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Surviving in a Metastable de Sitter SpaceTime
Sitender Pratap Kashyapa, Swapnamay Mondala, Ashoke Sena,b, Mritunjay Vermaa,c
aHarishChandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211019, India
bSchool of Physics, Korea Institute for Advanced Study, Seoul 130722, Korea
cInternational Centre for Theoretical Sciences, Malleshwaram, Bengaluru  560 012, India.
Email: sitenderpratap,swapno,sen,mritunjayverma@mri.ernet.in
Abstract
In a metastable de Sitter space any object has a ﬁnite life expectancy beyond which it
undergoes vacuum decay. However, by spreading into diﬀerent parts of the universe which
will fall out of causal contact of each other in future, a civilization can increase its collective
life expectancy, deﬁned as the average time after which the last settlement disappears due to
vacuum decay. We study in detail the collective life expectancy of two comoving objects in de
Sitter space as a function of the initial separation, the horizon radius and the vacuum decay
rate. We ﬁnd that even with a modest initial separation, the collective life expectancy can
reach a value close to the maximum possible value of 1.5 times that of the individual object
if the decay rate is less than 1% of the expansion rate. Our analysis can be generalized to
any number of objects, general trajectories not necessarily at rest in the comoving coordinates
and general FRW spacetime. As part of our analysis we ﬁnd that in the current state of the
universe dominated by matter and cosmological constant, the vacuum decay rate is increasing
as a function of time due to accelerated expansion of the volume of the past light cone. Present
decay rate is about 3.7 times larger than the average decay rate in the past and the ﬁnal decay
rate in the cosmological constant dominated epoch will be about 56 times larger than the
average decay rate in the past. This considerably weakens the lower bound on the halflife of
our universe based on its current age.
1
arXiv:1506.00772v2 [hepth] 1 Aug 2015
Contents
1 Introduction 2
2 Independent decay 6
3 Vacuum decay in 1+1 dimensional de Sitter space 7
3.1 Isolatedcomovingobject.............................. 8
3.2 Apairofcomovingobjects............................. 9
4 Vacuum decay in 3+1 dimensional de Sitter space 16
4.1 Isolatedcomovingobject.............................. 17
4.2 Apairofcomovingobjects............................. 17
4.3 The case of small initial separation . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Generalizations 23
5.1 Multiple objects in de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2 Realistictrajectories ................................ 25
5.3 Mattereﬀect .................................... 26
6 Discussion 30
A Decay rate for equation of state p=w ρ 33
1 Introduction
The possibility that we may be living in a metastable vacuum has been explored for more that
ﬁfty years [1–6]. Discovery of the accelerated expansion of the universe [7,8] and subsequent
developments in string theory leading to the construction of de Sitter vacua [9–11] suggest that
the vacuum we are living in at present is indeed metastable. Unfortunately our understanding
of string theory has not reached a stage where we can make a deﬁnite prediction about the
decay rate of our vacuum. The only information we have about this is from the indirect
observation that our universe is about 1.38 ×1010 years old. Therefore, assuming that we have
not been extremely lucky we can conclude that our inverse decay rate1is at least of the same
1For exponential decay the inverse decay rate diﬀers from halflife by a factor of ln 2. In order to simplify
terminology, we shall from now on use only inverse decay rate and life expectancy – to be deﬁned later – as
2
order.2
Typically the decay of a metastable vacuum proceeds via bubble nucleation [1–6] (see [14]
for a recent survey). In a small region of spacetime the universe makes transition to a more
stable vacuum, and this bubble of stable vacuum3then expands at a speed that asymptotically
approaches the speed of light, converting the rest of the region it encounters also to this stable
phase. Due to this rapid expansion rate it is impossible to observe the expanding bubble before
encountering it – it reaches us when we see it. However, due to the existence of the future
horizon in the de Sitter space, even a bubble expanding at the speed of light cannot ﬁll the
whole space at future inﬁnity. Indeed, it has been known for quite some time that in de Sitter
space if the expansion rate of the universe exceeds the decay rate due to phase transition then
even collectively the bubbles of stable vacuum cannot ﬁll the whole space [15] and there will
always be regions which will continue to exist in the metastable vacuum. Nevertheless, any
single observer in the metastable vacuum will sooner or later encounter an expanding bubble
of stable vacuum, and the probability of this decay per unit time determines the inverse decay
rate of the observer in the metastable vacuum.
This suggests that while any single observer will always have a limited average life span
determined by the microscopic physics, a civilization could collectively increase its longevity
by spreading out and establishing diﬀerent civilizations in diﬀerent parts of the universe [16].
If the bubble of stable vacuum hits the civilization – henceforth refered to as object – in the
initial stages of spreading out then it does not help since the same bubble will most likely
destroy all the objects. However, with time the diﬀerent objects will go outside each other’s
horizon and a single bubble of stable vacuum will not be able to destroy all of them. This will
clearly increase the life expectancy of the objects collectively – deﬁned as the average value of
the time at which the last surviving object undergoes vacuum decay – although there will be
no way of telling a priori which one will survive the longest. A simple calculation shows that
if we could begin with 2 objects already far outside each other’s horizon so that their decay
measures of longevity.
2We shall in fact see in §5.3 that the actual lower bound for the current inverse decay rate is weaker by a
factor of 3.7, making it comparable to the time over which the earth will be destroyed due to the increase in
the size of the sun. Allowing for the possibility that we could have been extremely lucky reduces the lower
bound on the inverse decay rate by about a factor of 10 [12,13].
3We shall refer to the more stable vacuum as the stable vacuum, even if this vacuum in turn could decay to
other vacua of lower energy density. In any case since this vacuum will have negative cosmological constant, the
spacetime inside the bubble will undergo a gravitational crunch [6]. We shall ignore the possibility of decay to
Minkowski vacua or other de Sitter vacua of lower cosmological constant since the associated decay rates are
very small due to smallness of the cosmological constant of our vacuum.
3
1
5
10
50
100
500
1000
T
1.1
1.2
1.3
1.4
1.5
Gain
Figure 1: The ﬁgure showing the ‘gain’ in the life expectancy for two objects compared to that
of one object as a function of Tfor r= .0003, .001,.003,.01,.03,.1 and .3.
probabilities can be taken to be independent, then the life expectancy of the combined system
increases by a factor of 3/2 compared to the life expectancy of a single isolated object. In the
case of ncopies the life expectancy increases by a factor given by the nth harmonic number.
However, in actual practice we cannot begin with copies of the object already outside each
other’s horizon. As a result the increase in the life expectancy is expected to be lower.
The goal of this paper will be to develop a systematic procedure for computing the increase
in the life expectancy of the object as a result of making multiple copies of itself. For two
objects we obtain explicit expression for the life expectancy in terms of three parameters: the
Hubble constant Hof the de Sitter spacetime determined by the cosmological constant, the
vacuum decay rate or equivalently the life expectancy Tof a single isolated object and the
initial separation rbetween the two objects. In fact due to dimensional reasons the result
depends only on the combination HT and Hr, so we work by setting H= 1. In Fig. 1we
have shown the result for the ratio of the life expectancy of two objects to that of a single
object – called the ‘gain’ – as a function of Tfor diﬀerent choices of r. From this we see that
even for a modest value of r= 3 ×10−4the gain in the life expectancy reaches close to the
maximum possible value of 1.5 if Tis larger that 100 times the horizon size of the de Sitter
4
1
10
100
1000
a
0.2
0.4
0.6
0.8
1.0
R
Figure 2: Growth of the relative decay rate R– deﬁned as the ratio of the decay rate to
its asymptotic value – of a single object in FRW spacetime as a function of the scale factor
a. Value of Rat a= 1 represents the decay rate today relative to what it would be in the
cosmological constant dominated epoch.
space, i.e. the decay rate is less than 1% of the expansion rate. T= 100 corresponds to about
1.7×1012 years. r= 3 ×10−4corresponds to a physical distance of the order of 5 ×106light
years and is of the order of the minimal distance needed to escape the local gravitationally
bound system of galaxies. If T= 10 – i.e. of order 1.7×1011 years – the gain is about 20% for
r= 3 ×10−4. These time scales are shorter than the time scale by which all the stars in the
galaxy will die. Therefore, if Tlies between 1011 years and the life span of the last star in the
local group of galaxies which will be gravitationally bound and will remain inside each other’s
horizon, then we gain a factor of 1.2  1.5 in life expectancy even by making one additional
copy of the object at a distance larger than about 107light years from us. On the other hand
if Tis larger than the life span of the last star in the galaxy then our priority should be to
plan how to survive the death of the galaxy rather than vacuum decay. Some discussion on
this can be found in [17].
