The understanding of voids formation , which is at the origin of the foam like patterns in the distribution of galaxies within scale up to 100 Mpc , has become an important challenge for the large scale formation theory [ 23 ] .
Such a structure has been observed since three decades and confirmed by recent surveys [ 27 , 15 , 16 , 7 , 25 , 5 , 14 , 4 ] .
Investigations has been performed – on their statistical properties by improving identification techniques [ 24 ] , by exploring their formation process in a \Lambda CDM model through N-body simulations [ 12 , 26 , 11 , 19 , 3 , 29 , 22 ] , by probing their origins [ 21 ] ; – on the kinematics of giant voids [ 17 ] and the dynamics by testing models of void formation [ 1 , 10 , 2 ] .
Herein , we investigate the effect of the cosmological constant \Lambda on the evolution of a spherical void through an exact solution of Euler- ( modified ) Â Poisson equations system ( EPES ) [ 9 ] .
Let us remind that Friedmann-Lemaître models , which provide us with a suitable description of the universe at large scales ( thanks to their stability with respect to linear perturbations [ 18 ] ) , can be described within a Newtonian approach by means of EPES solutions for whom kinematics satisfy Hubble ( cosmological ) law .
The void consists of three distinct media : a material shell ( S ) with null thickness and negligible tension-stress , an empty inside and a uniform dust distribution outside which expands according to Friedmann equation .
We use a covariant formulation of EPES [ 28 , 6 ] for deriving the evolution with time of { S } acting as boundaries condition for the inside and outside media . 1 The corrective factor y to Hubble expansion .
It results from a void that initially expands with Hubble flow at expansion parameter a _ { i } = 0.003 , with \Omega _ { \circ } = 0.3. and \Omega _ { \lambda } = \lambda = 0 , 0.7 , 1.4 . Figure 1 The corrective factor y to Hubble expansion .
It results from a void that initially expands with Hubble flow at expansion parameter a _ { i } = 0.003 , with \Omega _ { \circ } = 0.3. and \Omega _ { \lambda } = \lambda = 0 , 0.7 , 1.4 .

As a result , S expands with a huge initial burst that freezes asymptotically up to matching Hubble flow .
The related perturbation on redshift of sources located on S does not exceed \Delta z \sim 10 ^ { -3 } .
In the Friedmann comoving frame , its magnification increases nonlinearly with \Omega _ { \circ } and \Lambda .
These effects interpret respectively by the gravitational attraction from the outer parts and repulsion ( of vacuum ) from the inner parts of S , with a sensitiveness on \Omega at primordial epochs and on \Lambda later on by preserving the expansion rate from an earlier decreasing .
This dependence of the expansion velocity \vec { v } = yH \vec { r } on \Lambda is shown on Fig . 1 through the corrective factor y to Hubble expansion , where \vec { r } and H stand respectively for radius of S and Hubble parameter at time t .
It is characterised by a protuberance at redshift z \sim 1.7 , the larger the \Lambda the higher the bump .
It is due to the existence of a minimum value of Hubble parameter H which is reached during the cosmological expansion ( also referenced as a loitering period ) .
It characterises spatially closed Friedmann models that expands for ever , they offer the property of sweeping out the void region , what interprets as a stability criterion .