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Draft version August 10, 2015
Preprint typeset using LATEX style emulateapj v. 5/2/11
STATISTICAL SIGNATURES OF PANSPERMIA IN EXOPLANET SURVEYS
Henry W. Lin1,2
, Abraham Loeb2
Draft version August 10, 2015
ABSTRACT
A fundamental astrobiological question is whether life can be transported between extrasolar sys-
tems. We propose a new strategy to answer this question based on the principle that life which arose
via spreading will exhibit more clustering than life which arose spontaneously. We develop simple
statistical models of panspermia to illustrate observable consequences of these excess correlations.
Future searches for biosignatures in the atmospheres of exoplanets could test these predictions: a
smoking gun signature of panspermia would be the detection of large regions in the Milky Way where
life saturates its environment interspersed with voids where life is very uncommon. In a favorable
scenario, detection of as few as ∼ 25 biologically active exoplanets could yield a 5σ detection of
panspermia. Detectability of position-space correlations is possible unless the timescale for life to
become observable once seeded is longer than the timescale for stars to redistribute in the Milky Way.
Subject headings: planets: extrasolar — astrobiology
1. INTRODUCTION
The question of where life originated is centuries old
(for a review, see Miller & Orgel 1974; Wesson 2010),
but to date the only experimentally viable method of
detecting panspermia is the detection of biomaterial on
an asteroid or comet. Unless a significant fraction of in-
terplanetary objects are biologically active, this method
will not yield positive results or falsify the hypotheses of
panspermia because the enormous number of objects in
our solar system (Moro-Mart´ın et al. 2009) may permit
a significant number of panspermia events, even if the
fraction of objects which contain life is miniscule. Al-
though previous estimates suggested that lithopansper-
mia events should be quite rare (Melosh 2003; Adams
& Spergel 2005), more recent proposals Belbruno et al.
(2012) yield considerably more optimistic rates. Given
the experimental difficulties of testing the hypotheses of
panspermia and poor constraints on the theoretical di-
versity of life, one may even question whether pansper-
mia is truly falsifiable. In this Letter, we answer this
question in the affirmative. Under certain conditions,
panspermia leads to statistical correlations in the distri-
bution of life in the Milky Way. If future surveys detect
biosignatures in the atmospheres of exoplanets, it will be
possible to devise statistical tests to detect or constrain
panspermia event rates while remaining agnostic to the
biological mechanisms of panspermia.
This Letter is organized as follows. In §2 we describe a
simple class of panspermia models that qualitatively cap-
tures the statistical features of any panspermia theory.
We discuss observable signatures of panspermia in §3.
We conclude with further implications of our pansper-
mia models in §4.
2. A MODEL FOR PANSPERMIA
Consider an arbitrary lattice L in two or three dimen-
sions. (The two dimensional model corresponds to a thin-
1 Harvard College, Cambridge, MA 02138, USA
2 Harvard-Smithsonian Center for Astrophysics, 60 Garden
St., Cambridge, MA 02138, USA
Email: henry.lin@cfa.harvard.edu, aloeb@cfa.harvard.edu
disk approximation of the Milky Way). While a lattice
model is a crude approximation to reality, lattices are an-
alytically tractable, and the conclusions we will draw will
hold in the continuum limit.4
Each lattice point repre-
sents a habitable extrasolar system. Viewed as a graph,
the number of edges N associated with each lattice point
represents the average number of panspermia events per
extrasolar system. Associated with each point x ∈ L is
a state variable h(x) which is either 0 or 1, representing
the biologically uninhabited and inhabited states, respec-
tively.
The initial state of the lattice is h(x) = 0 for all x ∈ L.
The system is updated as follows. At each discrete time
step, neighbors of each inhabited site become inhabited.
Furthermore, some fraction 0 < ps < 1 of the uninhab-
ited sites are switched to h = 1. This describes pansper-
mia in the regime where life spontaneously arises at a
very gradual constant rate. It is also possible to study
the opposite regime, where life spontaneously arises sud-
denly. We will refer to the two regimes as the “adiabatic”
and “sudden” scenarios.
We consider the adiabatic case first. Consider the
regime where ps 1 and N ≥ 1. Pictorially, bubbles
form in the lattice as shown in Figure 1. At each time
step, the bubbles grow linearly in size, as new bubbles
are formed. After a while, there are bubbles of many
sizes. Well before the overlap time, namely for t to,
the probability that a lattice site is contained in a bubble
of radius R is given by
p(R) ≈ p0(1 − V (R + 1)p(R + 1)), (1)
where Rmax = t and p(Rmax) = p0 and V (R) in two
(three) dimensions is the area (volume) of a bubble of
radius R. Since bubbles of size R + 1 already occupy
4 A slightly more realistic model would distribute the points
randomly and consider circles or spheres of influence surrounding
these points where panspermia events could take place. This setup
is known as a continuum percolation problem in the mathematics
literature (Meester & Roy 1996). In the regime where most of the
spheres of influence overlap, the lattice approximation described
above gives similar results.
arXiv:1507.05614v2[astro-ph.EP]6Aug2015
2
Time!
Space
inhabited
uninhabited
Time!