Even though most of our analysis focusses on the case of a pair of objects in de Sitter space
at ﬁxed comoving coordinates, our method is quite general and can be applied to arbitrary
5
number of objects in a general FRW metric moving along general trajectories. We discuss these
generalizations in §5. In particular considering the case of a single object in an FRW metric
dominated by matter and cosmological constant, as is the case with the current state of our
universe, we ﬁnd that the vacuum decay rate increases as a function of time due to accelerated
expansion of the volume of the past light cone. This has been shown in Fig. 2. This rate
approaches a constant value as the universe enters the cosmological constant dominated era,
but we ﬁnd for example that this asymptotic decay rate is about 15 times larger than the
decay rate today, which in turn is about 3.7 times larger than the average decay rate in the
past. Now given that the universe has survived for about 1.38 ×1010 years, we can put a lower
bound of this order on the inverse of the average decay rate in the past.4This translates to a
lower bound of order 3.7×109years on the inverse decay rate today and 2.5×108years on
the asymptotic inverse decay rate.
The rest of the paper is organised as follows. In §2we describe the case of the decay of n
objects assuming that their decay probabilities are independent of each other, and show that
the life expectancy of the combined system goes up by a factor equal to the nth harmonic
number. In §3we carry out the complete analysis for two observers in 1+1 dimensional de
Sitter space. The ﬁnal result for the life expectancy of the combined system can be found in
(3.30). This is generalized to the case of two observers in 3+1 dimensional de Sitter space
time in §4. Eq.(4.17) together with (4.16) and (4.11) gives the probability that at least one
of the two objects survives till time t, which can then be used to compute the life expectancy
of the combined system using (4.18). In §5we discuss various generalizations including the
case of multiple observers, general trajectories and general FRW type metric. We conclude
in §6with a discussion of how in future we could improve our knowledge of possible values
of the parameters rand Twhich enter our calculation. In appendix Awe compute the time
dependence of the decay rate for a general equation of state of the form p=wρ.
2 Independent decay
Let us suppose that we have two independent objects, each with a decay rate of cper unit
time. We shall label them as C1and C2. If we begin with the assumption that both objects
4One must keep in mind that this is not a strict bound since we could have survived till today by just being
lucky.
6
exist at time t= 0 then the probability that the ﬁrst object exists after time tis
P1(t) = e−c t .(2.1)
Therefore, the probability that it decays between time tand t+δt is −˙
P1(t)δt where ˙
P1denotes
the derivative of P1with respect to t, and its life expectancy, is
¯
t1=−Z∞
0
t˙
P1(t)dt =Z∞
0
P1(t)dt =c−1.(2.2)
Independently of this the probability that the second object exists after time tis also given by
exp[−c t] and it has the same life expectancy.
Now let us compute the life expectancy of both objects combined, deﬁned as the average of
the larger of the actual life time of C1and C2. To compute this note that since the two objects
are independent, the probability that both will decay by time tis given by (1 −P1(t))(1 −
P2(t)) = (1 −P1(t))2. Therefore, the probability that the last one to survive decays between t
and t+δt is d
dt (1 −P1(t))2δt. This gives the life expectancy of the combined system to be
¯
t12 =Z∞
0
td
dt(1 −P1(t))2dt =3
2c−1.(2.3)
Therefore, we see that by taking two independent objects we can increase the life expectancy
by a factor of 3/2. A similar argument shows that for nindependent objects the life expectancy
will be
¯
t12···n=Z∞
0
td
dt(1 −P1(t))ndt =1 + 1
2+1
3+· ·· +1
nc−1.(2.4)
3 Vacuum decay in 1+1 dimensional de Sitter space
Consider 1+1 dimensional de Sitter space
ds2=−dt2+e2tdx2.(3.1)
Note that we have set the Hubble constant of the de Sitter space and the speed of light to
unity so that all other time / lengths appearing in the analysis are to be interpreted as their
values in units of the inverse Hubble constant. We introduce the conformal time τvia
τ=−e−t(3.2)
7

6
x
τ
τ= 0
τ=−1
τ
τ+δτ
σ
Figure 3: A comoving object in de Sitter space and its past light cones at conformal times τ
and τ+δτ.
in terms of which the metric takes the form
ds2=τ−2(−dτ2+dx2).(3.3)
At t= 0 we have τ=−1 and comoving distances coincide with the physical distances.
We shall use this spacetime as a toy model for studying the kinematics of vacuum decay.
We shall assume that in this spacetime there is a certain probability per unit time per unit
volume of producing a bubble of stable vacuum, which then expands at the speed of light
causing decay of the metastable vacuum. We shall not explore how such a bubble is produced;
instead our goal will be to study its eﬀect on the life expectancy of the objects living in this
space. In §4we shall generalize this analysis to 3+1 dimensional de Sitter space.
3.1 Isolated comoving object
Consider a single object in de Sitter space at rest in the comoving coordinate x(say at x= 0),
shown by the vertical dashed line in Fig. 3. We start at t= 0 (τ=−1) and are interested in
calculating the probability that it survives at least till conformal time τ. If we denote this by
P0(τ) then the probability that it will decay between τand τ+δτis −P0
0(τ)δτwhere 0denotes
derivative with respect to τ. On the other hand this probability is also given by the product
of P0(τ) and the probability that a vacuum bubble is produced somewhere in the past light
cone of the object between τand τ+δτ, as shown in Fig.3. The volume of the past light cone
of this interval can be easily calculated to be
2δτZτ
−∞
dσ
σ2=−2
τδτ.(3.4)
8

6
x
τ
r
τ= 0
τ=−1
1 2
Figure 4: Two comoving objects in de Sitter space separated by physical distance rat τ=−1.
Therefore, if Kis the probability of producing the bubble per unit spacetime volume then the
probability of producing a bubble in the past light cone of the object between τand τ+δτis
given by −2Kδτ/τ. The previous argument then leads to the equation
P0
0(τ)=2Kτ−1P0(τ).(3.5)
This equation, together with the boundary condition P0(τ=−1) = 1, can be integrated to
give
ln P0(τ) = 2Kln(−τ).(3.6)
In terms of physical time twe have5
P0(t) = e−2Kt .(3.7)
From this we can calculate the life expectancy, deﬁned as the integral of tweighted by the
probability that the object undergoes vacuum decay between tand t+dt. Since the latter is
given by −˙
P0(t)dt, we have the life expectancy
T=−Z∞
0
t˙
P0(t)dt =Z∞
0
P0(t)dt =1
2K,(3.8)
where in the second step we have used integration by parts. We shall express our ﬁnal results
in terms of Tinstead of K.
