Assuming that human civilization in the future will spread from earth into the galaxy, how long would it take for the human population to exceed one trillion (10¹²)? I doubt this is possible if humans remain on Earth, and it may require humans to colonize exoplanets, depending on how many planets in our Solar System are suitable for colonization.
Edit: since a timeframe is hard to estimate, let me modify the question to ask "what is required for the population to exceed 1 trillion?"
Not as long as you might think
It depends on the assumptions you're willing to make.
With the world population growing at a vigorous 2% per year, there would be a trillion people alive in around the year 2260. The fastest human population has ever grown in recent history was 2.2% per year in the early sixties. This type of scenario would correspond to a strongly pronatalist policy by the powers-that-be, perhaps going as far as banning birth control and mandating early marriage.
With the world population growing at a more moderate 1% per year, there would be a trillion people alive in around the year 2500. This rate is about where we are currently. As the population ages and becomes more urban, most demographers think the population growth rate will slow even further. From an overall perspective, many industrial democracies in fact have strongly negative growth rates if you exclude immigration. However, there are fast-growing subpopulations with strong pronatalist proclivities that will become dominant if current growth rates are maintained.
With the world population growing at a glacial 0.1% per year, you wouldn't reach a trillion until almost the year 7000. This would be a slow growth scenario, and the one that gives mankind the most time to adapt. Perhaps a society where aging has been cured and very few people choose to have children at a particular given point might look like this.
The energy and food requirements of such a host would be far beyond our current technologies, and might require extensive industrialization and agriculturalization of space.
Imagine that view: a billion miles of space-farms, their bright green glittering against the dark sky, surrounding Earth like a gigantic set of Saturn-like rings. The plants within are made of organic material harvested from Titan, and are powered by the abundant 24h/day inflow of solar energy. A dense mesh of thousands of massive space elevators carry food to Earth and 'fertilizer' from Earth.
If you're not limiting yourself to Earth, but allow for space farming habitats, space industries and even extraterrestrial colonies, the cosmic resources available in the solar system can easily support a trillion people. To give a simple example, the energy output from the sun is so enormous, that if all of it were captured, each one of the trillion people would have 100 times as much energy as all of today's humanity combined. We have all the organic volatiles we could ever need on Titan, oceans of it, metals in the asteroid belt, quadrillions upon quadrillions of tons. Our solar system's resources are literally on cosmic scale.
A trillion people sounds like a lot. Nonetheless, humans are pretty small and Earth is pretty large. Plus its nice to have the air already here, not leaking into space, a magnetosphere, comfortable 1g gravitation ready made, all that good stuff. So Earth is pretty damn convenient. Let's see how much space they would take. If we use the density of a place like Mumbai, India, (30,000 people/sq.km) and extrapolate from there, you could fit a trillion people in an arcology covering about 23% of Earth's land area. That's about the area of Earth's deserts. So with a trillion people on it most of Earth could be a lush unspoiled garden... Moreover, since such an arcology whould presumably be multilevel, each individual could have a vastly larger living area than a current resident of Mumbai.
So we can definitely fit a trillion people on Earth, but Earth alone probably does not have the energy resources to feed them and meet all of their future needs.
NASA in 2005 published a report that stated if we were to take all the Earth orbit crossing asteroids and the asteroids in the Asteroid Belt, and process them for materials we could build living space equal to 30,000 X the land area of Earth(58 million sq. miles)and be able to house a half a quadrillion people at half the current population/sq. mile we now have. Think Stanford torus, O'Neil cylinder, McKendree cylinder. If we tear apart all the planets, comets, asteroid we could build enough habitat for 2 quadrillion people. We don't need to find planets around nearby stars. We just need the material. This link http://www.science20.com/robert_inventor/blog/asteroid_resources_could_create_space_habs_trillions_land_area_thousand_earths-116541 tells how that future might come about. No, we don't have the tech to build space habitat equal to 58 million sq. miles, but I believe we have the tech to build 100,000 sq. mile torus or cylinders. NASA is considering a 1g component(a rotating assemblage) to the next space station. That would be the beginning of humankinds conquest of the Solar System, and nothing like what we have now. There is a point in our species future when we might consider the resources in the Earth too valuable(equal to 20,000 earths of living space) to remain a habitat for only the richest 10-20 billion people in the Solar System. There are 3 things pushing the future, zombie apocalypse not being one of these. They are: 1.) humans conquering space(here and around at least nearby stars) 2.) everlasting health/youth(stopping aging in the 11-14 yr. old range because aging would be far easier to control then) 3.) becoming one with our tech(cyborgism, not borg) nothing obvious more like super advances on the cellular level. Replacing cell parts with far advanced cellular machinery. Here are more Links on the subject of population: http://blog.nss.org/?cat=35also and here http://www.ignorancedenied.com/threads/8479-The-Wealth-Of-Ages?p=56390 All we lack as a species is imagination! There are 520 stars within a 100lys of Earth if only half are suitable for development we could increase the human species to 260 quadrillion people! If all those star systems are suitable for development(meaning: lacking intelligent life) we could use the resources to support a half a quintillion eternally young humans/cyborgs with open ended lifespans! All we lack is imagination! Make it so ..
