Role of stellar physics in regulating the critical steps for life

Role of stellar physics in regulating the critical steps for life

Manasvi Lingam Electronic address: Institute for Theory and Computation, Harvard University, 60 Garden St, Cambridge MA 02138, USA Abraham Loeb Electronic address: Institute for Theory and Computation, Harvard University, 60 Garden St, Cambridge MA 02138, USA

We use the critical step model to study the major transitions in evolution on Earth. We find that a total of five steps represents the most plausible estimate, in agreement with previous studies, and use the fossil record to identify the potential candidates. We apply the model to Earth-analogs around stars of different masses by incorporating the constraints on habitability set by stellar physics including the habitable zone lifetime, availability of ultraviolet radiation for prebiotic chemistry, and atmospheric escape. The critical step model suggests that the habitability of Earth-analogs around M-dwarfs is significantly suppressed. The total number of stars with planets containing detectable biosignatures of microbial life is expected to be highest for K-dwarfs. In contrast, we find that the corresponding value for intelligent life (technosignatures) should be highest for solar-mass stars. Thus, our work may assist in the identification of suitable targets in the search for biosignatures and technosignatures.

1 Introduction

In less than a decade, our understanding of exoplanets has improved dramatically thanks to the Kepler mission, which was launched in 2009 (Borucki et al., 2010; Borucki, 2016). The fields of exoplanetary science and astrobiology also received two major boosts over the last couple of years as a result of two remarkable discoveries. The first was the discovery of the potentially habitable planet Proxima b around Proxima Centauri, the star nearest to our Solar system (Anglada-Escudé et al., 2016). The second was the discovery of at least seven Earth-sized planets orbiting the star TRAPPIST-1 at a distance of about pc (Gillon et al., 2016, 2017), some of which may be capable of hosting liquid water on their surfaces. Looking ahead, there are a wide range of space- and ground-based telescopes that will become operational within the next - years with the purpose of hunting for exoplanet biosignatures (Fujii et al., 2017).

The Search for Extraterrestrial Intelligence (SETI) has also received an impetus in this period on both the theoretical and observational fronts. Theoretically, many innovative technosignatures have been proposed for identifying artifacts of extraterrestrial species, both extant and extinct (Bradbury et al., 2011; Wright et al., 2014, 2016). From the observational standpoint, the recently established Breakthrough Listen project (Worden et al., 2017; Isaacson et al., 2017) has injected new funding and rejuvenated SETI,111 after the unfortunate demise of NASA funding in 1993.

Thus, from the perspective of searching for biosignatures and technosignatures, it is therefore necessary to understand what are the constraints imposed on planetary habitability by the host star. This will help in facilitating the optimal selection of suitable target stars and planets, where the prospects for life may be maximized (Horner and Jones, 2010; Lingam and Loeb, 2018a; Kite et al., 2018). In this paper, we will therefore use a model originally developed by Carter (1983), where evolution is treated as a succession of critical steps, to assess the likelihood of primitive (microbial) and intelligent (technological) life, and the implications for detecting biosignatures and technosignatures. A brief description of the methodology is provided in Sec. 2, followed by an extended discussion of the critical step model in the context of Earth’s evolutionary history in Sec. 3. Next, we assess the likelihood of these critical steps being successfully attained on other exoplanets in Sec. 4. We conclude with a summary of our major points in Sec. 5.

2 Methodology

We begin with a brief summary of the mathematical preliminaries. A detailed derivation of these results can be found in Barrow and Tipler (1986), Carter (2008) and Watson (2008). In the critical-step model, the basic assumption is that there are critical (i.e. “hard”) steps in all. Each step is stochastic in nature, and has an associated probability of occurrence (denoted by with ), and the condition must be satisfied . Here, denotes the total period of habitability, and its value for the Earth and other exoplanets will be addressed later.

The central quantity of interest is the probability density function (PDF) for the case where the -th step takes place at time , and the remaining steps take place after . Denoting this quantity by , the PDF can be expressed as


Hence, the mean time taken for the -th step, represented by , is


and hence it follows that the average spacing between two consecutive steps is approximately equal,222This important fact - along with the idea that this methodology could be used to assess the accuracy of models describing the major evolutionary transitions on Earth - was first emphasized by Robin Hanson in his unpublished manuscript on evolutionary transitions: with


The cumulative probability that the -th step occurs at a time is given by


where is the incomplete beta function. For the limiting case , (4) reduces to .

3 Critical Steps and Major Transitions on Earth

We briefly discuss the use of multi-step model as a heuristic means of understanding the evolutionary history of the Earth.

3.1 How many critical steps were present?

Although this question has been explored recently by means of the critical steps approach (Carter, 2008; Watson, 2008; McCabe and Lucas, 2010), there are some major points of divergence in our analysis, as discussed below.

One notable difference is that we assume the Earth was habitable approximately Ga (Gyr ago), as opposed to previous treatments which specified the earliest point of habitability as Ga. The primary reason for the choice of Ga was motivated by the fact that the Late Heavy Bombardment (LHB) - a phase wherein the Earth was subjected to cataclysmic bombardment by a high number of impactors (Gomes et al., 2005) - was detrimental to habitability. However, there are several lines of evidence that now suggest that the LHB may not have been a significant impediment to habitability:

  • There is some evidence indicating that the cratering record may also be explained via a sustained declining bombardment, instead of the intense LHB (Bottke and Norman, 2017). If this hypothesis is correct, the prospects for habitability are improved, and the Earth may have been geologically habitable as early as Ga (Valley et al., 2002; Zahnle et al., 2007; Harrison, 2009; Arndt and Nisbet, 2012). For instance, if the bombardment was relatively moderate, it could even have served as a valuable energy source for prebiotic chemistry (Chyba and Sagan, 1992), leading to the synthesis of biomolecules such as amino acids, peptides and nucleobases (Martins et al., 2013; Furukawa et al., 2015).

  • Even if the LHB were present, numerical models which computed the extent of crustal melting indicate that hyperthermophiles may have survived in near-surface and subsurface environments (Abramov and Mojzsis, 2009; Grimm and Marchi, 2018); see also Sloan et al. (2017).