3.2 A pair of comoving objects
Next we shall consider two comoving objects C1and C2in de Sitter space separated by physical
distance rat t= 0 or equivalently τ=−1. We shall take r < 1, i.e. assume that the two
5Note that by an abuse of notation we have used the same symbol P0to denote the probability as a function
of talthough the functional form changes. We shall continue to follow this convention, distinguishing the
function by its argument (tor τ). Derivatives with respect to τand twill be distinguished by using P0
0to
denote τderivative of P0and ˙
P0to denote tderivative of P0.
9

6
x
τ
τ=−1
τ
τ+δτ
1 2
r
Figure 5: The past lightcome of C2at τ=−1 and the past light cone of C1between τand
τ+δτfor τ< r −1.
objects are within each other’s horizon at the time they are created. We denote by Pi(τ) the
probability that Cisurvives at least till conformal time τfor i= 1,2 and by P12(τ1,τ2) the
joint probability that C1survives at least till conformal time τ1and C2survives at least till
conformal time τ2. The boundary condition will be set by assuming that both objects exist at
τ=−1, so that we have
P1(−1) = 1, P2(−1) = 1, P12(−1,τ2) = P2(τ2), P12 (τ1,−1) = P1(τ1).(3.9)
First we shall calculate P1(τ) and P2(τ). They must be identical by symmetry, so let us
focus on P1(τ). The calculation is similar to that for P0(τ) above for a single isolated object,
except that the existence of C2at τ=−1 guarantees that no vacuum decay bubble is produced
in the past lightcome of C2at τ=−1, and hence while computing the volume of the past
light cone of the C1between τand τ+δτ, we have to exclude the region inside the past light
cone of C2at τ=−1. This has been shown in Fig. 5. This volume is given by
δτ"2Zτ
−∞
dσ
σ2−Z−1−r−1−τ
2
−∞
dσ
σ2#=δτ−2
τ−2
r+ 1 −τfor τ< r −1.(3.10)
However, for τ> r −1 the past light cone of C1between τand τ+δτdoes not intersect the
past light cone of C2at τ=−1 (see Fig. 6), and we get the volume to be
2δτZτ
−∞
dσ
σ2=−2δτ1
τfor τ> r −1.(3.11)
10

6
x
τ
τ=−1
τ
τ+δτ
 r
1 2
Figure 6: The past lightcome of C2at τ=−1 and the past light cone of C1between τand
τ+δτfor τ> r −1.
This leads to the following diﬀerential equation for P1(τ):
1
P1(τ)
dP1
dτ=(−K[−2/τ−2/(r+ 1 −τ)] for τ< r −1,
2K /τfor τ> r −1.(3.12)
Using the boundary condition P1(−1) = 1 and the continuity of P1(τ) across τ=r−1 we get
ln P1(τ) = (2K{ln(−τ)−ln(r+ 1 −τ) + ln(r+ 2)}for τ< r −1,
2K{ln(−τ)−ln 2 + ln(r+ 2)}for τ> r −1.(3.13)
Using the symmetry between 1 and 2 we also get the same expression for P2(τ). In terms of
the physical time twe have
P1(t) = P2(t) =
e−2Kt (r+1+e−t)−2K(r+ 2)2Kfor t < −ln(1 −r),
(r+ 2)/22Ke−2Kt for t > −ln(1 −r).(3.14)
Therefore, the life expectancy of C1is
¯
t1=−Z∞
0
t˙
P1(t)dt =Z∞
0
P1(t)dt
= (r+ 2)2KB1
2 + r; 2K, 0−B1−r
2; 2K, 0+(1 −r)2K
22K+1K(3.15)
where B(x;p, q) is the incomplete beta function, deﬁned as
B(x;p, q) = Zx
0
tp−1(1 −t)q−1dt =Zx/(1−x)
0
yp−1
(1 + y)p+qdy , (3.16)
the two expressions being related by the transformation t=y/(y+ 1). In terms of the life
expectancy T= 1/2Kof a single isolated object, we have
¯
t1= (r+ 2)1/T B1
2 + r;1
T,0−B1−r
2;1
T,0+T(1 −r)1/T
21/T .(3.17)
11

6
x
τ
τ=−1
τ1
τ1+δτ1
 r
1 2
Figure 7: The past lightcome of C2at τ2and the past light cone of C1between τ1and τ1+δτ1
for τ1<τ2−r.
C2also has the same life expectancy. (3.17) is somewhat larger than T, but that is simply a
result of our initial assumption that both objects exist at t= 0. If both objects had started
at the same spacetime point and then got separated following some speciﬁc trajectories, then
there would have been a certain probability that one or both of them will decay during the
process of separation; this possibility has been ignored here leading to the apparent increase
in the life expectancy. However, for realistic values of rand T, which corresponds to r << 1
and T>
∼1, the ratio ¯
t1/T remains close to unity.
Let us now turn to the computation of the joint survival probability P12(τ1,τ2). In this
case the probability that the ﬁrst object undergoes vacuum decay between τ1and τ1+δτ1and
the second object survives at least till τ2is given by −δτ1(∂P12(τ1,τ2)/∂τ1). On the other
hand the same probability is given by K×P12(τ1,τ2) times the volume of the past lightcome
of C1between τ1and τ1+δτ1, excluding the region inside the past light cone of C2at τ2. The
relevant geometry has been shown in Figs. 7,8and 9for diﬀerent ranges of τ1and τ2. The
results are as follows:
1. For τ1<τ2−rthe geometry is shown in Fig. 7. In this case C1at τ1(and hence the
whole of the past light cone of C1between τ1and τ1+δτ1) is inside the past light cone
of C2at τ2. Therefore, the decay probability is zero and we have the equation:
∂ln P12(τ1,τ2)
∂τ1
= 0 for τ1<τ2−r . (3.18)
2. For τ2−r < τ1<τ2+rthe geometry is as shown in Fig. 8. In this case C1at τ1and C2
at τ2are spacelike separated. The volume of the past light cone of C1between τ1and
12

6
x
τ
τ=−1
τ1
τ1+δτ1
1 2
r
Figure 8: The past lightcome of C2at τ2and the past light cone of C1between τ1and τ1+δτ1
for τ2−r < τ1<τ2+r.
τ1+δτ1outside the past light cone of C2at τ2is given by
Zτ1
−∞
dσ
σ2+Zτ1
1
2(τ1+τ2−r)
dσ
σ2=−2
τ1−2
r−τ1−τ2
.(3.19)
This gives
∂ln P12(τ1,τ2)
∂τ1
= 2K1
τ1
+1
r−τ1−τ2for τ2−r < τ1<τ2+r . (3.20)
3. For 0 <τ2+r < τ1, the geometry is shown in Fig. 9. In this case C2at τ2is inside the
past light cone of C1at τ1and there is no intersection between the past light cone of C1
between τ1and τ1+δτ1and the past light cone of C2at τ2. Therefore, the volume of
the past light cone of C1between τ1and τ1+δτ1is given by
2Zτ1
−∞
dσ
σ2=−2
τ1
,(3.21)
and we have ∂ln P12(τ1,τ2)
∂τ1
= 2K1
τ1
for τ2+r < τ1<0.(3.22)
We can now determine P12(τ1,τ2) by integrating (3.18), (3.20), (3.22) subject to the boundary
condition given in (3.9)
P12(−1,τ2) = P2(τ2) = P1(τ2),(3.23)
13

6
x
τ
τ=−1
τ1
τ1+δτ1
 r
1 2
Figure 9: The past light cone of C2at τ2and the past light cone of C1between τ1and τ1+δτ1
for τ2+r < τ1<0.
and using the fact that P12(τ1,τ2) must be continuous across the subspaces deﬁned by τ1=
τ2±r. The result of the integration is
ln P12(τ1,τ2) =
2K{ln(−τ2) + ln(r+ 2) −ln 2}for τ1<τ2−r ,
2K{ln(−τ2) + ln(−τ1)−ln(r−τ1−τ2) + ln(r+ 2)},for τ2−r < τ1<τ2+r
2K{ln(−τ1) + ln(r+ 2) −ln 2}for τ2+r < τ1<0.
