Machian strings as an alternative to dark energy

The expansion history of the Universe is calculated using a simple model in which the entire rest mass energy of a massive particle is distributed throughout the set of Machian strings connecting it to all the other particles in the observable Universe. With the assumption that the energy in a Machian string has the form of positive Newtonian potential energy, the deceleration rates in the radiation and matter eras are exactly the same as in the conventional $\Lambda$CDM model. The transition from deceleration to acceleration at the present time is obtained by making the simplest possible modification of the Newtonian potential energy to represent the effect of the cosmological expansion. The effective dark matter and dark energy densities are calculated in terms of the speed of the Hubble flow at the radius of the observable Universe.