Earth did not come under the rule of a single world government by intention, but rather by accident. The sheer cost and scope of colonizing the solar system, funding the terraforming of Mars, and eventually interstellar travel was so great that only by combining their resources through the United Nations was it possible. This lead to the birth of the precursor to world government, the Human Expansionary Fleet.
The HEF grew rapidly as its viability was proven, mining asteroids and making habitats proving to more than pay for itself, repay the creditor nations, and begin to expand itself. Soon, nearly every nation on Earth was contributing to it, no one wanting to be left behind in the wake of the wealth it was generating. Increasingly, however, power came not to rest on Earth at all, or even on Mars, the first colonized planet other than Earth, but rather in the habitats, space stations, and ships of the HEF that controlled most resource gathering and trade. Once the orbital rings were built, even travel around the Earth was largely controlled by them as they could do it far quicker and safer than anyone else.
This had an unexpected effect. For reasons that are still unknown, the effects of low gravity on men are stronger than they are on women, especially in the eyes. Men go blind after spending too long in space, where women do not. Over time, this lead to a significant advantage to the number of women serving in leadership positions of the HEF, and while the medical problem has long since been corrected the power inequality in leadership has not come close to stabilizing yet, since the same advances in medical technology have lead to people living much longer. To this day, nearly 80% of the officer corps of the HEF are women… and with the importance of the HEF in planet-bound society increasing every day the power dynamics of the last two thousands years found themselves suddenly quite reversed.
It was around the time of the founding of the second colony world that Earth began to realize their influence was slipping. Mars and Freya had been founded by a single government that represented their interests, and most of the other colonial sites in the system like Ceres and Titan did as well, while Earth still squabbled and fought and saw their prestige and power diminish with each year. After a series of soft-hegemonic moves and a few Wars of Unification with rogue states, Earth finally united under a single world government.
The Terran Federation is a democratic republic of planets centered around the orbital rings of Earth. Each planet has theoretically equal representation. In truth, Earth still holds almost a quarter of the population of humanity and access to the most resources, so out of the planets they tend to be the most in control. The true dominant power, however, is clearly the HEF, which at this point has grown beyond the control of any one of the human governments.
The Terran Federation currently has five colonies – Mars, Freya, Set, Athena, and Moro. Miranda would be the 6th colony. There are hundreds of space stations, mining stations, scientific research posts, and other such things scattered across human space.
The society is mostly matriarchal. It isn’t that men don’t have, legally, the same rights, its just that, with space travel being by far the more prestigious route to advancement, and women being so much more suited for it, that the vast majority of those in leadership positions are women.