Space
uninhabited
inhabited
Fig. 1.— Schematic diagrams of the topology of the bio-inhabited planets within the galaxy for the panspermia case (left) and no
panspermia case (right). In the panspermia case, once life appears it begins to percolate, forming a cluster that grows with time. Life can
ocassionally spontaneously arise after the first bio-event, forming clusters that are smaller than more mature clusters. (The limiting case
where life spontaneously arises once and then spreads to the rest of the galaxy would correspond to a single blue triangle. In the ”sudden”
scenario, all triangles start at the same cosmic time and are thus the same size.) As time progresses, the clusters eventually overlap and the
galaxy’s end state is dominated by life. In the no panspermia scenario, life cannot spread: there is no phase transition, but a very gradual
saturation of all habitable planets with life. Observations of nearby habitable exoplanets could statistically determine whether panspemia
is highly efficient (left), inefficient (right), or in some intermediate regime.
some of the lattice sites, new, smaller bubbles will have
less room to form, which is reflected in the second term
in equation (1). However, to linear order in p0, the distri-
bution is uniform on the interval [0, Rmax]. In the regime
where bubbles of all sizes are present with approximately
equal number densities, the system resembles a fluid at a
scale-invariant critical point (Stanley 1987), where bub-
ble nucleation converts one phase to the other. This sce-
nario is analogous to the formation of HII bubbles during
the epoch of reionization (Loeb & Furlanetto 2013) or the
production of bubbles during cosmic inflation (Turner
et al. 1992).
In the sudden case, the initial conditions are such that
each site is inhabited with probability ps. The dynam-
ical rule for updating the system is simply that neigh-
bors adjacent to an inhabited site become inhabited. In
this case, all bubbles are formed with the same size, and
grow linearly as a function of time. The dynamics of the
system can also be easily described via renormalization
group methods. Consider the operation of partitioning
the lattice into blocks containing a fixed number of
lattice sites. The renormalized probability p , e.g. the
probability that at least one of sites will be habitable
is p = 1 − (1 − ps) . Note that p > ps for 0 < ps < 1
and p is an increasing function of . The coarse grained
evolution of the system corresponds to incrementing
with each time step and setting all sites in a block of
size to h = 1 if a single site is inhabited. The dynam-
ical picture is one where clusters of life form and grow,
overlap, and eventually merge into a percolating cluster
when the renormalized probability p equals the percola-
tion threshold for the given lattice.
A qualitative difference between a simple lattice model
and the Milky Way is that stars in the Milky Way drift
relative to each other with a characteristic speed σv of
a few tens of km/s (Binney & Tremaine 2011). This
presents three interesting regimes which are character-
ized by the effective spreading speed of life v. The effects
of drifting should be negligible in the limit that pansper-
mia takes place at speeds v σv. This could be the
case if there exists an intelligent species which can spread
at high speeds. But even if panspermia takes place at
speeds comparable to the relative speeds of stars, the re-
sults from our lattice models still hold. To see this, con-
sider modifying the dynamical rule in the following way:
at each given time increment, a cell is randomly swapped
with one of its nearest neighbors. This simulates the
random motion of stars due to their velocity dispersion.
Consider a bubble which has already formed. Outside of
the bubble, swapping uninhabited lattice sites has no ef-
fect on the correlation function. Inside the bubble, swap-
ping uninhabited lattice sites also does not produce any
observable effects. Only the boundary of the bubble is
affected by swapping. If the dimensionality of the lattice
is large, most of the lattice sites on the boundary will
be swapped amongst themselves. In a two or three di-
mensional lattice, the boundary effectively grows by < 1
unit. Macroscopically, the effective speed of panspermia
increases by a factor of order unity. This regime may be
of particular interest to lithopanspermia, since ejected
rocks have velocities v ∼ σv. Finally, the third regime is
when panspermia takes place at a rate v σv. Although
a proper treatment of this regime requires numerically
integrating orbits of stars in the Milky Way, a tractable
approximation is given by a linearized reaction-diffusion
equation, which describes randomly-walking stars that
can spread life locally:
∂h
∂t
= D 2
h + Γh + J, (2)
where D is the diffusion constant that controls the rela-
tive drifting of stars, Γ is the infection rate, and a source
term J accounts for the spontaneous development of life.
In the adiabatic regime, J is the sum of delta functions
uniformly distributed on some region of space-time. In
the sudden regime, J consists of a single delta function.
To determine the evolution of this system, it suffices to
compute the Green’s function G(x, x0) for this equation:
G(x, x0) = eΓt
× ∆
x − x0
√
2Dt
, (3)
where ∆ is a Gaussian with vanishing mean and a stan-
dard deviation of unity. Note that the ∆ term is simply
the propagator for the diffusion equation. In this regime,
bubble formation is modified by the fact that bubbles
3
������������(�)
Adiabatic
Sudden
No Panspermia
0 2 4 6 8 10
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
������ �
Fig. 2.— Schematic plot of the spherically averaged correla-
tion function C(r) as a function of radius r during the panspermia
phase transition for two different panspermia scenarios and the no
panspermia scenario. In the adiabatic regime, life can percolate
in addition to spontaneously arising. In the sudden regime, life
arises once and then begins to percolate. Both correlation func-
tions define a characteristic scale radius ξ which reflects the time
elapsed t since the first life started percolating. In the case where
no panspermia occurs, the reduced correlation function is exactly
zero, so a measurement of C(r) > 0 (see equation (5) for a defini-
tion) would be compelling evidence for panspermia.
grow in size R ∝ (Dt)1/2
instead of R ∝ t.