(3.24)
Note that the result is symmetric under the exchange of τ1and τ2even though at the inter
mediate stages of the analysis this symmetry was not manifest.
Expressed in terms of physical time the above solution takes the form:
P12(t1, t2) =
{(r+ 2)/2}2Ke−2Kt2for t1<−ln(r+e−t2),
(r+ 2)2Ke−2K(t1+t2)(r+e−t1+e−t2)−2Kfor −ln(r+e−t2)< t1<−ln(e−t2−r),
{(r+ 2)/2}2Ke−2Kt1for t1>−ln(e−t2−r).
(3.25)
If e−t2−ris negative then the third case is not relevant and in the second case there will be no
upper bound on t1. Physically this can be understood by noting that in this case τ2>−rand
C2will never come inside the past light cone of C1even when τ1reaches its maximum value 0.
Our interest lies in computing the probability that at least one of the two objects survives
till time t. Let us denote this by e
P12(t). This is given by the sum of the probability that C1
survives till time tand the probability that C2survives till time t, but we have to subtract
from it the probability that both C1and C2survive till time tsince this will be counted twice
otherwise. This can be seen from the Venn diagram of two objects shown in Fig. 10. Therefore,
14
A
B
A∩B
P(A∪B) = P(A) + P(B)−P(A∩B)
Figure 10: Probability rule for N=2 using Venn Diagram.
we have
e
P12(t) = P1(t) + P2(t)−P12 (t, t).(3.26)
From this we can compute the probability that the last one to survive decays between tand
t+δt as
−δt d
dt e
P12(t).(3.27)
Therefore, the life expectancy of the combined system is given by
¯
t12 =−Z∞
0
dt t d
dt e
P12(t, t) = Z∞
0
dt {P1(t) + P2(t)−P12(t, t)},(3.28)
where in the second step we have integrated by parts and used (3.26). Each of the ﬁrst two
integrals gives the result ¯
t1computed in (3.17). For the last integral since we have to evaluate
P12(t1, t2) at t1=t2=tonly the middle expression in (3.25) is relevant, and we get
Z∞
0
P12(t, t)dt =Z∞
0
(r+ 2)2Ke−4Kt (r+ 2e−t)−2Kdt
= 2−4Kr2K(r+ 2)2KB2
2 + r; 4K, −2K.(3.29)
15
Combining this with the result for ¯
t1given in (3.17) and replacing Kby 1/2Twe get
¯
t12 = 2(r+ 2)1/T B1
2 + r;1
T,0−B1−r
2;1
T,0+T(1 −r)1/T
21/T
−2−2/T r1/T (r+ 2)1/T B2
2 + r;2
T,−1
T.(3.30)
We can now check various limits. First of all we can study the r→0 limit using the result
B(x; 2α, −α)'1
α(1 −x)−α,(3.31)
for xclose to 1. This gives limr→0¯
t12 =T. This is in agreement with the fact that if the two
objects remain at the same point then their combined life expectancy is the same as that of
individual objects.
If on the other hand we take the limit of large Tthen, using the result
B(x;α, β)'1
α(3.32)
for small α, we get ¯
t12 '3T/2. Therefore, the life expectancy of the two objects together is 3/2
times that of an isolated object. This is consistent with the fact that if the inverse decay rate
of individual objects is large then typically there will be enough time for the two objects to
go out of each other’s horizon before they decay. Therefore, we can treat them as independent
objects and recover the result (2.3). Mathematically this can be seen from the fact that when
Tis large and t∼Tthen P12(t, t) given in the middle expression of (3.25) approaches e−4Kt,
which in turn is approximately equal to the square of P1(t) given in (3.14).
4 Vacuum decay in 3+1 dimensional de Sitter space
In this section we shall repeat the analysis of §3for 3+1 dimensional de Sitter spacetime.
Since the logical steps remain identical, we shall point out the essential diﬀerences arising in
the two cases and then describe the results.
The metric of the 3+1 dimensional de Sitter space is given by
ds2=−dt2+e2t(dx2+dy2+dz2) = τ−2(−dτ2+dx2+dy2+dz2),τ≡ −e−t.(4.1)
There are of course various other coordinate systems in which we can describe the de Sitter
metric, but the coordinate system used in (4.1) is specially suited for describing out universe,
16
with (x, y, z) labelling comoving coordinates and tdenoting the cosmic time in which the
constant tslices have uniform microwave background temperature. This form of the metric
uses the observed ﬂatness of the universe. The actual metric at present is deformed due to the
presence of matter density, and also there is a lower cutoﬀ on tsince our universe has a ﬁnite
age of the order of the inverse Hubble constant. But both these eﬀects will become irrelevant
within a few Hubble time and we ignore them. In §5.3 we shall study these eﬀects, but at
present our goal is to get an analytic result under these simplifying assumptions.
4.1 Isolated comoving object
First consider the case of an isolated object. The calculation proceeds as in §3.1. However, in
computing the volume of the past light cone in Fig. 3we have to take into account the fact
that for each σ, the light cone is a sphere of radius (τ−σ). Since the coordinate radius of the
sphere is (τ−σ) and the spacetime volume element scales as 1/σ4we get the volume of the
past light cone of the object between τand τ+δτto be
δτZτ
−∞
dσ
σ44π(τ−σ)2=−4
3πτ−1δτ.(4.2)
This replaces the right hand side of (3.4). Therefore, (3.5) takes the form
P0
0(τ) = 4
3πτ−1K P0(τ),(4.3)
with the solution
ln P0(τ) = 4
3πK ln(−τ),(4.4)
P0(t) = exp −4
3πKt.(4.5)
From this we can calculate the life expectancy of the isolated object to be
T=Z∞
0
P0(t)dt =3
4πK .(4.6)
4.2 A pair of comoving objects
The additional complication in the case of two objects comes from having to evaluate the
contribution of the past light come of the ﬁrst object between τ1and τ1+δτ1in situations
depicted in Figs. 5and 8. Let us consider Fig. 8since Fig. 5can be considered as a special
17
case of Fig. 8with τ2=−1. Now in Fig. 8which occurs for τ2−r < τ1<τ2+r, the past
light cone of C1between τ1and τ1+δτ1lies partly inside the past light cone of C2. We need
to subtract this contribution from the total volume of the past light cone of C1between τ1
and τ1+δτ1, since the assumption that C2survives till τ2rules out the formation of a bubble
inside the past light cone of C2. Our goal will be to calculate this volume.
