排斥性暗物质 排斥性暗物质（p3）
...如果假设在两个褶皱中热速度相对于光速可以忽略不计，那么压力也可以忽略不计（无压尘埃宇宙模型）。第一步，当解耦刚刚发生时，我们假设r = r*，系统变为：
(12-a)
(12-b)
(12-c)
(12-d)
...从(12-b)和(12-d)可知，曲率指数k和k必须为负。演化规律是简单的线性关系：R = R » t
...在这一短暂时期之后（物质主导时期的开始阶段），我们回到系统(11-a)到(11-d)，曲率指数k = k* = -1：
(13-a)
(13-b)
(13-c)
(13-d)
由此我们立即得到：
(14-a)
(14-b)
引入两个褶皱中的质量守恒：
(15) r R³ = r* R*³ = ro Ro³ = 常数
ro和Ro为特征量。设：
(16)
(17)
那么系统变为：
(18-a)
(18-b)
...请注意，R = R意味着R" = R" = 0。另一方面，如果两个宇宙是“完全耦合”的，即R*/R = 常数，那么它们将对应于弗里德曼模型，具有“平行演化”。但我们认为它们通过(18-a)和(18-b)由引力场耦合，这表明线性膨胀是不稳定的。例如，如果R > R*，那么R" > 0 且 R*" < 0。该系统可以数值求解。典型解对应于图5。
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图5：宇宙和暗宇宙尺度参数的演化。
...与经典的弗里德曼模型相比，这个新模型增加了宇宙的年龄。这取决于所采用的哈勃常数Ho的值。1993年哈勃望远镜的测量将哈勃常数从45增加到70 km/s/Mpc，这在基于零宇宙常数的爱因斯坦-德西特模型下，宇宙的年龄在80亿到90亿年之间，这与我们银河系中最古老的恒星的年龄存在严重矛盾。但最近基于依巴谷卫星测量的造父变星校准降低了Ho。同时，一些人倾向于降低我们银河系中最古老恒星的年龄估计。
...请注意，排斥性暗物质起到了“宇宙常数”的作用，因为它在我们的褶皱中给出了正的加速度R"。
无论如何，关于宇宙年龄的讨论尚未结束。
原始版本（英文）
repulsive dark matter Repulsive dark matter (p3)
...If we assume the thermal velocities, in both folds, to be negligible with respect to the velocity of the light, the pressures can be neglected (zero-pressure dust universes model). In a first step, when decoupling has just occurred, we assume r = r* and the system becomes :
(12-a)
(12- b)
(12- c)
(12- d)
...From (12-b) and (12-d) the curvature indices k and k* must be negative. The evolution laws are simply linear : R = R* » t
...After that short period (the very beginning of the matter dominated era), we return to the system (11-a) to (11-d), with curvature indices k = k* = -1 :
(13-a)
(13- b)
(13- c)
(13- d)
from which we get immediately :
(14-a)
(14-b)
Introducing the mass-conservation in both folds :
(15) r R³ = r* R*³ = ro Ro³ = constant
ro and Ro being characteristic quantities. Let :
(16)
(17)
Then the system becomes :
(18-a)
(18-b)
...Notice that R = R* gives R" = R*" = 0. On another hand, if the two universes were "fully coupled", i.e. R*/R = constant, then they would correspond to Friedmann models, with "parallel evolutions". But we consider that they are coupled by the gravitational field, through (18-a) and (18-b) that show that the linear expansion is unstable. If, for example R > R* then R" > 0 and R* " < 0 . The system can be solved numerically. The typical solution corresponds to figure 5.
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Fig. 5 :The evolution of the scale parameters of the universe and dark universe.
...With respect to the classical Friedmann model, this new model enlarges the age of the universe. Depends on the retained value of the Hubble constant Ho. In 1993 measurements from Hubble telescope enlarged the constant from 45 to 70 km/s/Mpc which, based on the Einstein de Sitter model with zero cosmological constant, gave an age of the universe lying between 8 and 9 billions yrs, and arose a severe problem with respect to the age of the oldest stars of our galaxy. But more recent calibration of cepheids, from Hipparcos satellite measurements, lowered Ho . At the same time some people tended to lower the estimated age of the oldest stars of our galaxy.
...Notice that the repulsive dark matter plays the role of a "cosmological constant", for it gives a positive acceleration R" in our fold.
Anyway, the discussion about the age of the universe is not ended.