It should be noted that the results stated above will
hold until bubbles are too large to neglect the effects of
velocity shear in the Milky Way (Binney & Tremaine
2011). Once the bubbles grow to a significant fraction
the radius of the Milky Way, they will be sheared apart
on ∼ 100 Myr timescales. Hence, shearing effectively
disperses bubbles greater than some critical size. If the
density of the dispersed region is low, the separation be-
tween inhabited sites will be large, so the shearing will
effectively convert convert the larger bubble into smaller
bubbles. So long as the phase transition is not complete,
small bubbles will start to regrow, and the cycle starts
over again.
Finally, it is important to note that life may take a non-
negligible amount of time td to become detectable once
life is spread to it. We will not attempt to quantify td
except to note that td could in principle be very short if
a photosynthetic (or more exotically, an industrially pol-
luting) species can propagate between solar systems such
as cyanobacteria (Olsson-Francis et al. 2010). Of course,
for Earth’s history td (for currently proposed biomark-
ers) is very long td ∼ 109
yr. If td is much larger than the
timescale for stars to diffuse/shear (on the order of ∼ 108
yr), it will not be possible to detect any position-space
correlations, though more subtle phase-space correlations
(see §3) could in principle be detected.
3. OBSERVABLE SIGNATURES
An important observable consequence of panspermia
that is illustrated in both of these models. The two-point
correlation function
C(x − y) ≡ h(x)h(y) − h(x)
2
(4)
has the property C = 0 during the entire evolution of
the system. This is true unless we are unlucky enough
that the phase transition has already been completed and
h(x) = 1 everywhere. In particular, the timescale for the
phase transition to run to completion is most likely sev-
eral times the life crossing time of the Milky Way. For
v ∼ 10 km/s, the life crossing time corresponds to sev-
eral Gyr, so the phase transition could take of order the
Hubble time. For the sudden case, the correlation length
ξ ∼ min(max(vt,
√
2Dt), RMW) where RMW is a length
scale several times smaller than the radius of the Milky
Way. For the adiabatic case, the correlation function is
peaked at x − y = 0 and drops to zero over the same
characteristic length ξ. In general, ξ will always show a
cutoff at some scale radius ∼ RMW due to shearing ef-
fects. The schematic form of the correlation function is
displayed in Figure 2. A more complicated rule where the
rate ps varies with time will encode itself in the correla-
tion function; the important point is that any spreading
whatsoever will yield potentially observable deviations
from C = 0 which is the Poisson case.
For real observations, one must take into account the
fact that stars are not distributed uniformly on a lat-
tice and may themselves exhibit clustering. To take this
into account, we propose the following reduced correlation
function as a potentially robust indicator of panspermia:
C(x − y) ≡
h(x)h(y)
σ(x)σ(y)
−
h(x)
σ(x)
2
, (5)
where in the continuum limit σ(x) is the stellar density
and h(x) is the density of inhabited stellar systems. If
life arises independently among different stellar systems,
h ∝ σ, so C(x − y) accounts for the fact that stars do
not form a perfect lattice by “dividing out” the star-star
correlation. An even more sophisticated treatment could
replace the spatial densities with corresponding phase
space densities x → (x, p), since two stars which are
closer together in phase space will have more time to
transfer biomaterial than two stars which are close in
position space but far away in momentum space. In prin-
ciple, one could reverse-integrate the orbits of stars and
calculate the radius of closest approach rc for any two
given stars. Measuring the correlation as a function of
rc would be an alternative strategy for disentangling the
effects of stellar diffusion and panspermia, which may
be useful if life propagates at very low speeds, or if the
timescale td is longer than the stellar mixing timescale.
We note that the above discrete models can be gener-
alized to the continuous case by using a slightly differ-
ent formalism. If the number distribution of bubbles is
known, a power spectrum of inhabited star density fluc-
tuations can be derived that will reproduce the bubble
spectrum by retracing the steps of the Press-Schechter
formalism (Press & Schechter 1974). Once the power
spectrum is obtained, one can obtain the correlation
function via a Fourier transform. If the habitable stellar
density is not constant but fluctuates in space, the prob-
lem becomes analogous to the spread of disease on an
inhomogeneous medium (Lin & Loeb 2015), which again
makes use of the Press-Schechter formalism.
