1 Very Large Structure numerical simulation in a compact computational space. Jean-‐Pierre Petit1, Gilles d’Agostini, Xavier Lafont & Nicolas Borrallo ____________________________________________________________________________________________________
Keywords : bimetric model, VLS, joint soap bubbles like universe, negative mass, runaway phenomenon, coupled field equation system, compact computational space ____________________________________________________________________________________________________ Abstract : Considering the universe as a manifold M4 plus two metrics, coupled by field equations system, we use corresponding interaction laws to perform 2D simulation of VLS in a compact computational space, S2 sphere. Classical Newton’s law is replaced by gravitational force which is proportional to the inverse of the square of curvilinear distance, as measured along a geodesic. ____________________________________________________________________________________________________ The Very Large Structure of the Universe is currently considered as a giant bubble-‐like structure, with walls, filaments and voids. The first attempts to modelize this structure corresponds to the works of Ya.B. Zel’dovitch and F. Shandarin ([1], [2]). This so-‐called pancake theory is now abandonned. Today researchers try to figure VLS through numerical simulations based on cold dark matter behaviour. See figure 1.
Fig.1 : VLS dark matter 3D simulation 1 Former research manager at CNRS, France. Private adresse :
[email protected]
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Which is somewhat different from typical pattern corresponding to observational data. See figure 2 and 3.
Fig.2 : Typical VLS after observational data
Fig.3 : Typical VLS after observational data
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A more suitable model corresponds to joint bubbles’ like design. See figure 4.
Fig.4 : 3D VLS structure, shaped as joint bubbles In the middle of the nineties H. El-‐Ad and T. Piràn ([3],[4],[5],[6],[7],[8],[9])build a program which gives bubble-‐like structure, from IRAS observational data. See figure 5.
Fig.5 : After Piràn [6] : VLS Bubble structure, from IRAS Survey In a previous paper ([10], [11]) we suggested a scenario, implying interaction between positive and negative matter. The model is based on a bimetric description of the Universe, considered as a manifold M4 plus two metrics gµν(+ ) and gµν(− ) .
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Positive mass (and positive energy) particles follow geodesics corresponding to metric gµν(+ ) . Negative mass (and negative energy) particles follow geodesics corresponding to metric gµν(− ) . As those two families are disjoint those particles cannot encounter. In addition, photons follow null-‐geodesics. Positive energy photons follow null geodesics as derived from metric gµν(+ ) . Negative energy photons follow null geodesics as derived from metric gµν(− ) . Positive masses emit and receive positive energy photons. Negative masses emit and receive negative energy photons So that, on pure geometrical grounds an observer made of positive mass cannot observe structures made of negative mass, and vice-‐versa. It is assumed that the two species interact only through gravitational force, which is phrased through the coupled field equation system, introduced in ([10], [11]) : (1)
(2)
(+ ) Rµν −
1 (+ ) (+ ) ) (− ) R gµν = χ ( T (+ µν + T µν ) 2
The mentioned system of equations fits time-‐independent cosmological solutions. Applying Newtonian approximation method we can derive interaction laws : -‐
Positive masses mutually attract through Newton’s law
-‐
Negative masses mutually attract through Newton’s law
-‐
Masses with opposite signs mutually repel through « anti-‐ Newton’s law »
Fig. 6 : Bimetric interaction schema This is different from H.Bondi’s result ([12] , 1957 ). But it can be shown that the system (1)+(2) eliminates the prosterous so-‐called runaway phenomenon. In effect, when one considers the classical model of RG (a manifold M4, plus a single metric, solution of Einstein’s equation) the interaction laws are different. -‐
Positive masses attract anything
-‐
Negative masses repel anything
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Fig.7 : The runaway phenomenon So that when a positive mass encounters a negative mass, it escapes, and the negative mass runs after, which corresponds to so-‐called runaway phenomenon. This last disappears when we shift to a bimetric description of the universe. The bimetric interaction laws makes possible to build numerical simulations. As shown in 1995 [11] if one assumes that, after discoupling the absolute value of negative mass density is much higher that the one of positive mass, Jeans’ time is shorter for the negative species, which forms a series of species first. (3)
t J+ =
1 4π G ρ
+
t J− =
1 4 π G ρ−
If we choose ρ (− ) = 64 ρ (+ ) then t J+ = 8t J− . Negative matter drives the birth of VLS. In 1995 we showed, through 2D simulation, that coupled structures formed. Negative matter gives clumps, while positive matter is confined in the remnant place. See figure 8.