Examining Fig. 8we see that the intersection of the past light cones of C1at τ1and C2at
τ2occur at τ=σfor σ < (τ1+τ2−r)/2. At a value of σsatisfying this constraint, the past
light cone of C1at τ1is a sphere of coordinate radius r1= (τ1−σ) and the past light cone
of C2at τ2is a sphere of coordinate radius r2= (τ2−σ). The centers of these spheres, lying
at the comoving coordinates of the two objects have a coordinate separation of r. A simple
geometric analysis shows that the coordinate area of the part of the ﬁrst sphere that is inside
the second sphere is given by
πr1
r{r22−(r1−r)2}=π(τ1−σ)
r(τ2−τ1+r)(τ1+τ2−r−2σ).(4.7)
Taking into account the fact that physical volumes are given by 1/σ4times the coordinate
volume we get the following expression for the volume of the past light cone of C1between τ1
and τ1+δτ1that is inside the past light cone of C2:
π
r(τ2−τ1+r)δτ1Z(τ1+τ2−r)/2
−∞
dσ
σ4(τ1−σ)(τ1+τ2−r−2σ)
=2π
3r(τ2−τ1+r)δτ1
(3r−τ1−3τ2)
(r−τ1−τ2)2.(4.8)
As already mentioned the excluded volume in case of Fig. 5can be found by setting τ1=τ
and τ2=−1 in (4.8).
We are now ready to generalize all the results of §3. Let us begin with (3.12). Its general
ization to the 3+1 dimensional case takes the form
d
dτln P1(τ) =
2πK
32
τ+(−1 + r−τ)(3 + 3r−τ)
r(τ−r−1)2if τ< r −1
4πK
3τif τ> r −1
(4.9)
Its solution is given by
ln P1(τ) =
4πK
3ln(−τ) + τ
2r−ln(−τ+r+ 1) + 2(r+1)
r(τ−r−1) + ln(r+ 2) + 5r+6
2r(r+2) if τ< r −1,
4πK
3ln(−τ) + ln(r+ 2) −ln 2 −r
2(r+2) if τ> r −1.
(4.10)
18
1
5
10
50
100
500
1000
T
1.02
1.04
1.06
1.08
ratio
Figure 11: The ﬁgure showing the ratio ¯
t1/T for r= .0003, .001,.003,.01,.03,.1 and .3. For
r≤.003 the ratio is not distinguishable from 1 in this scale.
Expressing this in terms of tusing τ=−e−tand T≡3/(4πK) we get
P1(t) =
(e−t+r+ 1)−1
T(r+ 2) 1
Texp −t
T+1
T−e−t
2r−2(r+ 1)
r(e−t+r+ 1) +5r+ 6
2r(r+ 2)
for t < −ln(1 −r),
r+ 2
21
T
exp −t
T−r
2T(r+ 2)for t > −ln(1 −r).
(4.11)
The same expression holds for the survival probability P2(t) of C2. From this we can ﬁnd the
life expectancy of C1
¯
t1=Z∞
0
P1(t)dt . (4.12)
As in the 1+1 dimensional case, ¯
t1is slightly larger than Tbut this is simply due to the choice
of initial condition that both observers are assumed to exist at t= 0. In Fig. 11 we have
plotted the ratio ¯
t1/T as a function of Tfor various values of r, and as we can see the result
remains close to 1. More discussion on ¯
t1can be found below (4.23).
Next we consider the generalization of (3.18)(3.22). The analysis is straightforward and
19
we get the results
∂ln P12(τ1,τ2)
∂τ1
= 0 for τ1<τ2−r ,
∂ln P12(τ1,τ2)
∂τ1
=2πK
32
τ1
+(r−τ1+τ2)(3r−τ1−3τ2)
r(r−τ1−τ2)2for τ2−r < τ1<τ2+r
∂ln P12(τ1,τ2)
∂τ1
=4πK
3τ1
for τ2+r < τ1<0.(4.13)
The solution to these equations, subject to the boundary condition P12(τ1=−1,τ2) = P2(τ2) =
P1(τ2) is given by
ln P12(τ1,τ2) =
4πK
3hln(−τ2) + ln(r+ 2) −r
2(r+2) −ln 2iif τ1<τ2−r
4πK
3hln(−τ1) + ln(−τ2)−ln(−τ1−τ2+r) + τ1+τ2
2r−2τ1τ2
r(τ1+τ2−r)
+ ln(r+ 2) + 1
r+2 iif τ2−r < τ1<τ2+r
4πK
3hln(−τ1) + ln(r+ 2) −r
2(r+2) −ln 2iif τ2+r < τ1<0
(4.14)
In terms of the physical time, and T= 3/(4πK), this becomes
P12(t1, t2) =
{(r+ 2)/2}1/T exp −r
2T(r+ 2) −t2
Tif t1<−ln(r+e−t2)
(r+ 2)1/T (e−t1+e−t2+r)−1
T
×exp 1
T(r+ 2) −1
T(t1+t2)−1
2T r (e−t1+e−t2) + 2
T r
1
et1+et2+ret1+t2
if −ln(r+e−t2)< t1<−ln(e−t2−r)
{(r+ 2)/2}1/T exp −r
2T(r+ 2) −t1
Tif t1>−ln(e−t2−r)
(4.15)
This gives
P12(t, t)=(r+ 2)1/T e1
T(r+2) (2e−t+r)−1
Texp −2
Tt−1
T r e−t+2
T r
1
2et+re2t.(4.16)
In terms of this, and the functions P1=P2given in (4.11), we can calculate the probability
e
P12 of at least one of the two objects surviving till time tusing
e
P12(t) = P1(t) + P2(t)−P12 (t, t) (4.17)
20
and the combined life expectancy of two objects using the analog of (3.28)
¯
t12 =Z∞
0e
P12(t)dt =Z∞
0
dt {P1(t) + P2(t)−P12(t, t)}= 2 ¯
t1−Z∞
0
dt P12(t, t).(4.18)
For the integral of P12(t, t) one can write down an expression in terms of special functions
as follows. Deﬁning yvia
2 + ret=2 + r
y(4.19)
for r6= 0, we get
Z∞
0
dt P12(t, t)=[r(r+ 2)−1]1/T e1
(r+2)TZ1
0
dy y−1+2/T 1−2y
2 + r−1−1/T
exp −y
(2 + r)T.
(4.20)
Now, using the result
Z1
0
dy ya−1(1 −y)c−a−1
(1 −u y)bev y =B(a, c −a)Φ1(a, b, c;u, v) (4.21)
with Re c > Re a > 0,u<1, Bthe beta function and Φ1the conﬂuent hypergeometric series
of two variables (Humbert series), we get
Z∞
0
dt P12(t, t) = T
2r
(r+ 2)1/T
e1
T(r+2) Φ12
T,1 + 1
T,1 + 2
T;2
2 + r,−1
(2 + r)T.(4.22)
Φ1has a power series expansion
Φ1(a, b, c;u, v) =
∞
X
m,n=0
(a)m+n(b)m
(c)m+nm!n!umvn,u<1 (4.23)
where (a)m≡a(a+ 1) · ··(a+m−1).
Unfortunately we have not been able to ﬁnd an expression for ¯
t1=R∞
0dt P1(t) in terms of
special functions. However, we can write down a series expansion for this that will be suitable
for studying its behaviour for small r. The integral of (4.11) from t=−ln(1 −r) to ∞is
straightforward and yields
T(1 −r)1/T (r+ 2)1/T 2−1/T exp −r
2T(r+ 2).(4.24)
21
The integral of (4.11) from t= 0 to −ln(1−r) can be analyzed by making a change of variable
from tto yvia e−t= (1 −y r). In terms of this variable the integral can be expressed as
rZ1
0
dy 1−yr
r+ 2−1/T
(1−yr)−1+1/T exp "−2y2r(r+ 1)
T(2 + r)31−yr
2 + r−1#exp yr2
2T(2 + r)2.
(4.25)
Using series expansion of the second and third terms in the integrand we get
∞
X
m,n=0
1
m!n!1−1
Tm
(−1)n2nrm+n+1(r+ 1)n
Tn(2 + r)3n
Z1
0
dy ym+2n1−yr
2 + r−n−1/T
exp yr2
2T(2 + r)2.(4.26)
The integral over ycan be expressed in terms of Φ1using (4.21). Adding (4.24) to this we get
¯
t1=T(1 −r)1/T (r+ 2)1/T 2−1/T exp −r
2T(r+ 2)
+
∞
X
m,n=0
1
m!n!