Future surveys such as the TESS (Ricker et al. 2015)
will detect hundreds of earth-like exoplanets (Sullivan
et al. 2015). Ground-based and space-based (e.g. JWST)
follow ups that can characterize the exoplanet atmo-
spheres could test for biosignatures such as oxygen in
combination with a reducing gas (for a review, see
Kaltenegger et al. 2002). However, it is likely that only
a few earth-like exoplanets will be close enough to be
biologically characterized (Brandt & Spiegel 2014; Rein
4
et al. 2014) with next-decade instruments. Eventually,
surveys could test for more specific spectral signatures
such as the “red edge” of chlorophyl (Seager et al. 2005)
or even industrial pollution (Lin et al. 2014). It is also
possible that searches for extraterrestrial intelligence in
the radio or optical wavelengths could also yield detec-
tions that could be tested for clustering. Any positive
detections will yield first constraints on the correlation
function C(x−y). In a favorable scenario, our solar sys-
tem could be on the edge of a bubble, in which case a
survey of nearby stars would reveal that ∼ 1/2 of the sky
is inhabited while the other half is uninhabited. In this
favorable scenario, ∼ 25 targets confirmed to have biosig-
natures (supplemented with 25 null detections) would
correspond to a 5σ deviation from the Poisson case, and
would constitute a smoking gun detection of panspermia.
A more generic placement would increase the number of
required detections by a factor of a few, though an un-
usual bubble configuration could potentially reduce the
number of required detections. It should be noted that
the local environment of our solar system does not re-
flect the local environment ∼ 4 Gyr ago when life arose
on earth, so the discovery of a bubble of surrounding
earth should be interpreted as the solar system “drift-
ing” into a bubble which has already formed, or perhaps
the earth seeding its environment with life.
Finally, we note that since the efficiency of panspermia
is dictated by the number of transfer events between stel-
lar systems, which in turn will depend on the local stellar
density. For example, the transfer rates of rocky material
between stars grows with stellar density simply because
the mean distance between stars is smaller (Adams &
Spergel 2005; Belbruno et al. 2012), and the enhanced
frequency of close stellar encounters should further in-
crease these rates. One might therefore expect that an
inhomogeneous distribution of stars (i.e. the Milky Way)
might exhibit a “multi-phase” structure, where regions
of high stellar density have completed the phase transi-
tion and are saturated with life, whereas regions of low
stellar density will have little to no signs of life. How-
ever, regions of high stellar density may be inhospitable
to life due to a variety of factors (Gonzalez et al. 2001;
Lineweaver et al. 2004; Gowanlock et al. 2011). For ex-
ample, perturbations of Oort cloud analogs due to close
stellar encounters, could potentially make a habitable
zone planet inhospitable to life (Gonzalez et al. 2001).
The number density per unit time of supernovae explo-
sions (Svensmark 2012) and the local stellar metallicity
may also play a factor in determining the habitability
of planets (Gonzalez et al. 2001). It is therefore likely
that the number of panspermia events is a complicated,
non-monotonic function of stellar density and other as-
trophysical parameters.
4. IMPLICATIONS
In our simple formalism, life spreads from host to host
in a way that resembles the outbreak of an epidemic. A
key point is that the correlations that quickly arise imply
that the transition from an uninhabited galaxy to an in-
habited galaxy can occur much faster in the panspermia
regime than in the Poisson case. Panspermia implies a
phase transition, whereas a Poisson process will only lead
to a gradual build up of life. Said differently, the start
time for life for different stellar systems exhibits a very
small scatter in the panspermia case. A consequence of
the panspermia scenario is that the severity of the Fermi
paradox may be reduced somewhat. If life started ev-
erywhere at the same time, we expect fewer advanced
civilizations at the present time than if life could have
started much earlier on other stellar systems. It should
be noted, however, that this statement is predicated on
the somewhat controversial assumption that there is an
evolutionary bias towards increasing complexity (Adami
et al. 2000). A second consequence of panspermia is that
the Drake equation (Shklovsky & Sagan 1966) becomes
a lower bound on the number of civilizations, since the
multiplicative form of the equation is based on the as-
sumption that life arises independently everywhere. This
assumption may be strongly violated in the regime where
panspermia is highly efficient.
The mathematical similarity between panspermia and
disease spread may also represent a biological one: any
species which acquires panspermia abilities will have
enormous fitness advantages. Just as viruses evolved to
brave the “harsh” environment of “inter-host” space to
harness the energy of multiple biological hosts, perhaps
evolution has or will drive a class of organisms to brave
the harsh environment of interstellar space to harness
the energy of multiple stellar hosts. Whether or not
the organisms will be primitive (e.g., lithopanspermia,
cometary panspermia (Hoyle & Wickramasinghe 1981))
or intelligent (directed panspermia (Crick & Orgel 1973)
or accidental panspermia) remains to be determined.
Even if the earth is the only inhabited planet and prim-
itive life cannot survive an interstellar journey, interstel-
lar travel led by humans may one day lead to colonization
of the galaxy. As a zeroth order model, the same for-
malism should approximately describe the growth of the
colonies as they percolate through the galaxy, assuming
that the processes such as population diffusion also occur
in space (Newman & Sagan 1981). Although the ques-
tion is a purely astrobiological question today, in the dis-
tant future urban sociologists and astrophysicists might
be forced to work together (Lin & Loeb 2015). Indeed,
well after the colonization regime, models of panspermia
may continue to be relevant, as spaceships capable of
interstellar travel will provide the opportunity for primi-
tive life (e.g. domesticated life, diseases, and viruses) to
spread efficiently. The question that awaits is whether
primitive life has already spread efficiently, or whether it
will have to wait for “intelligent” life to make the voyage.
