1. MOND Dark Matter effects in non-local models. IVAN ARRAUT

2. THE UNIVERSE

3. What is the Cosmological Constant (CC)? 1). It produces local and global effects. 2). It produces the accelerated expansion of the Universe. 3). What is? Vacuum energy? Graviton mass? Scalar-tensor vector? Scalar- tensor? Etc? The cosmological Constant produces a repulsive effect, at the local physics.

4. What is Dark Matter? 1). It makes gravity stronger at some scales.

5. Gravitational lenses

6. Is there any relation between Dark Matter and Dark Energy? (MOND). The MOND theory. The fundamental equation is: Coming from the Lagrangian. In agreement with Bekenstein and Milgrom.

7. What is the MOND scale? The MOND equations written in this way, reproduce a logarithmic potential:

8. The MOND scale is an acceleration scale. Below of which, the Dark Matter effects appear: The scale of MOND, can however, become a distance scale given by the geometric average between the gravitational radius and the Cosmological Constant scale. Just replace this inside the standard Newtonian potential and you will obtain the Tully Fisher law in the form:

9. One of the succeses of MOND, is the reproduction of the baryonic Tully-Fisher law. However MOND has some problems too, like the reproduction of gravitational lenses and energy-momentum conservation. A Lagrangian formulation is required. Non-local gravity 1). If you ask me: Do you really believe that non-local gravity is the solution for Dark Matter and Dark Energy problem? A: No, but at least I can see what is going on in order to reproduce a better theory in the near future. 2). If you ask me: Do you believe in MOND? A: No, but at least it can show me some observational facts very important if I want to find a final theory.

10. The non-local equations Check Odintov and Sasaki.

11. Weak Field approximation in spherical symmetry. In the static situation, the field equations reduce to: If you compare with the MOND equations written in the form: Then:

12. In the weak field approximation If we solve the non-linear equations; we then obtain: For:

13. Plot of the C function, assuming :

14. From the ratio, depends the character of the potential (attractive or repulsive).

15. The gradient of the potential. For C=2/3, it reproduces an attractive potential, with a gradient falling like 1/r. This potential could in principle reproduce the Tully-Fisher law if we tune the constant A in agreement with the geometric average between the gravitational radius scale and the Cosmological Constant scale. But the potential seems to be repulsive if we normalize $f_0=1$. We would have to tune $f_0$ to some specific value in order to recreate the attractive behavior.

16. Two important cases. 1). The case where C=-1. In such a case, the non-linear differential equation just looks like the MONDian case; but the potential is repulsive. 2). The case where C=2/3. In such a case, takes two values and the potential reproduces the Tully-Fisher law, but the potential could be repulsive.

17. Is this really a MONDian theory? Correspondence with Bekenstein (October 2012). In Bekenstein words: “The idea of reproducing the Dark Matter effects with non-localities is nice as simple as it sounds……. However, in this non- local model, one of the characteristics of MOND is lost, namely, the scale independence”.

18. Difference between MOND and the non-local model. The interpolating function is a function of the potential, rather than a function of acceleration as it is the case in MOND. If the differential equation is the same as in the MONDian case, then the Tully Fisher law is not satisfied. The Tully Fisher law can be reproduced as C=3/2. But in such a case, the differential equation is not the same as that obtained from the AQUAL Lagrangian. The difference is a factor. Remember that the Tully-Fisher law is reproduced with baryonic matter.

19. Conclusions 1). The non-local model can reproduce Dark Matter effects, but also additional repulsive effects. 2). The Dark Matter effects are in agreement with MOND for some specific values of the parameter C. 3). However, MOND cannot be achieved completely. The reason is that the found interpolating function is a function of the potantial, rather than a function of the gradient of the potential.

20. Acknowledgements 1). Misao Sasaki for helping me to improve the idea. 2). Takahiro Tanaka. 3). The students of YITP for the continuous discussion every week about different topics. 4). Hideo Kodama for the collaboration in other projects. 5). Achim Kempf (University of Waterloo) for the collaboration in other project. 6). Carlos Segovia (Mathematician Post Doc at University of Heidelberg) who is helping me in some mathematically difficult problem related to the Kempf problem. 7). Jacob Bekenstein for the feedback about this work in Taipei and subsequent by e-mail. 8). Hagen Kleinert for the suggested bibliography which I have not yet read, but for sure in the near furture I will.

21. Thanks Korea