  • Yet another possibility is that life-bearing ejecta spawned during the LHB can return to the Earth, and thereby reseed it over short ( yr) timescales (Wells et al., 2003; Gladman et al., 2005), effectively ensuring that habitability was almost continuously prevalent during the Hadean-Archean eons.

Thus, we start our habitability “clock” at Ga. Several studies have attempted to assess the end of Earth’s habitability in the future. While early models yielded a value of Gyr in the future (Lovelock and Whitfield, 1982), more recent analyses have pushed forward this boundary to Gyr in the future (Caldeira and Kasting, 1992; Franck et al., 2000; Goldblatt and Watson, 2012). While the Gyr limit is conceivably valid for extremophiles, the limits for more complex organisms (including humans) could be closer to Gyr (Wolf and Toon, 2015). In addition, there may be other astrophysical risks posed to habitability over multi-Gyr timescales (Bailer-Jones, 2009; Melott and Thomas, 2011; Sloan et al., 2017). Hence, we will assume that the Earth becomes uninhabitable Gyr in the future, but we will address the Gyr case later in Sec. 3.4.

As per the preceding discussion, Gyr. Let us suppose that the evolution of technological intelligence (i.e. Homo sapiens) represents the -th step, and use the fact that the mean timescale for our emergence was Gyr. From (2) and the above values, it follows that . Thus, it seems plausible that a 4- or 5-step model may represent the best fit. This result is in good agreement with earlier studies that arrived at the conclusion (Watson, 2008; Carter, 2008; McCabe and Lucas, 2010), and we will adopt this value henceforth. Classical frameworks for understanding the course of evolution on Earth also seemingly indicate that the total number of critical steps was quite small (Szathmáry and Smith, 1995; de Duve, 2005), i.e. , and could have been to in number (Knoll and Bambach, 2000; Judson, 2017).

3.2 What were the five critical steps?

In order to determine the five critical steps, there are two routes that are open to us. The first approach assumes that these steps correspond to the major evolutionary transitions identified in the seminal work of Smith and Szathmáry (1995), wherein each step involves significant changes in the storage and transmission of information. This paradigm has been extensively utilized and extended by several authors (Calcott and Sterelny, 2011; Jablonka and Lamb, 2014; Bains and Schulze-Makuch, 2016). This strategy for identifying the critical steps was employed by Watson (2008), who observed that the temporal constraints on first three transitions (origin of replicating molecules, chromosomes, and the genetic code) indicate that not all of them are likely to be critical steps; instead, if the origin of prokaryotic cells is considered as a single critical step, the model can be formulated accordingly.

(1A) Origin of Prokaryotic Life: Of all the potential critical steps, dating the origin of life (abiogenesis) is the most difficult owing to the near-absence of sedimentary rocks and the action of processes like diagenesis and metamorphism (Knoll et al., 2016). We will adopt a conservative approach, and adopt the value of Ga for the earliest robust evidence of life. There are two independent lines of evidence that support this date. The first is the recent discovery of stromatolite-like structures in the Isua Supracrustal Belt (ISB) by Nutman et al. (2016). The second stems from the low C values in graphite globules from the ISB (Rosing, 1999; Ohtomo et al., 2014), which is conventionally indicative of biological activity. The oldest microfossils, which arguably display evidence of cell structure (e.g. lumen and walls), date from Ga (Wacey et al., 2011a; Brasier et al., 2015). Here, it should be noted that even older claims for life do exist - the potentially biogenic carbon in a Ga Jack Hills zircon (Bell et al., 2015) and putative microfossils Ga in the Nuvvuagittuq belt (Dodd et al., 2017) are two such examples - but they are not unambiguous. As per our discussion, the timescale for abiogenesis on Earth after the onset of habitability is Gyr. From (4), the cumulative probability is found to be .

(2A) Origin of Eukaryotes: The origin of the crown eukaryotes, which occurred through endosymbiosis between an archaeon (probably the Lokiarchaeota) and a proto-mitochondrion (Sagan, 1967; Embley and Martin, 2006; McInerney et al., 2014; Archibald, 2015), was apparently a very important one from the standpoint of bioenergetics and the eventual increase in biological complexity (Lane and Martin, 2010; Martin et al., 2015); see, however, Booth and Doolittle (2015) and Lynch and Marinov (2015). The oldest fossils that appear to be eukaryotic in origin are the vesicles from the Changzhougou Formation dated to Ga (Lamb et al., 2009; Li et al., 2013). There are several other ostensibly eukaryotic microfossils that have been dated to - Ga, and possibly as old as Ga (Han and Runnegar, 1992; Knoll, 2014; Javaux and Knoll, 2017; Bengtson et al., 2017). The use of phylogenetic molecular clock models has yielded ages for the Last Eukaryotic Common Ancestor (LECA) ranging between and Ga, although recent studies are closer to the latter value (Parfrey et al., 2011; Eme et al., 2014; López-García and Moreira, 2015; Sánchez-Baracaldo et al., 2017). Although earlier claims for eukaryotic microfossils exist, for e.g. in the Ga Francevillian B Formation (Albani et al., 2010), the Ga shales from the Pilbara Craton (Brocks et al., 1999), the Transvaal Supergroup sediments from - Ga (Waldbauer et al., 2009), and the - Ga lacustrine deposits of South Africa (Kaźmierczak et al., 2016),333It should also be noted that some molecular clocks have yielded ages Ga for the LECA (Hedges et al., 2004; Hedges and Kumar, 2009). we shall adopt the timing Ga for the origin of eukaryotes. This timescale of Gyr leads us to the cumulative probability of .