ACKNOWLEDGMENTS
We thank Ed Turner, Edwin Kite, and an anonymous
referee for useful discussions. This work was supported
in part by NSF grant AST-1312034, and the Harvard
Origins of Life Initiative.
REFERENCES
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National Academy of Science, 97, 4463
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Belbruno, E., Moro-Mart´ın, A., Malhotra, R., & Savransky, D.
2012, Astrobiology, 12, 754
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university press)
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Statistical signatures of_panspermia_in_exoplanet_surveys

  • 1. Draft version August 10, 2015 Preprint typeset using LATEX style emulateapj v. 5/2/11 STATISTICAL SIGNATURES OF PANSPERMIA IN EXOPLANET SURVEYS Henry W. Lin1,2 , Abraham Loeb2 Draft version August 10, 2015 ABSTRACT A fundamental astrobiological question is whether life can be transported between extrasolar sys- tems. We propose a new strategy to answer this question based on the principle that life which arose via spreading will exhibit more clustering than life which arose spontaneously. We develop simple statistical models of panspermia to illustrate observable consequences of these excess correlations. Future searches for biosignatures in the atmospheres of exoplanets could test these predictions: a smoking gun signature of panspermia would be the detection of large regions in the Milky Way where life saturates its environment interspersed with voids where life is very uncommon. In a favorable scenario, detection of as few as ∼ 25 biologically active exoplanets could yield a 5σ detection of panspermia. Detectability of position-space correlations is possible unless the timescale for life to become observable once seeded is longer than the timescale for stars to redistribute in the Milky Way. Subject headings: planets: extrasolar — astrobiology 1. INTRODUCTION The question of where life originated is centuries old (for a review, see Miller & Orgel 1974; Wesson 2010), but to date the only experimentally viable method of detecting panspermia is the detection of biomaterial on an asteroid or comet. Unless a significant fraction of in- terplanetary objects are biologically active, this method will not yield positive results or falsify the hypotheses of panspermia because the enormous number of objects in our solar system (Moro-Mart´ın et al. 2009) may permit a significant number of panspermia events, even if the fraction of objects which contain life is miniscule. Al- though previous estimates suggested that lithopansper- mia events should be quite rare (Melosh 2003; Adams & Spergel 2005), more recent proposals Belbruno et al. (2012) yield considerably more optimistic rates. Given the experimental difficulties of testing the hypotheses of panspermia and poor constraints on the theoretical di- versity of life, one may even question whether pansper- mia is truly falsifiable. In this Letter, we answer this question in the affirmative. Under certain conditions, panspermia leads to statistical correlations in the distri- bution of life in the Milky Way. If future surveys detect biosignatures in the atmospheres of exoplanets, it will be possible to devise statistical tests to detect or constrain panspermia event rates while remaining agnostic to the biological mechanisms of panspermia. This Letter is organized as follows. In §2 we describe a simple class of panspermia models that qualitatively cap- tures the statistical features of any panspermia theory. We discuss observable signatures of panspermia in §3. We conclude with further implications of our pansper- mia models in §4. 2. A MODEL FOR PANSPERMIA Consider an arbitrary lattice L in two or three dimen- sions. (The two dimensional model corresponds to a thin- 1 Harvard College, Cambridge, MA 02138, USA 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA Email: henry.lin@cfa.harvard.edu, aloeb@cfa.harvard.edu disk approximation of the Milky Way). While a lattice model is a crude approximation to reality, lattices are an- alytically tractable, and the conclusions we will draw will hold in the continuum limit.4 Each lattice point repre- sents a habitable extrasolar system. Viewed as a graph, the number of edges N associated with each lattice point represents the average number of panspermia events per extrasolar system. Associated with each point x ∈ L is a state variable h(x) which is either 0 or 1, representing the biologically uninhabited and inhabited states, respec- tively. The initial state of the lattice is h(x) = 0 for all x ∈ L. The system is updated as follows. At each discrete time step, neighbors of each inhabited site become inhabited. Furthermore, some fraction 0 < ps < 1 of the uninhab- ited sites are switched to h = 1. This describes pansper- mia in the regime where life spontaneously arises at a very gradual constant rate. It is also possible to study the opposite regime, where life spontaneously arises sud- denly. We will refer to the two regimes as the “adiabatic” and “sudden” scenarios. We consider the adiabatic case first. Consider the regime where ps 1 and N ≥ 1. Pictorially, bubbles form in the lattice as shown in Figure 1. At each time step, the bubbles grow linearly in size, as new bubbles are formed. After a while, there are bubbles of many sizes. Well before the overlap time, namely for t to, the probability that a lattice site is contained in a bubble of radius R is given by p(R) ≈ p0(1 − V (R + 1)p(R + 1)), (1) where Rmax = t and p(Rmax) = p0 and V (R) in two (three) dimensions is the area (volume) of a bubble of radius R. Since bubbles of size R + 1 already occupy 4 A slightly more realistic model would distribute the points randomly and consider circles or spheres of influence surrounding these points where panspermia events could take place. This setup is known as a continuum percolation problem in the mathematics literature (Meester & Roy 1996). In the regime where most of the spheres of influence overlap, the lattice approximation described above gives similar results. arXiv:1507.05614v2[astro-ph.EP]6Aug2015