1
m+ 2n+ 1 1−1
Tm
(−1)n2nrm+n+1(r+ 1)n
Tn(2 + r)3n
Φ1m+ 2n+ 1, n +1
T, m + 2n+ 2; r
2 + r,r2
2T(2 + r)2.(4.27)
It can be checked using (4.11), (4.12), (4.16) and (4.18) that for r→0 we get ¯
t12/¯
t1= 1
and for T→ ∞ we get ¯
t12/¯
t1= 3/2. The values of ‘gain’ ≡¯
t12/¯
t1for diﬀerent values of rhave
been plotted against Tin Fig. 1.
4.3 The case of small initial separation
Since from practical considerations the small rregion is of interest, it is also useful to consider
the expansion of ¯
t12/¯
t1for small r. For this we have to analyze the behaviour of ¯
t1as well as
that of R∞
0dt P12(t, t) for small r. Let us begin with ¯
t1given in (4.27). It can be easily seen
that this is given by T+O(r) with the contribution Tcoming from the ﬁrst term. However,
the contribution from R∞
0dt P12(t, t) has a more complicated behaviour at small r. This is
related to the fact that in the r→0 limit the fourth argument of Φ1in (4.22) approaches 1,
and in this limit the series expansion (4.23) diverges. To study the small rbehaviour we shall
go back to the original expression for P12(t, t) given in (4.16). We change variable to v=e−t/r
22
and write
Z∞
0
dt P12(t, t) = (r+ 2)1/T e1/T (r+2)r1/T Z1/r
0
dv
v(2v+ 1)−1/T v2/T exp −v
T(2v+ 1)
= (r+ 2)1/T e1/T (r+2)r1/T Z1/r
0
dv
vv2/T (2v+ 1)−1/T exp −v
T(2v+ 1)
−(2v)−1/T exp −1
2T+ (r+ 2)1/T 2−1/T exp 1
T(r+ 2) −1
2TT ,
(4.28)
where in the last step we have subtracted an integral from the original integral and compen
sated for it by adding the explicit result for the integral. This subtraction makes the integral
convergent even when we replace the upper limit 1/r by ∞. Taking the small rlimit we get
Z∞
0
dt P12(t, t) = 21/T r1/T Z∞
0
dv
vv2/T (2v+ 1)−1/T exp −v
T(2v+ 1) +1
2T−(2v)−1/T
+T+O(r).(4.29)
Combining this with the earlier result that for r→0, ¯
t1'T+O(r) and using (4.18) we get
¯
t12
¯
t1
= 2 −1
¯
t1Z∞
0
dt P12(t, t) = 1 + A(T)r1/T ,(4.30)
where
A(T) = T−121/T Z∞
0
dv v−1+2/T (2v)−1/T −(2v+ 1)−1/T exp −v
T(2v+ 1) +1
2T.(4.31)
The numerical values of A(T) are moderate – for example A(5) '0.439 and A(10) '0.457.
A plot of A(T) as a function of Thas been shown in Fig. 12. The 1/T exponent of rshows
that even if we begin with small r, for moderately large T(say T∼5) we can get moderate
enhancement in life expectancy.
5 Generalizations
In this section we shall discuss various possible generalizations of our results.
5.1 Multiple objects in de Sitter space
We shall begin by discussing the case of three objects C1,C2and C3placed at certain points
in 3 +1 dimensional de Sitter spacetime and analyze the probability that at least one of them
23
4
5
6
7
8
9
10
T
0.440
0.445
0.450
0.455
A
Figure 12: The coeﬃcient of the r1/T term in the expression for ¯
t12/¯
t1as a function of T.
will survive till time t. Let P123 (t1, t2, t3) denote the probability that C1survives till time t1,
C2survives till time t2and C3survives till time t3. Similarly Pij (ti, tj) for 1 ≤i, j ≤3 will
denote the probability that Cisurvives till tiand Cjsurvives till tjand Pi(ti) will denote the
probability that Cisurvives till ti. All probabilities are deﬁned under the prior assumption
that all objects are alive at t= 0. These probabilities can be calculated by generalizing the
procedure described in §3and §4by constructing ordinary diﬀerential equations in one of the
arguments at ﬁxed values of the other arguments. The geometry of course now becomes more
involved due to the fact that the past light cone of one object will typically intersect the past
light cones of the other objects which themselves may have overlaps, and one has to carefully
subtract the correct volume. But the analysis is straightforward.
The quantity of direct interest is the probability e
P123(t) that at least one of the objects
survives till time t. With the help of the Venn diagram given in Fig. 13 we get
e
P123(t) = P1(t) + P2(t) + P3(t)−P12 (t, t)−P13(t, t)−P23 (t, t) + P123(t, t, t).(5.1)
Using this we can calculate the life expectancy of the combined system as
−Z∞
0
dt t d
dt e
P123(t) = Z∞
0
dt e
P123(t).(5.2)
24
A
B
A∩B∩CC
Figure 13: The Venn diagram illustrating that the survival probability of one of A,Bor C,
denoted by P(A∪B∪C), is given by P(A) + P(B) + P(C)−P(A∩B)−P(B∩C)−P(A∩
C) + P(A∩B∩C).
The generalization to the case of Nobjects is now obvious. The relevant formula is
e
P12···N(t) = N
X
i=1
Pi(t)−
N
X
i<j
Pij(t, t) +
N
X
i<j<k
Pijk (t, t, t) + ··· (−1)N+1P12···N(t, t, · ··, t)(5.3)
where Pi1···ik(ti1,···tik) are again computed by solving ordinary diﬀerential equations in one
of the variables. Once e
P12···N(t) is computed we can get the life expectancy of the combined
system by using
¯
t12···N=Z∞
0e
P12···N(t)dt . (5.4)
5.2 Realistic trajectories
Another generalization involves considering a situation where multiple objects originate at
the same space time point and then follow diﬀerent trajectories, eventually settling down at
diﬀerent comoving coordinates. This has been illustrated in Fig. 14. This represents the
realistic situation since by deﬁnition diﬀerent civilizations of the same race must originate at
some common source. We can now generalize our analysis to take into account the possibility
of decay during the journey as well. Eqs.(5.3) and (5.4) still holds, but the computation of
Pi1···ik(ti1,···tik) will now have to be done by taking into account the details of the trajectories
of each object and the overlaps of their past light cones. The principle remains the same, and
we can set up ordinary diﬀerential equations for each of these quantities. The only diﬀerence
25

6
τ=−1
τ= 0
x
τ
Figure 14: Multiple objects originating from the same spacetime point. Diﬀerent dashed lines
represent the trajectories followed by diﬀerent objects.
is that the spatial separation between the ith object at τ=τiand the jth object at τ=τj
will now depend on τiand τjaccording to the trajectories followed by them.
This analysis can be easily generalized to the case where each of the descendant objects in
turn produces its own descendants which settle away from the parent object and eventually go
outside each other’s horizon due to the Hubble expansion. If this could be repeated at a rate
faster than the vacuum decay rate then we can formally ensure that some of the objects will
survive vacuum decay [16]. However, since within a few Hubble periods most of the universe
will split up into gravitationally bound systems outside each other’s horizon, in practice this
is going to be an increasingly diﬃcult task.