After thought: there are 600 million star systems within 5000lys( http://www.atlasoftheuniverse.com/5000lys.html ) of Earth if 10% are suitable for future development they would support 60 sextillion human/cyborg inhabitants, and we can add 60 sextillion for every additional 10% we find useful.
I don't accept either of your premises that 1) Humanity will spread from the Solar System or that 2) that it isn't possible to have 1E12 people on Earth. I encourage you to do some simple math. Solar irradience at the surface is about 1000 W/m². The Earth shadows the Sun over an area of about 1.3E14 m². This gives us about 1.3E17 Watts per year (without pollution and clouds, together they will reduce that considerably). According to Wikipedia Global Power Consumption was (estimated) 1.2E13 Watts in 2013. Say this is for 7½ billion people, or ~1650 Watts per person per year. That is about 1% of what's available to us on Earth from the Sun. OK, say we need to increase per capita power consumption by a factor of 5, so we're using 5% of impinging solar energy, ok so lets limit it to the visible spectrum, and that's 20% - for a trillion people. Yeah, the planet would be drastically different. It couldn't support 1 trillion peeps and all its current flora and fauna, but then again we've got fission power and geothermal power now (along with a little tidal power) and who knows about fusion. I do agree that 1 terapeeps would be a daunting challenge, and perhaps isn't possible, but perhaps it is...
Most predictions that was made were wrong. Scientists missed when we reached 6 billions, they were wrong about 7 billon border and current predictions are different. The population growth was different in different periods and there are many factors (and some of them are very new) so it's not possible to make good prediction
Your second question, what is required to exceed 1 trillion has simple answer: we don't know.
Optimist could say just let us some time and we could solve all current problems and reach this line.
Pessimist could say we'll meet with more and more complex problems and sometime problem become big enough to almost destroy humanitity and fall back to reptilies. Note that although dinosaurs became extinct, reptilies in whole are alive and don't pretend both for extincting and be dominating
Humans balance out at 10 billion populate for earth size planet.
Humans will reach this number around 2050.
Minimum distance between stars is about 1ly.
Humans use .10c travel speed.
Let's say 22 stars are within a 1ly radius of any other star.
Population start should be around 10,000.
Population Growth is generally between 2 and 5% per year, let's just assume 5%.
We have all these set for minimum time... and the resulting number should be fairly low...
First 1t/10b = 100, so 100 planets is all you need to support this. This means that no ship has to leave 2 light years from Earth and max travel time is 20 years. That being the case we can just say we start in 2050 with 10b, wait a decade and start counting from there... even though the 10 year lag is useless because we can alway just colonize the planets, moons, asteroids around us and that is more than enough space to do that. and in the next hundred or so year construction projects will be fairly cheap... The most expensive part is launching human bodies, soil, and land materials up... after the first few decades we can rapidly expand and transofrm the whole solar system into very large colony colony station-ship-bases
2060 : 10000010000
2061 : 10500010500
2062 : 11025011025
2063 : 11576261576
2064 : 12155074654
2065 : 12762828386
2066 : 13400969805
2067 : 14071018295
2068 : 14774569209
2069 : 15513297669
2070 : 16288962552
2071 : 17103410679
2072 : 17958581212
2073 : 18856510272
2074 : 19799335785
2075 : 20789302574
2076 : 21828767702
2077 : 22920206087
2078 : 24066216391
2079 : 25269527210
2080 : 26533003570
2081 : 27859653748
2082 : 29252636435
2083 : 30715268256
2084 : 32251031668
2085 : 33863583251
2086 : 35556762413
2087 : 37334600533
2088 : 39201330559
2089 : 41161397086
2090 : 43219466940
2091 : 45380440287
2092 : 47649462301
2093 : 50031935416
2094 : 52533532186
2095 : 55160208795
2096 : 57918219234
2097 : 60814130195
2098 : 63854836704
2099 : 67047578539
2100 : 70399957465
2101 : 73919955338
2102 : 77615953104
2103 : 81496750759
2104 : 85571588296
2105 : 89850167710
2106 : 94342676095
2107 : 99059809899
2108 : 104012800393
2109 : 109213440412
2110 : 114674112432
2111 : 120407818053
2112 : 126428208955
2113 : 132749619402
2114 : 139387100372
2115 : 146356455390
2116 : 153674278159
2117 : 161357992066
2118 : 169425891669
2119 : 177897186252
2120 : 186792045564
2121 : 196131647842
2122 : 205938230234
2123 : 216235141745
2124 : 227046898832
2125 : 238399243773
2126 : 250319205961
2127 : 262835166259
2128 : 275976924571
2129 : 289775770799
2130 : 304264559338
2131 : 319477787304
2132 : 335451676669
2133 : 352224260502
2134 : 369835473527
2135 : 388327247203
2136 : 407743609563
2137 : 428130790041
2138 : 449537329543
2139 : 472014196020
2140 : 495614905821
2141 : 520395651112
2142 : 546415433667
2143 : 573736205350
2144 : 602423015617
2145 : 632544166397
2146 : 664171374716
2147 : 697379943451
2148 : 732248940623
2149 : 768861387654
2150 : 807304457036
2151 : 847669679887
2152 : 890053163881
2153 : 934555822075
2154 : 981283613178
2155 : 1030347793836
So basically 130 years under the quickest situation Adjusting for the 2% growth rate it's 275 years.