(3A) Origin of Plastids: In the original list of major evolutionary transitions (Smith and Szathmáry, 1995), sexual reproduction was present in place of plastids. An important reason for this alteration was because there exists sufficiently compelling evidence that LECA was a complex organism that was capable of sexual reproduction (Koonin, 2010; Butterfield, 2015); in other words, the origin of sexual reproduction was possibly coincident with eukaryogenesis (Szathmáry, 2015; Speijer et al., 2015), although there is no a priori reason to believe that this apparent coincidence will always be valid on other inhabited exoplanets.
The importance of plastids stems from the fact that they enable eukaryotic photosynthesis. Eukaryotes acquired this ability by means of endosymbiosis with a cyanobacterium (Rodríguez-Ezpeleta et al., 2005; Archibald, 2009; Keeling, 2010), thereby giving rise to the “primary” plastids in algae and plants (Gould et al., 2008; Price et al., 2012). This endosymbiosis is believed to have occurred around - Ga (Yoon et al., 2004; Falkowski et al., 2004; Reyes-Prieto et al., 2007; Parfrey et al., 2011; Ochoa de Alda et al., 2014), and these estimates appear to be consistent with the recent discovery of multicellular rhodophytes from Ga (Bengtson et al., 2017). However, recent evidence based on molecular clock analyses favors the origin of the Archaeplastida (that possess plastids) by Ga (Sánchez-Baracaldo et al., 2017). We choose to err on the side of caution and use Ga as the origin of the primary plastids. Upon calculating the cumulative probability using (4), we find .

(4A) Origin of Complex Multicellularity: In this context, the rise of “complex multicellularity” refers to the emergence of plants, fungi and animals (Szathmáry and Smith, 1995). An important point worth noting here is that each of these clades could have originated at a different time. The earliest evidence for metazoan fossils has been argued to be at least Ga (Love et al., 2009; Maloof et al., 2010), but it cannot be regarded as wholly conclusive. Molecular clocks indicate that the last common ancestor of animals lived around Ga or earlier (Douzery et al., 2004; Wray et al., 1996; Erwin et al., 2011; Richter and King, 2013). The molecular clock evidence for plants suggests that their origins may extend as far back as - Ga (Heckman et al., 2001; Lewis and McCourt, 2004; Clarke et al., 2011; Magallón et al., 2013), although these methods are subject to significant variability; the direct fossil evidence for plants is much more recent (Knoll and Nowak, 2017). Lastly, the use of molecular clocks to determine the origin of fungi has led to the estimate of - Ga (Lücking et al., 2009). Thus, taken collectively, it seems plausible that the origin of complex multicellularity was about Ga (Rokas, 2008), although the discovery of Bangiomorpha (Butterfield, 2000) can be construed as evidence for an earlier divergence time. This hypothesis gains further credibility in light of the distinctive increase in eukaryotic diversity documented in the fossil record at Ga (Knoll et al., 2006; Knoll, 2011). The cumulative probability for this step is .

(5A) Origin of Humans: More accurately, the revised version, Szathmáry (2015) refers to the origin of “Societies with natural language”, thus emphasizing the role of language. Since anatomically modern humans evolved only yr ago (Klein, 1995; Tattersall, 2009), the timescale for the evolution of H. sapiens since the onset of habitability is Gyr. Hence, the cumulative probability is estimated to be by making use of (4).

Next, we shall outline the second strategy for identify the five critical transitions. In order to do so, let us recall that the spacing between each critical step is roughly equal. From (3), we find that Ga. Thus, if we can identify five important transitions during Earth’s geobiological and evolutionary history that have a spacing of Ga, they could potentially represent the critical steps leading to technological intelligence. We will present our five transitions below, and offer reasons as to why they might constitute critical steps.

(1B) Origin of Prokaryotic Life: Our choice of (1B) is the same as (1A). The issue of whether abiogenesis is an “easy” or a “hard” step remains currently unresolved (Walker, 2017), but it has important implications for the likelihood of extraterrestrial life (Lineweaver and Davis, 2002; Davies, 2003; Spiegel and Turner, 2012). However, in the spirit of most conventional analyses, we will suppose that abiogenesis does constitute one of the critical steps. In this case, the cumulative probability turns out to be .

(2B) Origin of Oxygenic Photosynthesis: The evolution of oxygenic photosynthesis, due to the origin of prokaryotes akin to cyanobacteria, had a profound impact on the Earth’s biosphere (Hohmann-Marriott and Blankenship, 2011). On metabolic grounds, there are strong reasons to posit the emergence of oxygenic photosynthesis as a major transition in its own right (O’Malley and Powell, 2016). The many advantages due to oxygenic photosynthesis have been succinctly summarized in Judson (2017). Most notably, the addition of oxygen to the atmosphere led to the formation of the ozone layer, caused an increase in the diversity of minerals, led to the creation of new ecological niches, and above all, aerobic metabolism releases about an order of magnitude more energy compared to anaerobic metabolism (McCollom, 2007; Koch and Britton, 2008). The origin of photoautotrophs is very poorly constrained (Allen and Martin, 2007) with chronologies ranging between Ga to Ga, with the former estimate arising from indirect evidence of environmental oxidation based on U–Th–Pb isotopic ratios (Buick, 2008) and the latter representing the oldest direct evidence from microfossils (Fischer et al., 2016). If we naively take the mean of these two values, we obtain Ga. There are several lines of evidence, not all of which are robust biomarkers (Rasmussen et al., 2008; Fischer et al., 2016), which indicate that oxygenic photosynthesis evolved approximately Ga or later (Eigenbrode and Freeman, 2006; Falcón et al., 2010; Stüeken et al., 2012; Planavsky et al., 2014; Schirrmeister et al., 2015, 2016; Shih et al., 2017), i.e. a few Myr before the onset of the Great Oxygenation Event (GOE). With the choice of Gyr (which corresponds to Ga) for oxygenic photosynthesis, we obtain a cumulative probability of after using (4).
As noted earlier, the origin of oxygenic photosynthesis is subject to much controversy and uncertainty. Hence, it is quite conceivable that the GOE served as a critical step in the origin of complex (eukaryotic) life, and the attainment of sufficient oxygen levels could serve as an evolutionary bottleneck on exoplanets (Knoll, 1985; Catling et al., 2005). The GOE was a highly significant event that led to a considerable increase in the oxygen level (to of the present-day value) around to Ga, and thereby shaped Earth’s subsequent history (Holland, 2006; Lyons et al., 2014; Knoll, 2015a; Luo et al., 2016). If we choose the onset of the GOE as our critical step, we find that .

