Nature has changed carbon dioxide a lot, but slowly, and climate has responded rapidly. Younger than the snowballs, over the last half-billion years or so, we have had an atmosphere recognizably similar to the modern one in having oxygen. (You need a lot of oxygen to allow big critters, and there is a rich fossil record of big critters over the last 500 million years. Too much oxygen and everything burns rapidly, but there is a rich fossil record of unburned things.) The rate at which geology recycles shells to make carbon dioxide and sends that carbon dioxide out to make volcanic rocks can change — big belches of hot rock from deep in the mantle can occur, for example (there is carbon dioxide down there, and if a hot-spot plume head hits the surface to feed giant flood basalts, a lot of carbon dioxide can come out), and the collisions between continents that make a lot of metamorphic rocks happen only occasionally. (If North America and Asia continue moving towards each other as rapidly as your fingernails grow, another big collision may occur in a couple-hundred million years!) If there are no big mountains, soil builds up and the carbon dioxide in the air may have trouble getting all the way down to attack rocks and cause weathering. If the mountains are high, much of the soil can wash or slide off, exposing rocks to faster weathering. And, the mere accidents of geology might matter — shales at the surface don’t weather very rapidly, carbonates weather to produce carbonate shells in the ocean with no net change, but weathering of many volcanic rocks can be fast and remove carbon dioxide from the air, so the geologic accidents that control what rocks are at the surface may affect the setting of the rock-weathering “thermostat”.
And, evolution can affect how rapidly carbon is stored to make fossil fuels (which naturally release their carbon back to the atmosphere when erosion brings them to the surface and living things “eat” them.) There is a fascinating hypothesis that the great coal beds of Pennsylvania and many other places, which formed during the “Carboniferous” — the Mississippian and Pennsylvanian Periods — record the evolution of successful plants containing really hard-to-eat woody structures, and that coal formation was rapid but then slowed greatly tens of millions of years later as termites and fungi and other things evolved to break down those woody structures. When fossil fuels are being formed, carbon is being transferred from atmospheric carbon dioxide to oil or coal or natural gas in the ground, and when fossil fuels are being burned, the carbon dioxide is going back into the air. The time scale for lots of evolution to occur, or for lots of rearrangement of continents to occur, is sort of 100 million years, so it is not surprising that changes between high-carbon-dioxide and low-carbon-dioxide times have typically taken about 100 million years. There is no evidence for true cycles (no tick-tick-tick of a clock, such as we see with day-night or summer-winter), but lots of evidence that the changes in carbon dioxide occurred over the time scales one would expect given knowledge of the causes — the world does make sense.
Carbon dioxide has been the main driver of climate change on this hundred-million-year time scale. A statement such as this involves pretty much all of climatology and paleoclimatology. The general path is:
Reconstruct the history of past temperatures, which requires reading the temperature history in sediments, and knowing the time when the sediments were deposited. This can be done with considerable confidence; old crocodile-like critters on Ellesmere Island very close to the North Pole are a pretty good indication that it wasn’t too cold there then.
Reconstruct the history of past carbon-dioxide concentration in the atmosphere, again requiring ages as well as indicators of the atmospheric composition. Before ice cores (and the oldest ice core is less than 1 million years), the indicators of carbon dioxide in the atmosphere are not as clear as we’d like, but considerable agreement from several lines of evidence allows us to tell in general when carbon dioxide was high or low, and to make some quantitative estimates. (For example, plants “prefer” the lighter carbon-12, which diffuses and reacts more rapidly, so when carbon dioxide is common, plants are especially enriched in carbon-12; when carbon dioxide is rare, plants have to use more of the carbon-13. Special cell-wall molecules in the ocean, and soil carbonates, and remains of some water plants from lakes, are used to learn the carbon-12:carbon-13 ratio and hence the carbon dioxide level. Leaves grow fewer “breathing holes” — stomata — when there is more carbon dioxide in the environment, because stomata lose water while gaining carbon dioxide, so when carbon dioxide is high, plants can save water. Rising carbon dioxide shifts the ocean toward greater acidity, and this affects whether the little bit of boron in the ocean is as B(OH)3 or B(OH)4-1. The charged form substitutes more easily into carbonates, so the ratio of boron to calcium in a shell increases as the carbon dioxide drops. In addition, the charged ion of boron preferentially holds the light isotope boron-10 in comparison to the heavy boron-11. The residence time of boron in the ocean is many millions of years. Over shorter times, a drop in carbon dioxide will shift most of the boron in the ocean to the charged form, so its isotopic composition must become heavier as it comes to match the whole-ocean value, and the charged form is included in carbonates. There are other ways to get paleo-carbon-dioxide as well.)
Assess the correlation. The simple answer is that the correlation is not perfect, but is pretty darned good. There is a broad and shallow “skeptic” literature that plays with the estimates and dates to get fairly poor correlations, but the reputable sources (e.g., the IPCC Working Group I Fourth Assessment Report, chapter 6, at IPCC) show a rather tight coupling.
