Bows have a number of things working in their favor, namely the fact that they are largely silent and good armor penetration. This is cancelled out by the fact that using a bow requires a lot of training, bows are large and a liability in melee combat, and the range, rate of fire, and ability to shoot on the move are massively inferior to modern firearms let alone the advanced weapons of the Cosmic Era. But the rule of cool says bows are cool, and who are we to argue with the likes of John 'Blowing up Soviet Helicopters with a compound bow' Rambo, Katniss 'Girl on Fire' Everdeen, Hawk-eye shooting down a flying aircraft carrier, and the mythos of Robin Hood?
The Bow as a Stealth Weapon
The bow is a stealthy weapon, and this remains so in the Cosmic Era. One of the drawbacks of many of the advanced weapons in the setting is that they generate electromagnetic signatures. A mag rifle, plasma cannon, and other high energy weapons can be detected if they are in an active or powered up state, making them vulnerable to EM sniffing counter-weapons like armed drones and hunter killer flybots. Likewise conventional firearms are loud, and have discernible heat and muzzle flash, giving away their positions. The bow makes little sound and doesn't generate an EM signature. This makes it useful for assassins working to eliminate a target in a highly hostile environment. The bow can be used with specialty arrows, giving it a powerful punch when needed.
Another variation is that the bow is a dumb weapon in a smart world. This can make it valuable when fighting a foe that relies on not eyes, but electronic sensors. Robots programmed and equipped to hunt down and kill armed humans might not recognize a bow as a dangerous weapon, and when calculating where they are being shot at from, the robot could attempt to apply rifle and magnetic ballistics to the path of an arrow, sending the machines off to find a target thousands of feet away from where the archer actually is.
The bow doesn't require anything to work other than an arm to pull it back. It doesn't require propellant, a power source, or anything else like that. This means that with minimal effort and modification (using entirely synthetic materials) that bows can be used in a vacuum, such as in space, on the surface of the Jovian moons, and so forth. It likewise can be used on ships and space stations where barring explosive and armor defeating arrows, it cannot harm the hull of the vessel. This requires special training, as archery is based around wind resistance and gravity. Exo-Archery is a very rare and very exclusive skill.
One of the staples of the bow in modern and semi-futuristic combat is the ability to fire a variety of arrows, ranging from Rambo's explosive tipped arrows to Hawk-Eye's acid, thermite, explosive, and other arrows disgorged from his semi-automatic quiver.
High Explosive - a staple of action and war movies, the High Explosive arrow packs a punch. The Cosmic Era HE arrow is stunningly potent, packing the same firepower as an RPG, but in a fraction of the size. These arrows are potent enough to take out the majority of autons in production, as well as damaging most power armors provided a vulnerable point is hit. A favorite among saboteurs, assassins, and spec ops looking to make a mess.
Progressive - a Cosmic Era specialty, the progressive arrow has a large bladed head with a micro sized hyperedge. While the charge and duration of the weapon is short, it is generally more than required to penetrate light and medium power armors, cut through armored hides of horrors, and otherwise puncture and cause a lot of damage to a target. The arrows are expensive, but are able to take out hard targets without the very real concern of over-penetration or collateral damage. Shooting an unarmored target with a progressive arrow does little extra damage, generally allowing the arrow to shoot through the target.
Chemical Payload - CP arrows can carry and release a chemical on contact. Obvious choices are acids and poisons. Less common choices include nerve gas, biotoxins, or nano-paste. These are dramatically more common in entertainment feeds than actual military applications.
EMP - or electromagnetic pulse, EMP arrows generate a small explosion and a pulse a few feet to a few dozen feet across. While this pulse is too weak to damage the systems of mecha, large fighters, and installations with hardened gear, it can scramble a robot's cortex, shot down a drone, disrupt the power systems in a power armor suit, and otherwise hinder the ability to high tech devices to function. Imagine the surprise of the commando who thinks the archer missed, when in fact the hear hit was close enough to knock out his SmartGun system, and force the CPU of his magrifle to reset.
Arcanotech - many arcanotech devices can be made very small, thus creating a ultra-tight niche for unique and ultra rare weapons. Examples include implosion heads, cryostatic explosives (ice-bomb), thunderstroke (lightning attack) and so forth. These weapons are almost exclusively the domain of entertainment, but there are a few real ones that exist. One example is the Federation's Anvil arrow. Using the same technology as the blitzhammer, the Anvil Arrow can increase it's mass dramatically for approximately one second. This ups it's kinetic potential as the mass can jump from a few ounces to close to three hundred pounds. The force is enough to knock an armored trooper to the ground, shatter the rib-cage of a cosmic horror, or turn a normal person inside out from the impact.
Grapple - another staple of the action/spy genre is using an arrow to shoot a rope into a wall or ledge and use it to escape a situation. Using ultra-light microfilament lines and special climbing gear, this is possible and much more common. The Federation uses a specialized pistol to shoot the grappling line, but the same tech can and is used by other groups with bows rather than guns.
The Bow Itself
The power of a bow is limited to the strength and skill of the user. In the Cosmic Era, both can quickly augmented. Strength is easily augmented, be it through stimulant use, muscle tracer/booster suits, powered armor, parapsychic ability, or genetic augmentation. Skill can be inserted and removed through skill-soft chips and the proper cranial augmentation. Thus two things can be done to improve the performance of the bow, increase it's draw weight and add smart technology.
Using new composite materials, the same sort that go into mecha and aerospace fighters, Cosmic Era bows can be made with very high draw weights. A bear hunting bow might have a 120 lb draw weight, a bow designed to be used by an augmented supersoldier with a combat suit might have a 300 lb or even absurd 500 lb draw weight. At this level, a basic steel arrow could be shot with force comparable to a high caliber gun.
The SmartGun system can be adapted for a bow, provided the bow has the correct sensors on it. While expensive, such technological additions can give a soldier, with the appropriate training or skillsoft chip, shooting ability on par with Robin Hood.
Power Suit Ballista
There is no reason that an oversized bow couldn't be manufactured for use by power armor troopers. These oversized bows would be made for hunting tanks, cosmic horrors, other power armor troopers, or harrying mecha. While certainly absurd, there is something rather cool about the notion of a guy in a power armor suit shooting an explosive arrow six feet long into a machine gun nest, terminator style robot, or the face of an opposing mecha.
No. Just no.
The bow is still present in the Cosmic Era, but it is a rarity, and only used in special situations.