(3B) Origin of Eukaryotes: We have already remarked previously as to why eukaryogenesis represented such an important step. The possible endosymbiosis of mitochondria (followed by plastids and other organelles) has been regarded as essential from the standpoint of phagocytosis, bioenergetics, organismal complexity and cellular evolution (Margulis, 1981; Payne et al., 2009; Yutin et al., 2009), and is often perceived as a major evolutionary transition (Calcott and Sterelny, 2011). The difference is that it constitutes the third step in our hypothesis, whereas it served as the second step in the original -step model. The cumulative probability in this case is since we have used the fact that eukaryogenesis occurred Ga based on our preceding discussion in step (2A).

(4B) Origin of Complex Multicellularity: Our choice of (4B) is the same as (4A). This is primarily motivated by the fact that the origin of these organisms (especially plants and animals) have led to a radical transformation of Earth’s biosphere. More specifically, Earth’s energy balance, biomass productivity, biogeochemical cycles, ecological niches and macroevolutionary processes have been shaped by the emergence of complex multicellular organisms (Lewontin, 2000; Odling-Smee et al., 2003; Post and Palkovacs, 2009; Butterfield, 2011; Judson, 2017). Hence, in this case, we obtain the same cumulative probability, i.e. .
An alternative possibility is to consider the Neoproterozoic Oxygenation Event (NOE) as the critical step. The NOE is akin to the GOE since it also entailed a rise in the atmospheric oxygen (to near-modern levels), but its exact timing and causes are unclear. In particular, it remains ambiguous as to whether the NOE served as a cause or a consequence of the origin of animals (Och and Shields-Zhou, 2012; Lyons et al., 2014). The timing is also very variable, with evidence from selenium isotopes not ruling out the onset of the NOE as early as Ga (Pogge von Strandmann et al., 2015) while iron-based proxies do not demonstrate significant oxygenation even as late as Ga (Sperling et al., 2015). If we take the mean of these two quantities, the NOE would have taken place Ga and this estimate is roughly consistent with recent analyses that have yielded values of - Ga (Chen et al., 2015; Knoll and Nowak, 2017).444We have argued earlier that the diversification of metazoans commenced at Ga, while the NOE has been assigned a timing of Ga. Hence, this raises the question as to how animal evolution took place in the presence of low oxygen levels. This discrepancy can be explained by the fact that the oxygen requirements for early animals (akin to modern demosponges) were also sufficiently low (Mills et al., 2014; Knoll and Sperling, 2014). If we assume the NOE to be a critical step instead, and use the value of Gyr (i.e. Ga), we obtain .

(5B) Origin of Humans (Technological Intelligence): Our fifth step is the same as the previous model on account of the following reasons. In addition to the distinctive ability to construct and employ sophisticated tools (giving rise to technology), other attributes such as foresight, recursion and syntactical-grammatical language are also widely cited as being unique to humans (Tomasello, 1999; Penn et al., 2008; Tomasello, 2008; Corballis, 2011; Suddendorf, 2013; Berwick and Chomsky, 2016).555In contrast, we observe that other “human” characteristics such as culture, intelligence, morality and consciousness have been, to varying degrees of controversy, associated with other species (Griffin, 2001; Whiten and van Schaik, 2007; Bekoff and Pierce, 2009; Whitehead and Rendell, 2015; Roth, 2015; De Waal, 2016). Lastly, humans have also caused major (perhaps irrevocable) large-scale shifts in the functioning of Earth’s biosphere (Barnosky et al., 2011, 2012; Ellis et al., 2013) to the extent that the Earth’s latest epoch, the Anthropocene, has been primarily shaped by us (Steffen et al., 2011; Lewis and Maslin, 2015; Steffen et al., 2015). The cumulative probability for this step is given by .

McCabe and Lucas (2010) introduced a parameter to estimate the goodness of fit:


and a lower value of corresponds to a better fit. If each of the cumulative probabilities were close to either or , it would mean that the events are clustered towards the beginning or the end, thereby constituting a poor fit. For the -step model (1A-5A), we find . In contrast, if we use the -step model (1B-5B), we find ; even if use the GOE and the NOE in place of the steps (2B) and (4B) respectively, we find . Thus, we find that the second -step model (1B-5B) is approximately twice more accurate than the first model (1A-5A).

Before proceeding further, we wish to emphasize that even sequences of evolutionary transitions that yield relatively high values of could still be “correct”, since the real issue could lie with the theoretical framework employed herein (involving critical steps). For instance, the classification of evolutionary steps as “easy” or “hard” (i.e. critical) is both abstract and binary in nature. Second, in the “long fuse” paradigm (Bogonovich, 2011), a series of likely steps (each with a non-negligible timescale) unfold, culminating in slow, but near-inevitable, evolution - as per this framework, the emergence of any particular evolutionary innovation essentially becomes a matter of time.666Hence, in this event, low-mass stars would be ideally suited for the evolution of intelligent life because of their longer main-sequence lifetimes.

3.3 A six-step model

Carter (2008) concluded that a 5- or 6-step model represented the best fit for the total number of critical steps on our planet. Here, we will outline a 6-step model based on the “megatrajectories” paradigm introduced by Knoll and Bambach (2000) and assess whether it constitutes a good fit for the critical step model.

  • From the Origin of Life to the Last Common Ancestor (LCA) of Extant Life: As with the steps (1A) and (1B), we note that there is insufficient evidence to properly date the age when abiogenesis occurred and when the LCA lived. However, as we have argued in Sec. 3.2, the earliest definitive evidence for life appears to be around Ga. In this scenario, with Gyr and Gyr, we use (4) to obtain .

  • The Metabolic Diversification of Bacteria and Archaea: The first evidence for methanogens arguably comes from hydrothermal precipitates Ga (Ueno et al., 2006), although molecular clock analyses lead to the even earlier date of at least Ga (Battistuzzi et al., 2004). The earliest iron- and sulfate-reducing microbes also potentially appear in the fossil record at approximately the same time (Shen et al., 2001; Ueno et al., 2008; Wacey et al., 2011b; Bontognali et al., 2012). There is also some evidence suggesting that methanotrophy or the Wood-Ljungdahl pathway was operational at Ga (Flannery et al., 2018). The record for nitrogen fixation implies that it was present by Ga (Stüeken et al., 2015), or perhaps even earlier (Stüeken, 2016). Thus, taken collectively there is considerable evidence indicating that metabolic diversification had occurred by - Ga (Noffke et al., 2013; Knoll, 2015b; Moore et al., 2017). We will therefore adopt Gyr (i.e. Ga), which results in .