Attribute the correlation. Does the correlation match expectation from physical understanding? And, is there any other plausible explanation for the correlation, such that the correlation is a fluke, or the correlation arises because something else is controlling both temperature and carbon dioxide? This is the hardest one, and is never complete, because there always might be a new explanation that we haven’t thought of. But, we have known for more than a century that more carbon dioxide should make it warmer, based on fundamental physics that just won’t go away. The reconstructed warmings of the past actually are just about the size expected from our understanding of the effects of carbon dioxide (if there is a problem, the world changed a bit more than we might have expected). And no plausible hypothesis has been proposed that explains what happened without including the carbon dioxide. Moving continents around on the planet, opening and closing “gateways” to affect oceanic circulation, changing land albedo with plants, and other possibilities appear to be “fine-tuning” knobs on the climate, all mattering, but not mattering enough to explain the history by themselves or combined but ignoring carbon dioxide. Whether calculated on the back of an envelope or in a full Earth-system model, these non-carbon-dioxide effects do not suffice to explain the changes reconstructed from the features of the rock record, nor do other possible causes correlate well in time with the changes that happened in the climate.
Changes in carbon dioxide and other things can matter a lot to life. The early geologists named time intervals in geologic history, and the rocks deposited during those time intervals. Name changes were chosen at key times. The end of the Mesozoic, for example, is now known to have been caused by a huge meteorite impact that killed the dinosaurs. The end of the Paleozoic killed even more living things, and seems to have been linked to carbon dioxide. The last Period of the Paleozoic Era is the Permian, and the end-Permian extinction was the biggest mass extinction. Some uncertainty remains, but the leading hypothesis now is that a “plume head”, the mushroom-shaped top of a new hot spot bringing heat and mass from deep in the mantle, produced the Siberian traps, a vast basaltic lava-flow province, the biggest known. Carbon dioxide released by this volcanism increased the Earth’s temperature. The new rocks were easily weathered, fertilizing the ocean. Sulfur released by this affected chemistry. The warming from the carbon dioxide reduced the oxygen content of the ocean, and the warming caused the surface waters to “float” more strongly, reducing the ocean circulation taking oxygen to the deep ocean. Large areas became anoxic and euxinic, producing hydrogen sulfide, which is poisonous to many, many things. Certain bacteria, called Chlorobiaceae, or green sulfur bacteria, use hydrogen sulfide instead of water in photosynthesis, and make distinctive organic molecules. These molecules are found in sediments from shallow oceans at the end of the Permian, indicating that poisonous hydrogen sulfide was widespread. (No serious science yet suggests that human carbon dioxide could cause such a disaster, but our actions can contribute to spread of “dead zones” in the ocean that are in some ways analogous. And, note that we are releasing carbon dioxide faster than we believe the volcanoes released it at that time.)
Perhaps without going all the way to poisonous hydrogen sulfide, other times have produced low-oxygen marine environments that allowed deposition of organic-rich material that would have been eaten and burned if oxygen had been higher. The sediments are often black shales, and the “fracking” for natural gas now going on is exploiting the carbon in these deposits. Warm temperatures favor such anoxic events, including the oceanic anoxic events (OAEs) of the saurian sauna of the Cretaceous Period. Note that such deposition tends to lower the carbon dioxide in the air, leading to subsequent cooling. Coal formation also will tend to lower carbon dioxide in the air and favor cooling.
Faster changes in carbon dioxide have occurred, again with higher carbon dioxide causing warming. The best-documented of these is the Paleocene-Eocene Thermal Maximum (PETM). Temperature indicators show warming over a few thousand years, with warmth persisting for 200,000 years or so. Carbon dioxide shows the same history. Isotopic indicators suggest that the carbon dioxide came from volcanic and biological sources. The rapid warming and carbon-dioxide increase came with an acidification of the ocean (carbon dioxide and water make a weak acid), and with a major extinction event for bottom-dwelling types; extinction appears to have been in response to the climate change, with no plausible way that the extinction could have somehow caused the climate change. The most-likely source of the carbon was a large amount of volcanic activity, linked to the “unzipping” of the North Atlantic especially between Greenland and Europe, with melted rock squirting into sediments loaded with organic material (oil, coal and gas). And, the warming then seems to have released more carbon that was stored in plants, or soils, or sea-bed methane deposits. (At present, plants hold about as much carbon as does the atmosphere, soils somewhat more, and seabed methane more. Anything that caused a notable transfer of carbon from one of those other reservoirs to the atmosphere is in principle capable of explaining the event, including permafrost in Antarctica at the time. Note that the PETM is the biggest and fastest such event over very long times, so a coincidence may have been involved — if causing the PETM was easy, more PETMs would have happened over the vast span of Earth’s history.) The PETM and other abrupt events of the past point to the importance of carbon dioxide in temperature (they were far too fast for continental drift to have mattered, for example), and provide time scales for possible feedbacks in the carbon cycle (not fast enough to control the atmosphere on the time scales of decades to centuries over which human societies operate, but fast enough to matter on those time scales).