  • 2. 2 Time! Space inhabited uninhabited Time! Space uninhabited inhabited Fig. 1.— Schematic diagrams of the topology of the bio-inhabited planets within the galaxy for the panspermia case (left) and no panspermia case (right). In the panspermia case, once life appears it begins to percolate, forming a cluster that grows with time. Life can ocassionally spontaneously arise after the first bio-event, forming clusters that are smaller than more mature clusters. (The limiting case where life spontaneously arises once and then spreads to the rest of the galaxy would correspond to a single blue triangle. In the ”sudden” scenario, all triangles start at the same cosmic time and are thus the same size.) As time progresses, the clusters eventually overlap and the galaxy’s end state is dominated by life. In the no panspermia scenario, life cannot spread: there is no phase transition, but a very gradual saturation of all habitable planets with life. Observations of nearby habitable exoplanets could statistically determine whether panspemia is highly efficient (left), inefficient (right), or in some intermediate regime. some of the lattice sites, new, smaller bubbles will have less room to form, which is reflected in the second term in equation (1). However, to linear order in p0, the distri- bution is uniform on the interval [0, Rmax]. In the regime where bubbles of all sizes are present with approximately equal number densities, the system resembles a fluid at a scale-invariant critical point (Stanley 1987), where bub- ble nucleation converts one phase to the other. This sce- nario is analogous to the formation of HII bubbles during the epoch of reionization (Loeb & Furlanetto 2013) or the production of bubbles during cosmic inflation (Turner et al. 1992). In the sudden case, the initial conditions are such that each site is inhabited with probability ps. The dynam- ical rule for updating the system is simply that neigh- bors adjacent to an inhabited site become inhabited. In this case, all bubbles are formed with the same size, and grow linearly as a function of time. The dynamics of the system can also be easily described via renormalization group methods. Consider the operation of partitioning the lattice into blocks containing a fixed number of lattice sites. The renormalized probability p , e.g. the probability that at least one of sites will be habitable is p = 1 − (1 − ps) . Note that p > ps for 0 < ps < 1 and p is an increasing function of . The coarse grained evolution of the system corresponds to incrementing with each time step and setting all sites in a block of size to h = 1 if a single site is inhabited. The dynam- ical picture is one where clusters of life form and grow, overlap, and eventually merge into a percolating cluster when the renormalized probability p equals the percola- tion threshold for the given lattice. A qualitative difference between a simple lattice model and the Milky Way is that stars in the Milky Way drift relative to each other with a characteristic speed σv of a few tens of km/s (Binney & Tremaine 2011). This presents three interesting regimes which are character- ized by the effective spreading speed of life v. The effects of drifting should be negligible in the limit that pansper- mia takes place at speeds v σv. This could be the case if there exists an intelligent species which can spread at high speeds. But even if panspermia takes place at speeds comparable to the relative speeds of stars, the re- sults from our lattice models still hold. To see this, con- sider modifying the dynamical rule in the following way: at each given time increment, a cell is randomly swapped with one of its nearest neighbors. This simulates the random motion of stars due to their velocity dispersion. Consider a bubble which has already formed. Outside of the bubble, swapping uninhabited lattice sites has no ef- fect on the correlation function. Inside the bubble, swap- ping uninhabited lattice sites also does not produce any observable effects. Only the boundary of the bubble is affected by swapping. If the dimensionality of the lattice is large, most of the lattice sites on the boundary will be swapped amongst themselves. In a two or three di- mensional lattice, the boundary effectively grows by < 1 unit. Macroscopically, the effective speed of panspermia increases by a factor of order unity. This regime may be of particular interest to lithopanspermia, since ejected rocks have velocities v ∼ σv. Finally, the third regime is when panspermia takes place at a rate v σv. Although a proper treatment of this regime requires numerically integrating orbits of stars in the Milky Way, a tractable approximation is given by a linearized reaction-diffusion equation, which describes randomly-walking stars that can spread life locally: ∂h ∂t = D 2 h + Γh + J, (2) where D is the diffusion constant that controls the rela- tive drifting of stars, Γ is the infection rate, and a source term J accounts for the spontaneous development of life. In the adiabatic regime, J is the sum of delta functions uniformly distributed on some region of space-time. In the sudden regime, J consists of a single delta function. To determine the evolution of this system, it suffices to compute the Green’s function G(x, x0) for this equation: G(x, x0) = eΓt × ∆ x − x0 √ 2Dt , (3) where ∆ is a Gaussian with vanishing mean and a stan- dard deviation of unity. Note that the ∆ term is simply the propagator for the diffusion equation. In this regime, bubble formation is modified by the fact that bubbles