  • Evolution of the Eukaryotic Cell: This megatrajectory is essentially the same as steps (2A) and (3B). Using the timing identified therein, we find .

  • Multicellularity: It is well-known that multicellularity has evolved repeatedly over Earth’s history, and has been therefore characterized as a “minor” major transition (Grosberg and Strathmann, 2007). On the other hand, organisms that fall under the bracket of “complex multicellularity” belong to only six clades (Knoll, 2011). If this serves as the actual critical step, we have already discussed its timing in steps (4A) and (4B) and we end up with .

  • Invasion of the Land: Although the first land-dwelling organisms appeared in the Precambrian (Wellman and Strother, 2015), the Paleozoic radiation of the land plants (embryophytes) facilitated a major ecological expansion. The earliest fossil evidence dates from the mid Ordovician (Gensel, 2008), although it is conceivable that land plants may have originated in the Cambrian (Knoll and Nowak, 2017). Consequently, the fossil record is in good agreement with molecular clock evidence that dates land plants to - Ga (Sanderson et al., 2004; Smith et al., 2010; Morris et al., 2018).777However, there are other molecular clock studies that favor a Proterozoic origin of land plants instead (Heckman et al., 2001; Clarke et al., 2011; Magallón et al., 2013). Thus, by choosing Gyr, we find .

  • Intelligence and Technology: This megatrajectory is essentially the same as steps (5A) and (5B). The corresponding cumulative probability is .

By using (5), we compute the goodness of fit for this -step model. We find that , which is virtually identical to (although lower than by a factor of about ). Hence, this demonstrates that the megatrajectories considered herein are a fairly good fit insofar our model is concerned; the resultant value of is lower than the - or -step model analyzed in McCabe and Lucas (2010).

3.4 The ramifications of an extended habitability interval

As noted in Sec. 3.1, recent theoretical studies indicate that the Earth may remain habitable (modulo anthropogenic change) to Gyr in the future. With this revised estimate, the value of now becomes Gyr. As before, let us assume that humans represent the -th step. By calculating the value of using (2), we find . Hence, this estimate suggests that a 2-step model (or possibly a 3-step one) has the greatest likelihood of being valid; Carter (2008) also reached a similar conclusion.

We are confronted with the question as to what was the first critical step. The spacing between the critical steps must be approximately Gyr as seen from (3). Since the advent of humans at Gyr (i.e. Ga) constitutes the second step, the timing of the first critical step must have been approximately Ga. As noted in Sec. 3.2, the timing of the GOE (between to Ga) falls within this range. The GOE had profound consequences for Earth’s subsequent evolutionary history, and therefore represents a strong contender for the first critical step. Other notable candidates that lie approximately within the same time frame include the evolution of (i) oxygenic photosynthesis and (ii) eukaryotes. In the -step model, the origin of life (abiogenesis) is not likely to have been a critical step; in this regard, the -step model is akin to the original -step model proposed by Carter (1983).

However, we can ask ourselves the following question: if the origin of life was a critical step, how many steps were there in total? If we assume that abiogenesis was the first step and that the mean time for this step was equal to the abiogenesis timescale of Gyr, from (2) we find . In contrast, if we had assumed that Gyr and repeat the calculation, we arrive at . This leads us to the following conclusions:

  • For the case where habitability ends Gyr in the future, a -step model would be favored, although the -step model may also be plausible (Carter, 2008). The choice of is consistent with the discussion in Sec. 3.1.

  • When the habitability boundary extends to Gyr in the future, a -step model would represent a good fit. Let us assume that Gyr, i.e. that humans are the -th critical step. From (2), we obtain , implying that the evolution of humans could have been either the fifth or sixth critical step. In other words, there are still or critical steps ahead in the future, which we will discuss shortly hereafter.

As noted above, there is a possibility that humans are not the -th critical step, but merely the -th one (with ). In Secs. 3.1 and 3.2, we have seen that there are compelling reasons to believe that humanity was the fifth critical step. Therefore, with and assuming Gyr, we can estimate the value of using (2).

  • If Gyr, and using the above values, we find . In other words, when Earth’s habitability ends about Gyr in the future, the -step model is relatively favored and the evolution of humans is the last critical step.

  • Using the above parameters in conjunction with Gyr leads us to . Hence, if the Earth becomes uninhabitable Gyr in the future, the -step model seems the most likely. In this case, since humans are the fifth critical step, there is one critical step that is yet to occur.

Based on our discussion thus far, two broad inferences can be drawn. First, assuming that the habitability window ends Gyr in the future, the critical step model with is likely to be valid and humans represent the final critical step. In contrast, if the habitability window is extended to Gyr in the future, we suggest that the -step model is the best fit and that humans represent the fifth critical step. In other words, there is still one step which is unaccounted for. Naturally, it is not possible to identify this step prior to its occurrence.

One possibility is the emergence of superintelligence (Bostrom, 2014), especially in light of recent advancements (and concerns) in Artificial Intelligence (AI). However, the major issue from the standpoint of the critical step model is that the timescale between the emergence of H. sapiens and AI superintelligence is very low, i.e. on the order of - yrs, compared to the characteristic separation between successive critical steps ( yrs). This discrepancy might be resolved if the origin of superintelligence entails a much longer time than currently anticipated.

An underlying assumption pertaining to the above discussion is that we have automatically presupposed that the “biological complexity” (Carroll, 2001; McShea and Brandon, 2010) increases monotonically with time. The pitfalls of subscribing to implicit teleological arguments, certain theories of orthogenesis, and the “March of Progress” are many and varied (Gould, 1996; Ruse, 1996),888Yet, many of the critical step models discussed in the literature take it for granted that the evolution of humans constitutes the last critical step regardless of the duration of the habitable period of the Earth. and therefore it does not automatically follow that the sixth step alluded to earlier will lead to species of greater complexity. Instead, it seems quite plausible that the contrary could occur, especially if the environmental conditions Gyr in the future are less clement than today.

