A recent paper written by Avi Loeb, Rafael Batista and myself, available to read at the preprint arXiv and published in the Journal of Cosmological and Astroparticle Physics, has garnered quite a bit of attention from various media outlets, with a press release from Harvard. The interpretations in the press range from the somewhat speculative to the outright ridiculous.

What we wanted to find is when is it most likely that any given civilization will be around? Of course, in order to do this we have to take into account a lot of unknown parameters - we only have one example of an inhabited planet to work from! Fortunately, as we will see, if you ask questions about proportions rather than overall numbers a lot of these factors cancel out.

First let's deal with the truly unknown: What is the probability that life forms on an Earth-like planet - $p(Life|ELP)$ ? We do not know how life started on Earth - it may have required some very special conditions, it may happen all the time without us noticing it. However, we do assume that all Earth-like planets are made equal, and so have an equal chance of life starting. This may seem unreasonable at first, but if life is really a function of planetary conditions, given two identical planets there should be no preference for one forming life over the other. Therefore we assume this number to be a constant. It may be high - the galaxy teeming with life, or low - a lonely place to be, but when we talk only about proportions this doesn't matter. As an example consider people who are dominantly left-handed. If we didn't know the probability of being born left-handed this may occur in 1/10 of the population, or 1 in 100, or even a million. Lefties may be all around us, or very rare. However, if the chance of being born left handed is the same in all countries, we know there will be roughly 4 times as many lefties in China (pop 1.3 billion) as the USA (pop. 310 million).

So, how can we know how many habitable planets there are at a given time? Like with all difficult questions, it helps to break things down into smaller chunks that are more manageable. We keep splitting these questions into pieces until we get down to those that we can handle:

Number of inhabited planets = Sum over all combinations of attributes: Number of planets with attributes x,y,z * probability life forms on a planet with these attributes.

Immediately we run into an unknown - we do not know how habitability changes with type of planet. However, if we make the conservative assumption that we require an 'Earth-like' planet, for earth like life, we can make progress. For life to exist on a planet, we assume that there must be liquid water on the planet's surface. There may indeed be other types of life out there, living deep in the methane seas of Titan, for example, but for now we restrict to 'life as we know it (Jim)'. This requirement, the presence of liquid water, gives us an inner and outer radius around each region, the Habitable Zone, or 'Goldilocks region' (not too hot, not too cold). Thus we shift the question from being one about planets, to being about stars:

Number of earth-like planets = sum over mass of stars: number of stars of mass M * fraction of stars of mass M that have an earth-like planet in the habitable zone.

Remarkably the last factor, known as $\eta_{Earth} $ in the astrophysics literature, can be estimated from observations of exoplanets. In some surveys it's estimated to be as high as 25% for lower mass stars, and around 10% for stars like our sun. However, since we're trying to find an order of magnitude here, we can assume from this that it is a constant.

How do we deal with the number of stars of a given mass? We know the star formation rate from observations. This is denoted $\dot{\rho_*}(m,t)$ - the rate of formation of stars of mass m at time t. . We also know the star lifetime as a function of its mass. Therefore to find out how many stars of a given mass there are at any time, we integrate the star formation rate before this time, and multiply by a 'window function' that checks if the star is still alive.

Star lifetime is a function of the mass of a star, with low mass stars living a long time, and high mass stars burning out quickly. Our own sun has a lifetime of about 10 billion years, but a star of mass just 3.5 times the sun's will die within 200 million years. This is significant, because that is about the minimum length of time for an earth-like planet to accrete, form and cool to the point where water can exist on its surface. So it seems highly unlikely that life will exist around such heavy stars. Lower mass stars, such as red dwarfs, last much, much longer, as long as 10 trillion years for the lowest mass stars, about 8% of the sun's mass (below this mass, stars cannot hold together).

So we arrive at the 'master equation' - 2.1 in the paper.

$$ \frac{dP}{dt} = \frac{1}{N} \int_o^T dt' \int_{mass} dm' \dot{\rho_*}(t',m') \eta_{Earth} p(Life|ELP) Window(t-t',m') $$

Since we are looking at proportions, this is normalized - the $\frac{1}{N}$ in front of the equation. This allows us to say that the total fraction of civilizations must become 1 across all time, and so find out what proportion exist at a given time, $\frac{dP}{dt}$. We can calculate the proportion of inhabited planets at a given time from our master equation because these constants, such as the probability that life forms at all, fall out into the normalization. As $p(Life|ELP)$ and $\eta_{Earth}$ are constant, we pull them out of the front of the integral, without needing to know their exact values. The results are shown in figure 4. Those long lived dwarf stars push the most likely time for existence far into the future, partly because they live so long, and partly because some of them are yet to form.

Figure 4 from the paper: The proportion of a all civilizations existing at a given cosmic time, with different lowest mass stars allowed. If we keep masses close to our sun (red line), today is highly likely, but allowing low mass stars (green and black lines) pushes the majority of life well into the future.

We can't tell you how many alien civilizations are out there today. We have a sample size of 1. This may become clearer once we can do some spectroscopy of exoplanets, but for now we cannot claim that we are alone or that aliens will certainly exist in the future. It could be that low mass stars stop life forming (tidally locking planets, too much radiation etc) but we simply don't know.

What we have found is a new puzzle: Why are we now?