5.3 Matter eﬀect
A third generalization will involve relaxing the assumption that the universe has been de Sitter
throughout its past history. While de Sitter metric will be a good approximation after a few
Hubble periods, within the next few Hubble periods we shall still be sensitive to the fact that
the universe had been matter dominated in the recent past and had a beginning. This will
change the form of the metric (4.1) to
ds2=−dt2+a(t)2(dx2+dy2+dz2) (5.5)
where a(t) is determined from the Friedman equation
1
a
da
dt =r8πG
3ρΛ+ρm
a3(5.6)
26
in the convention that the value of ais 1 today and ρΛand ρmare the energy densities
due to cosmological constant and matter today. Since we have chosen the unit of time so
that the Hubble parameter in the cosmological constant dominated universe is 1, we have
p8πGρΛ/3 = 1. Deﬁning6
c≡ρm/ρΛ'0.45 (5.7)
we can express (5.6) as
1
a
da
dt =√1 + ca−3.(5.8)
Let τbe the conformal time deﬁned via
dτ=dt/a(t) (5.9)
with the boundary condition τ→0 as t→ ∞. Then (5.6) takes the form
1
a2
da
dτ=√1 + ca−3,(5.10)
whose solution is
τ=−Z∞
a
db
b2√1 + cb−3=−Z1/a
0
dv
√1 + cv3=−1
a2F11
3,1
2;4
3,−c
a3.(5.11)
This implicitly determines aas a function of τ. The metric is given by
ds2=a(τ)2(−dτ2+dx2+dy2+dz2).(5.12)
Using the experimental value c'0.45 we get that τ→τ0' −3.7 as a→0, showing that
the big bang singularity is at τ' −3.7.7We also have that at a= 1, τ' −0.95. This is
not very diﬀerent from the value τ=−1 for pure de Sitter spacetime with which we have
worked. However, we shall now show that the decay rate in the matter dominated epoch
of the universe diﬀers signiﬁcantly from that in the cosmological constant dominated epoch.
The decay rate of an isolated observer at some value of the conformal time τis given by the
following generalization of (4.2), (4.3):
d
dτln P0(τ) = −4πK Zτ
τ0
dσ a(σ)4(τ−σ)2
=−4πK Za(τ)
b=0
db
b2√1 + cb−3b4τ+1
b2F11
3,1
2;4
3,−c
b32
(5.13)
6We use cosmological parameters given in [18].
7Of course close to the singularity the universe becomes radiation dominated but given the short span of
radiation dominated era we ignore that eﬀect for the current analysis.
27
where in the second step we have changed the integration variable from σto b=a(σ). From
this we can compute the decay rate:
D(t)≡ − d
dt ln P0=−1
a(t)
d
dτln P0(τ)
=4πK
a(t)Za(t)
b=0
db
√1 + cb−3b2τ(t) + 1
b2F11
3,1
2;4
3,−c
b32
.(5.14)
Using the information that today a= 1 and τ' −0.95 we get
D(t)today'4πK
3×0.067 '0.067
T.(5.15)
This is lower than the corresponding rate T−1in the de Sitter epoch by about a factor of 15.
The growth of the decay rate with scale factor has been shown in Fig. 2.
Our analysis of §4can now be repeated for two or more observers and also for general tra
jectory discussed in §5.2 with this general form of the metric to get more accurate computation
of the life expectancy. These corrections will be important if T<
∼1 and the decay takes place
within a few Hubble period from now. On the other hand if Tis large (say >
∼10) then the
decay is likely to take place suﬃciently far in the future by which time the eﬀect of our matter
dominated past will have insigniﬁcant eﬀect on the results.
The fact that the decay rate increases with time till it eventually settles down to a constant
value in the de Sitter epoch has some important consequences:
1. We have already seen from (5.15) that the decay rate today is about 15 times smaller
than the decay rate in the de Sitter epoch. Eq.(5.14) for a(t) = 2 shows that even when
the universe will be double its size compared to today, the decay rate will remain at about
27% of the decay rate in the de Sitter epoch. Since most of the journeys to diﬀerent parts
of the universe – if they take place at all – are likely to happen during this epoch, we see
that the probability of decay during the journey will be considerably less than that in the
ﬁnal de Sitter phase. This partially justiﬁes our analysis in §4where we neglected the
probability of decay during the journey. This also shows that if we eventually carry out
a detailed numerical analysis taking into account the eﬀect discussed in §5.2, it should
be done in conjunction with the analysis of this subsection taking into account the eﬀect
of matter.
2. It is also possible to see from (5.14), (5.15) (or Fig. 2) that the decay rate in the past
was even smaller than that of today. If D(t) denotes the decay rate at time tdeﬁned in
28
(5.14), then the average decay rate in our past can be deﬁned as
1
t1Zt1
0
D(t)dt , (5.16)
where t1denotes the current age of the universe given by
t1=Z1
0
da da
dt −1
=Z1
0
da
a√1 + ca−3'0.79 .(5.17)
In physical units t1is about 1.38 ×1010 years. The evaluation of (5.16) can be facilitated
using the observation that D(t)dt is Ktimes the volume enclosed between the past light
cones of the object at times tand t+dt. Therefore, Rt1
0dtD(t) must be Ktimes the total
volume enclosed by the past light cone of the object at t1. This can be easily computed,
yielding
Zt1
0
D(t)dt =4
3π K Zτ
τ0
dσ a(σ)4(τ−σ)3
=1
TZa(τ)
b=0
db
b2√1 + cb−3b4τ+1
b2F11
3,1
2;4
3,−c
b33
.(5.18)
For c'0.45 and τgiven by today’s value −0.95 this gives
Zt1
0
D(t)dt =0.014
T.(5.19)
Using (5.16), (5.17) we get the average decay rate to be 0.018/T . This is about 3.7 times
smaller than the present decay rate given in (5.15) and 56 times smaller than the decay
rate 1/T in the de Sitter epoch.
Integrating the equation dP0/dt =−D(t)P0(t) we get
ln P0(t1) = −Zt1
0
dtD(t) = −0.014
T.(5.20)
Requiring this to be not much smaller than −1 (which is equivalent to requiring that the
inverse of the average decay rate (5.16) be not much smaller than the age of the universe
t1) gives T>
∼0.014. This is much lower than what one might have naively predicted by
equating the lower bound on Tto the age of the universe i.e. T>
∼t1∼0.79. Recalling
that the unit of time is set by the Hubble period in the de Sitter epoch which is about
1.7×1010 years, the bound T>
∼0.014 translates to a lower bound of order 2.5×108
29
years. Since the current decay rate is about 15 times smaller than that in the ﬁnal de
Sitter epoch, we see that the lower bound on the current inverse decay rate is of order
3.7×109years. This is comparable to the period over which the earth is expected to be
destroyed due to the expanded size of the Sun.
3. Finally we note that the above analysis was based on the assumption that the bubbles
continue to nucleate and expand in the FRW metric at the same rate as they would do
in the metastable vacuum. This will be expected as long as the matter and radiation
density and temperature are small compared to the microscopic scales involved in the
bubble nucleation process, e.g. the scale set by the negative cosmological constant of
the vacuum in the interior of the bubble. Some discussion on the eﬀect of cosmological
spacetime background on the bubble nucleation / evolution can be found in [19,20].
6 Discussion
We have seen that the result for how much we can increase the life expectancy by spreading
out in space depends on the parameters rand K, which in turn are determined by the Hubble
parameter of the de Sitter spacetime, the initial spread between diﬀerent objects and the
inverse decay rate. Therefore, the knowledge of these quantities is important for planning our
future course of action if we are to adapt this strategy for increasing the life expectancy of the
human race. In this section we shall discuss possible strategies for determining / manipulating
these quantities.