4 Critical steps on exoplanets

We will now study some of the salient features of multi-step models on exoplanets, and discuss the resulting implications. A similar topic was analyzed recently (using the Bayesian framework) by Waltham (2017) recently, but we incorporate additional constraints on habitability imposed by stellar physics in our treatment.

4.1 The Habitable Zone of Earth-Analogs

Figure 1: Number of critical steps as a function of the stellar mass based on (7).

The habitable zone (HZ) is defined as the region around the host star where liquid water can exist on the planet’s surface. The HZ is dependent on both planetary and stellar parameters, and evolves dynamically over time (Kasting et al., 1993; Kasting and Catling, 2003; Kopparapu et al., 2013, 2014). The HZ is typically computed for “Earth-analogs”, i.e. planets whose basic physical, chemical and geological parameters are similar to that of Earth. In our subsequent discussion, we will implicitly deal with Earth-analogs in the HZ of their host stars.999Thus, we shall not focus on habitable worlds outside the HZ, which are expected to be much more commonplace compared to those within the HZ (Lingam and Loeb, 2018c).

Clearly, the upper bound on the habitability of a planet is the stellar lifetime. However, the maximum duration that the planet remains habitable is less than the stellar lifetime for a simple reason: the stellar luminosity increases over time, and the planet will eventually enter a runaway greenhouse phase and become uninhabitable (like Venus). Thus, the duration of habitability is essentially specified by the temporal extent of the continuously habitable zone (denoted here by CHZ). By using the knowledge about the inner and outer boundaries of the HZ in conjunction with stellar evolution models, it is feasible to estimate the total duration of time that an Earth-analog will remain inside the HZ as a function of the stellar mass . This effort was undertaken by Rushby et al. (2013), and by making use of Fig. 11 and Table 5 in that paper, we introduce the scaling:


where Gyr and is the solar mass. Our choice of normalization constant differs from Rushby et al. (2013) since we have adopted the more conservative habitability duration of Gyr for the Earth-Sun system. An inspection of (4.1) reveals that low-mass stars are characterized by longer CHZs, which is along expected lines since they have longer main-sequence lifetimes.

Now, let us consider the highly simplified model wherein we suppose that the timescale for abiogenesis is the same on all exoplanets, and that the duration of habitability is given by . Since abiogenesis is taken to be the first critical step, from (2) with we find


where Gyr is the habitability duration of the Earth and is the number of critical steps on Earth, while represents the corresponding number of steps for Earth-analog orbiting a star of mass . We will henceforth use as this value has been advocated by several authors. Moreover, as we have seen from Sec. 3, there are reasons to believe that constitutes a fairly good fit. Thus, from (4.1), can be estimated as a function of , and this plot is shown in Fig. 1. The value of decreases when is increased, and shortly after the value of drops below unity.

4.2 Constraints on habitability imposed by stellar physics

The preceding analysis implicitly assumed that the only timescale for habitability was . In reality, there are a number of factors governed by stellar physics that influence habitability. In particular, there has been a growing appreciation of the role of space weather in governing habitability, i.e. for e.g. the role of flares, coronal mass ejections (CMEs), stellar energetic particles (SEPs) and stellar winds to name a few (Vidotto et al., 2013; Kay et al., 2016; Garraffo et al., 2016, 2017; Airapetian et al., 2017; Dong et al., 2017a, b, 2018; Lingam and Loeb, 2017d, 2018b; Lingam et al., 2018).

Since the presence of an atmosphere is necessary for maintaining liquid water on the surface of a planet, its complete depletion would lead to the termination of habitability insofar surficial life is concerned. For Earth-analogs that are closer to their low-mass host stars, they are subjected to intense stellar winds that can deplete their atmospheres over short timescales. The significance and magnitude of stellar wind erosion has been thoroughly documented in our Solar system (Brain et al., 2016; Jakosky et al., 2017). The approximate timescale associated with total atmospheric escape due to stellar wind erosion (Lingam and Loeb, 2017c) is


for an Earth-analog assuming that the star’s rotation rate is similar to the Sun; the model displayed decent agreement with numerical simulations (Dong et al., 2018). Thus, the actual duration of habitability should be defined as . In other words, if , the planet loses its atmosphere before it exits the CHZ and vice-versa. From (4.1) and (8), we find


Figure 2: The likelihood of a planet to host life (i.e. the ratio of its habitability duration to the abiogenesis timescale) as a function of the stellar mass .

Our next stellar constraint stems from the availability of biologically active ultraviolet (UV) radiation. The importance of UV radiation is because of the fact that it constitutes the most dominant energy source for prebiotic synthesis on Earth (Deamer and Weber, 2010). Although theories of abiogenesis are many and varied, there is a strong case that can be made for UV radiation as the driver of prebiotic chemistry (Rapf and Vaida, 2016; Sutherland, 2017), especially with regards to the RNA world (Gilbert, 1986; Orgel, 2004), on account of the following reasons:

  • Laboratory experiments have shown that UV light provides a selective advantage to RNA-like biopolymers due to the presence of nitrogenous bases, and may therefore play an important in their polymerization (Mulkidjanian et al., 2003).

  • The tendency in origin-of-life experiments towards the formation of complex organic mixtures is referred to as the “asphalt problem”. Recent experiments have shown that this issue might be bypassed in suitable geological environments (intermountain valleys), and that UV radiation can facilitate the synthesis of nucleosides, nucleotides, and perhaps RNA (Benner et al., 2012).

  • The synthesis of important biomolecules without excessive human intervention and under conditions that resembled Hadean-Archean environments has proven to be challenging. However, there have been several breakthroughs in recent times that are reliant on UV light (Sutherland, 2016). More specifically, the biologically relevant compounds produced include: (i) pyrimidine ribonucleotides and -ribonucleosides (Powner et al., 2009; Xu et al., 2017), (ii) building blocks of sugars, namely glycolaldehyde and glyceraldehyde (Ritson and Sutherland, 2012), (iii) precursors of nucleic acids, amino acids, lipids and carbohydrates (Patel et al., 2015), and (iv) iron-sulfur clusters (Bonfio et al., 2017).

  • RNA nucleotides have been shown to be stable when radiated by UV photons, and this has been argued to be evidence that they could have originated in the high-UV environments of Hadean-Archean Earth (Beckstead et al., 2016).

