New probe of modified gravity A. Boyarsky 1,2 and O. Ruchayskiy 1 $^{1}$Ecole Polytechnique F´ed´erale de Lausanne, FSB/ITP/LPPC , BSP 720, CH-1015, Lausanne, Switzerland $^{2}$Bogolyubov Institute of Theoretical Physics, Kyiv, Ukraine ( Dated: January 4, 2010) We suggest a new efficient way to constrain a certain class of l arge scale modifications of gravity. We show that the scale-free relation between density and size of Dark Matter halos, predicted within the Λ CDM model with Newtonian gravity, gets modified in a wide class of theor ies of modified gravity. Models with the large scale modification of gravity are ac- tively discussed in the recent years in connection with the observed accelerated expansions of the Universe and because they can be related to the existence of extra dimensions [1, 2]. However, the general principles of gauge invariance and uni- tarity strongly constrain possible theories of gravity, modify- ing the Newton’s law at large distances (see analysis of [3]). Thus, in addition to its phenomenological applications, this problem is related to the fundamental questions of particle physics, field theory and gravity. It is therefore importantto search for large scale modifications of gravity experimenta lly. A possible set of consistent (as a spin-2 field theory) large scale modifications of gravity is described by two parame- ters – scale r$_{c}$ and a number 0 ≤ α < 1 [3–5]. r$_{c}$ marks the distances at which at the linearized level gravitational law changes from 1 /r 2 to some other power 1 /r $^{n}$, and the pa- rameter α determines the value of n . Phenomenologically, deviations from Newton’s law we are looking for may be rep- resented in this parameter space. A significant fraction of t his space is excluded by precision measurements of the Moon or- bit [6]. Other natural probes of such modifications are cosmo - logical observables (see e.g. [7–12] and refs. therein). In this work we identify a new observable sensitive to the large scale modifications of gravity. We demonstrate that un i- versal properties of individual dark matter halos are also af- fected by the modifications of gravity, and provide novel way to probe them. Namely, we show that the scale-invariant rela- tion between density and size of dark matter halos, predicted by Newtonian gravity within the Λ CDM model [13] and found to hold to a good precision in observed dark matter halos [14], may receive non-universal (size-dependent) corrections for a wide range of parameters r$_{c}$ and α . Formation of structures in the Universe is an interplay between gravitational (Jeans) instability and overall Freed- man expansion. The gravitational collapse does not start un- til the potential energy U of a gravitating dark matter sys- tem overpowers the kinetic energy of the Hubble expansion K ∼ 1 $_{2}$H$^{2}$R $^{2}$. Once the gravitational collapse has began, at any moment of time t a dark halo is confined within a sphere of zero velocity or a turn-around sphere . As Hubble expan- sion rate H ( t ) decreases with time, the turn-around radius R$_{ta}$ ( t ) grows. In the Newtonian cosmology with potential φ$_{N}$ ( r ) = − GM/r the turn-around radius R$_{ta}$ is where 0 ≤ α < 1 . Here the characteristic ( Vainstein ) radius r$_{V}$ is defined as [3, 4, 20] where ρ$_{⋆}$ is some constant with dimension of density. In par- ticular, Λ -term obeys this property (with ρ$_{⋆}$ ∝ Λ /G ). All the theories of the form (4) obey the property that relative cor- rection to the Newtonian potential φ$_{N}$ depends only on the density ρ ( r ) within a radius r (and not on the mass or the size of objects). Next, we consider the modifications of gravity [3, 4]. The gravitational potential of a spherically symmetric system of mass M there has the form (c.f. [13]). The relation (1) still holds and is again independent on the mass of the halo. What is the most general form of gravitational potential, for which the property (2) remains true? Clearly, it will hold for all the gravitational potentials of the form The relation (2) continues to hold in the Universe where gravity is modified by the cosmological constant Λ . The grav- itational energy of a body of mass M at distance r becomes U Λ = − GM r − Λ r 2 6 . Comparing it with the kinetic energy of the Hubble flow K one arrives once again to the relation (2) It was shown in [13] that the property (2) leads to a uni- versal relation between characteristic scales and densities of dark matter halos. This relation holds in wide class of dark- matter dominated objects (from dwarf galaxies to galaxy clus- ters) [14] (see also [15–17]). The relation is in a very good agreement with pure dark matter simulations [18, 19], sug- gesting that baryonic feedback can be neglected in this case. Therefore, this relation can serve as a new tool of probing properties of dark matter and gravity at large scales. 1Mpc ). Notice that at any moment of time the average density within a turn-around radius (1) is proportional to the cosmo- logical density and is the same for halos of all masses: R$_{ta}$ ∝ ( GM H 2 ) 1 / 3 (1) (today for masses ∼ 10 $^{12}$M ⊙ the turn-around radius is ∼ φ ( r ) = − GM r F ( ρ ( r ) ρ$_{⋆}$ ) (4) φ$_{α}$ ( r ) = − GM r π ( r r$_{V}$ ) (5) r$_{V}$ = ( 2 GMr β c ) 1 1+ β where β $^{= 4(1}$− α ) (6) ρ$_{ta}$ ( t ) ∝ Λ G (3) ρ$_{ta}$ ∝ H 2 G ∝ ¯ ρ$_{tot}$ ( t ) (2) arXiv:1001.0565v1 [hep-th] 4 Jan 2010