a bit of a radical departure from the traditional paradigm of cosmology since it implies we could possess some position and motion relative to the original Planck box - depending on where we sit in relation to the center of the original Gaussian packet, which presumably would be what we see as the CMB rest frame. One can show an array of adjacent Gaussian boxes yield an infinite uniform density so isotropy would at least still be preserved. Cheers Allan E.