Another major theory for the origin of life posits that it occurred in submarine hydrothermal vents (Baross and Hoffman, 1985; Martin et al., 2008). This theory does have many advantages of its own (McCollom and Seewald, 2007; Russell et al., 2014; Sojo et al., 2016), and recent evidence suggesting that the LCA was thermophilic in nature is consistent with hydrothermal vents being the sites of abiogenesis (Akanuma et al., 2013; Weiss et al., 2016). However, it cannot be said at this stage that the LCA was definitively a thermophile, since other studies point to a mesophilic origin (Miller and Lazcano, 1995; Bada and Lazcano, 2002; Cantine and Fournier, 2018). A recent study by Deamer and Damer (2017) assessed seven factors ostensibly necessary for life’s origination, and concluded that submarine hydrothermal vents could potentially face difficulties in fulfilling all of these criteria.

Hence, in our subsequent discussion, we will assume that the origin of life on Earth-analogs was driven by UV radiation. In this scenario, the rate of prebiotic chemical reactions is constrained by the available bioactive UV flux at the surface (Ranjan et al., 2017). The latter can be estimated solely as a function of ,101010An important limitation is that the UV radiation from stellar flares, which can have both beneficial and detrimental effects (Dartnell, 2011; Lingam and Loeb, 2017d), is not taken into consideration. thereby leading us to the abiogenesis timescale (Lingam and Loeb, 2017b):


where Gyr. Next, we consider the ratio because of its significance. If , then the duration of habitability is lower than the timescale for abiogenesis, thus implying that such Earth-analogs are not likely to host life. We have plotted as a function of in Fig. 2. A couple of conclusions can be drawn from this figure. For , we find that indicating that planets in the HZ of these stars have a lower chance of hosting life. Second, we find that the curve flattens out when but it does attain a slight peak at . Although this maximum is attained exactly at due to the ansatzen used in this paper, the peak of the curve has a high likelihood of being in the vicinity of , thereby suggesting that Sun-like stars may represent the most appropriate targets in the search for life (Lingam and Loeb, 2018a).

Figure 3: Number of critical steps as a function of the stellar mass based on (11).

We turn our attention now to the -step model introduced in Sec. 3. Since we have argued that abiogenesis was the first critical step, using (2) along with Gyr for the Earth leads us to Gyr. Thus, we see that the timescale specified for the origin of life on Earth ( Gyr) obeys . As noted earlier, this is not surprising since the mean time taken for the -th critical step in a viable model is comparable to its actual timescale (Carter, 1983). If we assume that this condition is also valid on other potentially habitable planets, we have . Using this relation in conjunction with (2), (4.2) and (4.2), the value of is found to be


Fig. 3 depicts the dependence of on the stellar mass. From this plot, we see that occurs only for and this result is in agreement with Fig. 2, since planets orbiting such stars have a habitability duration that is shorter than the abiogenesis timescale. Second, we observe a clear peak at , and this behavior is also observed in some of the subsequent figures. This result is consistent with our earlier discussion: although the peak arises due to the scaling relations employed herein, there is still a strong possibility that the maximum number of critical steps occur when . It lends further credibility to the notion that G-type stars are the optimal targets in the search for life-bearing planets. Our results are qualitatively consistent with the Bayesian analysis by Waltham (2017), who concluded that: (i) the likelihood of life around M-dwarfs must be selectively suppressed, (ii) G-type stars are the most suitable targets for SETI (Search for Extraterrestrial Intelligence) observations, and (iii) the number of critical steps leading to intelligence is not likely to exceed five.

An important point to recognize here is that although the value of occurs in the vicinity of , this does not altogether preclude stars outside this range from hosting planets with technologically sophisticated species. This is because the total number of critical steps leading to the emergence of life and intelligence on other planets is unknown, and there are no reasons to suppose the total number of critical steps will always be the same.111111In light of the undoubted evolutionary and ecological significance of the breakthroughs discussed in Sec. 3, it may be tempting to conclude that they are sufficiently general, and argue that the convergent evolution of humanoids is “inevitable” if all these transitions are successful (Morris, 2003). However, in spite of the impressive and rapidly increasing list of convergent mechanisms and organs (McGhee, 2011), this standpoint appears to be overly anthropocentric. On the other hand, once the number of critical steps drops below unity, it becomes rather unlikely that such stars (with ) would have planets where intelligence can arise.

Figure 4: The probability of attaining the -step as a function of the stellar mass based on (13).

With these caveats in mind, we will, nevertheless, hypothesize that the critical steps discussed in Sec. 3 (for Earth) are sufficiently general, and therefore applicable to other exoplanets. As we have seen, there are two constraints that were employed in our analysis: (I) the steps must occur in the interval , and (II) the first step (), namely abiogenesis, must occur at . Hence, it follows that the remaining steps must unfold in remaining time interval. We introduce the expression for the PDF of the -th step (in a sequence of steps) in the time :


with being a constant, based on Sec. 2. By integrating this PDF over the interval , we obtain the probability, denoted by , for all steps to occur (since we have already imposed the constraint that the first step is attained at ). The constant of proportionality is calculated by demanding that when because of criterion (I). This yields


and the same formula can be obtained from (4) with , and ; see also Barrow and Tipler (1986). Note that this formula is valid only when , which automatically excludes stars with . There are two important scenarios worth considering from the standpoint of detecting the fingerprints of life:

  • The probability of technological intelligence: This requires based on the above assumptions. In this case, it will be theoretically possible to detect signs of intelligent life by searching for technosignatures because they are more distinctive.

  • The probability of detectable primitive life: From the standpoint of microbial life, most of the well-known biosignatures like oxygen and ozone are not detectable until they have attained a certain level (Meadows, 2017; Krissansen-Totton et al., 2018). Hence, although Earth had life throughout most of its history, the low concentrations of oxygen and ozone until the GOE would have led to a “false negative” (Reinhard et al., 2017). Based on our discussion in Sec. 3, the evolution of oxygenic photosynthesis and the GOE correspond to (or ).

We have plotted (13) as a function of the stellar mass in Fig. 4. It is seen that the peak is at , and that the curves rise sharply at . The figure indicates that an Earth-analog around a G-type star would have the highest probability of achieving the critical steps necessary for detectable primitive or intelligent life.