  • 3. 3 ������������(�) Adiabatic Sudden No Panspermia 0 2 4 6 8 10 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ������ � Fig. 2.— Schematic plot of the spherically averaged correla- tion function C(r) as a function of radius r during the panspermia phase transition for two different panspermia scenarios and the no panspermia scenario. In the adiabatic regime, life can percolate in addition to spontaneously arising. In the sudden regime, life arises once and then begins to percolate. Both correlation func- tions define a characteristic scale radius ξ which reflects the time elapsed t since the first life started percolating. In the case where no panspermia occurs, the reduced correlation function is exactly zero, so a measurement of C(r) > 0 (see equation (5) for a defini- tion) would be compelling evidence for panspermia. grow in size R ∝ (Dt)1/2 instead of R ∝ t. It should be noted that the results stated above will hold until bubbles are too large to neglect the effects of velocity shear in the Milky Way (Binney & Tremaine 2011). Once the bubbles grow to a significant fraction the radius of the Milky Way, they will be sheared apart on ∼ 100 Myr timescales. Hence, shearing effectively disperses bubbles greater than some critical size. If the density of the dispersed region is low, the separation be- tween inhabited sites will be large, so the shearing will effectively convert convert the larger bubble into smaller bubbles. So long as the phase transition is not complete, small bubbles will start to regrow, and the cycle starts over again. Finally, it is important to note that life may take a non- negligible amount of time td to become detectable once life is spread to it. We will not attempt to quantify td except to note that td could in principle be very short if a photosynthetic (or more exotically, an industrially pol- luting) species can propagate between solar systems such as cyanobacteria (Olsson-Francis et al. 2010). Of course, for Earth’s history td (for currently proposed biomark- ers) is very long td ∼ 109 yr. If td is much larger than the timescale for stars to diffuse/shear (on the order of ∼ 108 yr), it will not be possible to detect any position-space correlations, though more subtle phase-space correlations (see §3) could in principle be detected. 3. OBSERVABLE SIGNATURES An important observable consequence of panspermia that is illustrated in both of these models. The two-point correlation function C(x − y) ≡ h(x)h(y) − h(x) 2 (4) has the property C = 0 during the entire evolution of the system. This is true unless we are unlucky enough that the phase transition has already been completed and h(x) = 1 everywhere. In particular, the timescale for the phase transition to run to completion is most likely sev- eral times the life crossing time of the Milky Way. For v ∼ 10 km/s, the life crossing time corresponds to sev- eral Gyr, so the phase transition could take of order the Hubble time. For the sudden case, the correlation length ξ ∼ min(max(vt, √ 2Dt), RMW) where RMW is a length scale several times smaller than the radius of the Milky Way. For the adiabatic case, the correlation function is peaked at x − y = 0 and drops to zero over the same characteristic length ξ. In general, ξ will always show a cutoff at some scale radius ∼ RMW due to shearing ef- fects. The schematic form of the correlation function is displayed in Figure 2. A more complicated rule where the rate ps varies with time will encode itself in the correla- tion function; the important point is that any spreading whatsoever will yield potentially observable deviations from C = 0 which is the Poisson case. For real observations, one must take into account the fact that stars are not distributed uniformly on a lat- tice and may themselves exhibit clustering. To take this into account, we propose the following reduced correlation function as a potentially robust indicator of panspermia: C(x − y) ≡ h(x)h(y) σ(x)σ(y) − h(x) σ(x) 2 , (5) where in the continuum limit σ(x) is the stellar density and h(x) is the density of inhabited stellar systems. If life arises independently among different stellar systems, h ∝ σ, so C(x − y) accounts for the fact that stars do not form a perfect lattice by “dividing out” the star-star correlation. An even more sophisticated treatment could replace the spatial densities with corresponding phase space densities x → (x, p), since two stars which are closer together in phase space will have more time to transfer biomaterial than two stars which are close in position space but far away in momentum space. In prin- ciple, one could reverse-integrate the orbits of stars and calculate the radius of closest approach rc for any two given stars. Measuring the correlation as a function of rc would be an alternative strategy for disentangling the effects of stellar diffusion and panspermia, which may be useful if life propagates at very low speeds, or if the timescale td is longer than the stellar mixing timescale. We note that the above discrete models can be gener- alized to the continuous case by using a slightly differ- ent formalism. If the number distribution of bubbles is known, a power spectrum of inhabited star density fluc- tuations can be derived that will reproduce the bubble spectrum by retracing the steps of the Press-Schechter formalism (Press & Schechter 1974). Once the power spectrum is obtained, one can obtain the correlation function via a Fourier transform. If the habitable stellar density is not constant but fluctuates in space, the prob- lem becomes analogous to the spread of disease on an inhomogeneous medium (Lin & Loeb 2015), which again makes use of the Press-Schechter formalism. Future surveys such as the TESS (Ricker et al. 2015) will detect hundreds of earth-like exoplanets (Sullivan et al. 2015). Ground-based and space-based (e.g. JWST) follow ups that can characterize the exoplanet atmo- spheres could test for biosignatures such as oxygen in combination with a reducing gas (for a review, see Kaltenegger et al. 2002). However, it is likely that only a few earth-like exoplanets will be close enough to be biologically characterized (Brandt & Spiegel 2014; Rein
