Reconstructing the distortion function for non-local gravity

We develop Maggiore and Mancarella non-local model by using new function, $f\left(m^2\frac{R}{{◻}^2}\right)$ and we obtained analytical hyperbolic tangent function. This new model has expansion history exactly as same as ΛCDM (Lambda Cold Dark Matter) with same matter content, but without cosmological constant or dark energy. However, background evolution in our model and ΛCDM are the same, but these models may be distinguishable in structure formation or observation that will contain additional information more than background level. (paper)