Fig. 8 : 2D simulation of VLS [11]
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Such VLS is very stable in time. In effect the positive mass lacunar structure prevents negative clusters’merging, which those last behave like ankors, with respect to lacunar structure. Compact computational space In the above simulation our computational area was a square. We did not pay very much attention to border’s conditions. In the following we deal with closed computational space : S2 sphere. We replace classical Newton’s law by an interaction law based on the inverse of the square of the curvilinear distance between points. At short distance, this is equivalent to Newton’s law in a plane (the tangent plane of the sphere).
Fig.9 : Direction of the forces along geodesic lines. The distance is computed, using quaternion technique. In order to check the robustness of the program we computed the path of a small positive test mass orbiting around a fixed mass M, and we got the figure 10.
Fig.10 : Precession phenomenon due to space curvature.
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VLS simulation in computational close space. Using a PC we take 10,000 positive mass-‐points. Each one represents 1012 solar masses and may be considered as clusters of stars. In order to shorten the duration of the calculation we concentrate negative matter in only 49 mass-‐point, each one figuring 1016 (negative) solar masses. It correponds to initial ratio : (4) t J+ = 7t J− Perimeter of the computational space : 3 Mpc Duration of a step : 106 years. After 2 h 30’ :
Fig.11 : Net-like structure of positive matter. At the center of each cell, a negative mass clump
Of course, the chosen characteristic length is too short to figure real Universe conditions. This is just to check, through qualitative study that the computational technique on compact space fits previous results for VLS [11]. This is the preliminary to a study of the impact of dynamical friction of a 2d galaxy, surrounded by negative mass, which gives good looking barred spiral, stable over 20 turns (unpublished work). This has already been checked in compact S2 computational space and will be soon extended to 3D simulations, thanks to the opportunity to use a more powerful system.
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Fif.12 : Unpublished result. 2D barred spiral galaxy orbiting in surrounding negative mass, due to dynamical friction. Stable over 20 turns.
Conclusion :
This result is encouraging, although schematic. Thanks to the help of Japanese researchers we are now going to use a much powerfull system, in order to build 3D simulations, in a S3 closed computational space for both 3D VLS simulation and spiral galaxy formation and evolution. References : [1] Y. B. Zeldovich, « Gravitational instability : an approximate theory for large density perturbations », Astronomy and Astrophysics, vol. 5, 1970, p. 84–89 [2] Sergueï F. Shandarin et Ya. B. Zeldovich, « The large-‐scale structure of the universe: Turbulence, intermittency, structures in a self-‐gravitating medium », Reviews of Modern Physics, vol. 61, no 2, 1989, p. 185–220 [3] Piran T. : On Gravitational Repulsion, Gen. Relat. and Gravit. Vol. 29 , N° 11 , 1997 [4] El-‐Ad H. , Piran T. , and da Costa L.N. , (1996) Astrophys. J. Lett. 462 L13 [5] El-‐Ad H. , Piran T. , and da Costa L.N. , (1997) Mon. Not. R. Astro. Soc. [6] Piran T. : On Gravitational Repulsion, Gen. Relat. and Gravit. Vol. 29 , N° 11 , 1997 [7] El-‐Ad H. , Piran T. , and da Costa L.N. , (1996) Astrophys. J. Lett. 462 L13 [8] El-‐Ad H. , Piran T. , and da Costa L.N. , (1997) Mon. Not. R. Astro. Soc.
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[9] El-‐Ad H. , Piran T. ( 1997 ) Astrophys. J. [10] J.P.Petit : The missing mass problem. Il Nuovo Cimento B, 109: 697–710 [11] J.P. Petit (1995). Twin Universe Cosmology. Astrophysics and Space Science (226): 273–307. [12] H. Bondi: Negative mass in General Relativity, [: Negative mass in General Relativity.] Rev. of Mod. Phys., Vol 29, N°3,1957