  • 4. 4 et al. 2014) with next-decade instruments. Eventually, surveys could test for more specific spectral signatures such as the “red edge” of chlorophyl (Seager et al. 2005) or even industrial pollution (Lin et al. 2014). It is also possible that searches for extraterrestrial intelligence in the radio or optical wavelengths could also yield detec- tions that could be tested for clustering. Any positive detections will yield first constraints on the correlation function C(x−y). In a favorable scenario, our solar sys- tem could be on the edge of a bubble, in which case a survey of nearby stars would reveal that ∼ 1/2 of the sky is inhabited while the other half is uninhabited. In this favorable scenario, ∼ 25 targets confirmed to have biosig- natures (supplemented with 25 null detections) would correspond to a 5σ deviation from the Poisson case, and would constitute a smoking gun detection of panspermia. A more generic placement would increase the number of required detections by a factor of a few, though an un- usual bubble configuration could potentially reduce the number of required detections. It should be noted that the local environment of our solar system does not re- flect the local environment ∼ 4 Gyr ago when life arose on earth, so the discovery of a bubble of surrounding earth should be interpreted as the solar system “drift- ing” into a bubble which has already formed, or perhaps the earth seeding its environment with life. Finally, we note that since the efficiency of panspermia is dictated by the number of transfer events between stel- lar systems, which in turn will depend on the local stellar density. For example, the transfer rates of rocky material between stars grows with stellar density simply because the mean distance between stars is smaller (Adams & Spergel 2005; Belbruno et al. 2012), and the enhanced frequency of close stellar encounters should further in- crease these rates. One might therefore expect that an inhomogeneous distribution of stars (i.e. the Milky Way) might exhibit a “multi-phase” structure, where regions of high stellar density have completed the phase transi- tion and are saturated with life, whereas regions of low stellar density will have little to no signs of life. How- ever, regions of high stellar density may be inhospitable to life due to a variety of factors (Gonzalez et al. 2001; Lineweaver et al. 2004; Gowanlock et al. 2011). For ex- ample, perturbations of Oort cloud analogs due to close stellar encounters, could potentially make a habitable zone planet inhospitable to life (Gonzalez et al. 2001). The number density per unit time of supernovae explo- sions (Svensmark 2012) and the local stellar metallicity may also play a factor in determining the habitability of planets (Gonzalez et al. 2001). It is therefore likely that the number of panspermia events is a complicated, non-monotonic function of stellar density and other as- trophysical parameters. 4. IMPLICATIONS In our simple formalism, life spreads from host to host in a way that resembles the outbreak of an epidemic. A key point is that the correlations that quickly arise imply that the transition from an uninhabited galaxy to an in- habited galaxy can occur much faster in the panspermia regime than in the Poisson case. Panspermia implies a phase transition, whereas a Poisson process will only lead to a gradual build up of life. Said differently, the start time for life for different stellar systems exhibits a very small scatter in the panspermia case. A consequence of the panspermia scenario is that the severity of the Fermi paradox may be reduced somewhat. If life started ev- erywhere at the same time, we expect fewer advanced civilizations at the present time than if life could have started much earlier on other stellar systems. It should be noted, however, that this statement is predicated on the somewhat controversial assumption that there is an evolutionary bias towards increasing complexity (Adami et al. 2000). A second consequence of panspermia is that the Drake equation (Shklovsky & Sagan 1966) becomes a lower bound on the number of civilizations, since the multiplicative form of the equation is based on the as- sumption that life arises independently everywhere. This assumption may be strongly violated in the regime where panspermia is highly efficient. The mathematical similarity between panspermia and disease spread may also represent a biological one: any species which acquires panspermia abilities will have enormous fitness advantages. Just as viruses evolved to brave the “harsh” environment of “inter-host” space to harness the energy of multiple biological hosts, perhaps evolution has or will drive a class of organisms to brave the harsh environment of interstellar space to harness the energy of multiple stellar hosts. Whether or not the organisms will be primitive (e.g., lithopanspermia, cometary panspermia (Hoyle & Wickramasinghe 1981)) or intelligent (directed panspermia (Crick & Orgel 1973) or accidental panspermia) remains to be determined. Even if the earth is the only inhabited planet and prim- itive life cannot survive an interstellar journey, interstel- lar travel led by humans may one day lead to colonization of the galaxy. As a zeroth order model, the same for- malism should approximately describe the growth of the colonies as they percolate through the galaxy, assuming that the processes such as population diffusion also occur in space (Newman & Sagan 1981). Although the ques- tion is a purely astrobiological question today, in the dis- tant future urban sociologists and astrophysicists might be forced to work together (Lin & Loeb 2015). Indeed, well after the colonization regime, models of panspermia may continue to be relevant, as spaceships capable of interstellar travel will provide the opportunity for primi- tive life (e.g. domesticated life, diseases, and viruses) to spread efficiently. The question that awaits is whether primitive life has already spread efficiently, or whether it will have to wait for “intelligent” life to make the voyage. ACKNOWLEDGMENTS We thank Ed Turner, Edwin Kite, and an anonymous referee for useful discussions. 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