We begin with the Hubble expansion parameter H. This is determined by the cosmological
constant which has been quite well measured by now. Assuming that the current expansion
rate is of order 68Km/sec/Mpc and accounting for the fact that the cosmological constant
accounts for about 69% of the total energy density we get H−1'1.7×1010 years. Future
experiments will undoubtedly provide a more accurate determination of this number, but
given the uncertainty in the other quantities, this will not signiﬁcantly aﬀect our future course
of action. Of course we may discover that the dark energy responsible for the accelerated
expansion of the universe comes from another source, in which case we have to reexamine the
whole situation.
Next we turn to the initial separation between diﬀerent objects which determine the value
of r. Since in order for the Hubble expansion to be eﬀective in separating the objects they
have to be unbound gravitationally, a minimum separation between the objects is necessary for
30
overcoming the attractive gravitational force of the home galaxy. For example the size of our
local gravitationally bound group of galaxies is of order 5 million light years which correspond
to r∼3×10−4. The question is whether larger values of rcan be accessed. An interesting
analysis by Heyl [21] concluded that by building a spaceship that can constantly accelerate /
decelerate at a value equal to the acceleration due to gravity, we can reach values of rclose
to unity in less than 100 years viewed from the point of view of the spacetraveller. Of course
this will be close to about 1010 years viewed from earth, and roughly the reduction of time
viewed from the spaceship can be attributed to the large time dilation at the peak speed of
the spaceship reaching a value close to that of light. However, this large time dilation will
also increase the eﬀective temperature of the microwave background radiation in the forward
direction and without a proper shield such a journey will be impossible to perform. If one
allows a maximum time dilation of the order of 100 then the microwave temperature in the
forward direction rises to about the room temperature. Even then we have to worry about
the result of possible collisions with intergalactic dust and othe debris in space. Even if these
problems are resolved, we shall need a time of order 108years from the point of view of the
spaceship to travel a distance of order 1010 light years. Even travelling the minimum required
distance of order 107light years will take 105years in such a spaceship. Such a long journey
in a spaceship does not seem very practical but may not be impossible.
Another interesting suggestion for populating regions of spacetime which will eventually be
outside each other’s horizon has been made by Loeb [22]. Occasionally there are hypervelocity
stars which escape our galaxy (and the cluster of galaxies which are gravitationally bound)
and so if we could ﬁnd a habitable planet in such a star we could take a free ride in that planet
and escape our local gravitationally bound system. In general of course there is no guarantee
that such a star will reach another cluster of galaxies where we could spread out and thrive,
but some time we may be lucky. It has been further suggested in [23,24] that the merger of
Andromeda and the Milky Way galaxies in the future [25] could generate a large number of
such hypervelocity stars travelling at speeds comparable to that of light and they could travel
up to distances of the order of 109light years by the time they burn out. This could allow us
to achieve values of rof order 10−1or more.
Let us now turn to the value of Tor equivalently the decay rate of the de Sitter vacuum
in which we currently live. This is probably the most important ingredient since we have
seen that for T<
∼1 we do not gain much by spreading out, while for large enough Twe can
achieve the maximum possible gain, given by the harmonic numbers, by spreading out even
31
over modest distances of r∼10−3. At the same time if Tis so large that it exceeds the period
over which galaxies will die then vacuum decay may not have a signiﬁcant role in deciding our
end and we should focus on other issues. For this reason estimating the value of Tseems to be
of paramount importance. Unfortunately, due to the very nature of the vacuum decay process
it is not possible to determine it by any sort of direct experiment since such an experiment will
also destroy the observer. It may be possible in the future to device clever indirect experiments
to probe vacuum instability without actually causing the transition to the stable vacuum, but
no such scheme is known at present.
At a crude level the current age of the universe – which is about 0.79 times the asymptotic
Hubble period in the cosmological constant dominated epoch – together with the assumption
that we have not been extremely lucky to survive this long, suggests that the inverse decay
rate of the universe is >
∼0.8. However, (5.18) shows that this actually gives a lower bound of
T>
∼0.014 reﬂecting the fact that the decay rate in the de Sitter epoch will be about 56 times
faster than the average decay rate in the past. If we allow for the possibility that we might
have been extremely lucky to have survived till today, then we have indirect arguments that
lower the bound by a factor of 10 [13]. Clearly these rates are too fast and if Treally happens
to be less than 1, then there is not much we can do to prolong our collective life.
Is there any hope of computing Ttheoretically? Unfortunately any bottom up approach
based on the analysis of low energy eﬀective ﬁeld theory is insuﬃcient for this problem. The
reason for this is that the vacuum decay rate is a heavily ultraviolet sensitive quantity. Given
a theory with a perfectly stable vacuum we can add to it a new heavy scalar ﬁeld whose eﬀect
will be strongly suppressed at low energy, but which can have a potential that makes the
vacuum metastable with arbitrarily large decay rate. For this reason the only way we could
hope to estimate Tis through the use of a top down approach in which we have a fundamental
microscopic theory all of whose parameters are ﬁxed by some fundamental principle, and then
compute the vacuum decay rate using standard techniques. In the context of string theory this
will require ﬁnding the vacuum in the landscape that describes our universe. Alternatively, in
the multiverse scenario, we need to carry out a statistical analysis that establishes that the
overwhelming majority of the vacua that resemble our vacua will have their decay rate lying
within a narrow range. This can then be identiﬁed as the likely value of the decay rate. There
have been attempts in this direction [26–29], but it is probably fair to say that we do not yet
have a deﬁnite result based on which we can plan our future course of action.
Acknowledgement: We would like to thank Adam Brown, Rajesh Gopakumar and Abra
32
ham Loeb for useful discussions. This work was supported in part by the DAE project 12
R&DHRI5.020303. The work of A.S. was also supported in part by the J. C. Bose fellowship
of the Department of Science and Technology, India and the KIAS distinguished professorship.
The work of MV was also supported by the SPM research grant of the Council for Scientiﬁc
and Industrial Research (CSIR), India.
A Decay rate for equation of state p=w ρ
In this appendix we shall compute the growth of decay rate with time for a general equation
of state of the form p=w ρ with w > −1. In this case the ρand aare related as
ρ=ρ0a−3(w+1) ,(A.1)
for some constant ρ0. As a result the dependence of aon tand τare determined by the
equations
1
a
da
dt =r8πG
3ρ=C a−3(w+1)/2, C ≡r8πG
3ρ0,(A.2)
and 1
a2
da
dτ=C a−3(w+1)/2.(A.3)
The solutions to these equations are
t=2
3C(w+ 1)a3(w+1)/2,τ=2
(3w+ 1)Ca(3w+1)/2.(A.4)
As aranges from 0 to ∞, both tand τalso range from 0 to ∞.
We can now compute the decay rate using the ﬁrst equation of (5.13):
D(t)≡ −1
a
d
dτln P0(τ)
=1
a4πK Zτ
0
dσ a(σ)4(τ−σ)2,(A.5)
where a(σ) denotes the scale factor at conformal time σ. Changing integration variable to
b=a(σ), which due to (A.4) corresponds to
σ=2
(3w+ 1)Cb(3w+1)/2,(A.6)
33
we can express (A.5) as
D(t) = 4πK a−12
(3w+ 1)C33w+ 1
2Za
0
db b4b(3w−1)/2a(3w+1)/2) −b(3w+1)/22
=32πK
3C3a9(w+1)/21
(w+ 3)(3w+ 5)(9w+ 11) .(A.7)
Using the relation between aand tgiven in (A.4), this can be expressed as
D(t) = 36 π K (w+ 1)3
(w+ 3)(3w+ 5)(9w+ 11) t3.(A.8)
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