We must however point out an important caveat here. For and , we obtain a probability of when we make use of (13). In other words, this should imply that the likelihood of attaining the fifth critical step (intelligence), even with the constraints (I) and (II), is about . This value appears to be very high, but it must recognized that we have merely calculated the mathematical probability. In reality, there will be a vast number of other criteria - for example, the availability of liquid water, the concentration of nutrients such as phosphorus, the maintenance of a stable climate over Gyr timescales (Lammer et al., 2009; Cockell et al., 2016) - that must be simultaneously fulfilled in order for each critical step to occur.121212Although the number of necessary and sufficient conditions that must be fulfilled for the major transitions in Earth’s evolutionary history to occur is probably very high, we cannot say for certain whether all of these steps obey the Anna Karenina principle, i.e. the premise that the absence (or breakdown) of even a single factor is capable of dooming a particular process to failure (Diamond, 1997). Hence, it is more instructive to view (13) as an upper bound, and use it to assess the relative chances of life-bearing planets existing around stars of differing masses.

Figure 5: The relative number of life-bearing stars which have completed critical steps as a function of the stellar mass .

Hitherto, we have only discussed the prospects for life on an Earth-analog around a given star. However, it should be recognized that the total number of stars also varies depending of their mass, i.e. low-mass stars are more numerous than high-mass ones. Thus, we can calculate the relative number of stars with detectable primitive or intelligent life, with the total number of stars defined as follows:


where represents the number of stars per logarithmic mass interval, and is calculated from the stellar initial mass function (IMF); here, we will use the IMF proposed by Kroupa (2001). Note that is given by (13) and can be viewed as a measure of the probability of planets with life (and have passed through the critical steps) per star.131313We have not included an additional factor for the number of planets in the HZ of the host star because this quantity appears to be mostly independent of the stellar mass (Kaltenegger, 2017).

We have plotted as a function of in Fig. 5. Let us begin by observing that when by definition. For , we find that the peak occurs at , implying that solar-mass stars in our Galaxy are the most numerous in terms of planets with intelligent life. On the other hand, for , the peak is seen at (and at when ) indicating that K-type stars are potentially the most numerous in terms of having primitive, but detectable, life. For the values of considered herein, we find that is almost constant in the range suggesting that these stars are the best targets in the search for life.

5 Conclusion

We began by outlining a simple mathematical model wherein evolution is effectively modeled as a series of independent “hard” steps. One of the primary objectives was to study the ramifications of this model for the timing and likelihood of primitive and intelligent life on Earth and Earth-like exoplanets around other stars.

We began our analysis by focusing on the Earth and studying the total number of critical steps () that are likely on Earth based on the latest developments in geobiology. We found that the result depended on the time at which the Earth becomes uninhabitable in the future. For the more conservative estimate of Gyr, we found that probably represents the best fit, in agreement with previous studies. Unlike the standard -step model based on the classical evolutionary transitions (Smith and Szathmáry, 1995), we proposed that the following five steps could have represented the major breakthroughs in the history of life on Earth: (i) abiogenesis, (ii) oxygenic photosynthesis, (iii) eukaryogenesis (endosymbiosis involving mitochondria and plastids), (iv) complex multicellularity, and (v) genus Homo (and H. sapiens in particular). On the other hand, if habitability ends Gyr in the future, we suggested that a -step model could represent the best fit, and that humans constitute the fifth critical step (with one further transition in the future).

Subsequently, we applied this model to study the prospects for life on Earth-analogs orbiting stars of different masses. Our analysis took into account constraints based on: (i) the duration of the continuously habitable zone, (ii) atmospheric escape due to stellar wind erosion, and (iii) availability of bioactive UV flux to promote abiogenesis. We found that the timescale for abiogenesis is longer the duration of habitability for , strongly suggesting that such stars are not likely to host life-bearing planets. The prospects for primitive or intelligent life are highest for a generic Earth-analog around a solar-mass star based on this analysis.

Next, we computed the total number of stars (relative to the solar value) that could give rise to detectable signatures of primitive and intelligent life. With regards to the former, we found that the number peaks in the range -, implying that certain K- and G-type stars should potentially be accorded the highest priority in the hunt for biosignatures. Our analysis is in agreement with previous studies (Kasting et al., 1993; Heller and Armstrong, 2014; Tian and Ida, 2015; Cuntz and Guinan, 2016; Lingam and Loeb, 2017c). On the other hand, the total number of stars with intelligent life exhibited a peak near , thereby implying that Sun-like stars represent the best targets for SETI. This could also serve to explain why technological intelligence like our own finds itself in the vicinity of a solar-mass star, despite the fact that low-mass stars are more numerous and long-lived (Loeb et al., 2016; Haqq-Misra et al., 2018).

Naturally, there are a number of caveats that must be borne in mind with regards to the above conclusions. It is by no means clear as to whether evolution truly proceeds through a series of “hard” steps, and that the number and nature of these steps will be similar on other exoplanets. Our analysis has dealt solely with the stellar mass, although other stellar parameters (e.g. activity, rotation, metallicity) play an important role. Moreover, by focusing exclusively on Earth-analogs, we have not taken the plethora of biogeochemical factors that have shaped Earth’s evolutionary history. Our discussion also ignored the possible transfer of life between stars. Such transfer may involve lithopanspermia (Burchell, 2004) or interstellar travel by technological species (Crick and Orgel, 1973). If such transfer events are common enough, which might be the case in some instances (Lingam and Loeb, 2017a), they could blur the quantitative conclusions that were reached in our paper.

In spite of these limitations, it seems plausible that the critical step framework can be used to assess the relative merits of different models of the major evolutionary transitions on Earth. Furthermore, it also provides a useful formalism for gauging the relative likelihood of life on Earth-like planets orbiting different stars given the sparse data available at the current stage. Lastly, it offers testable predictions in the future, and, in principle, can therefore be falsified.


ML is grateful to Andrew Knoll for the insightful and illuminating conversations. This work was supported in part by the Breakthrough Prize Foundation for the Starshot Initiative, Harvard University’s Faculty of Arts and Sciences, and the Institute for Theory and Computation (ITC) at Harvard University.


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