UFIFT-QG-23-04

Big Steve and the State of the Universe

R. P. Woodard†

Department of Physics, University of Florida,
Gainesville, FL 32611, UNITED STATES

ABSTRACT

I share some reminiscences of the late Steven Weinberg. Then I discuss a topic
in quantum field theory which he taught me: the role of state wave functionals
in deriving the $i\epsilon$ term of the Feynman propagator when using the
functional formalism. This is perhaps a curiosity for in-out scattering
amplitudes on flat space backgrounds, but it is has much greater significance
for the in-in amplitudes of the Schwinger-Keldysh formalism in cosmology. It
also touches on the fate, about which Weinberg wondered, of the large
logarithms one sometimes finds in quantum corrections from inflationary
particle production.

PACS numbers: 04.50.Kd, 95.35.+d, 98.62.-g

† e-mail<EMAIL_ADDRESS>
##  1 Reminiscences of Big Steve

Physicists recall the oddest things about Steven Weinberg, often having
nothing to do with his towering scientific achievements. I first heard him
speak at a colloquium at MIT, during September of 1977. I had just met Nick
Tsamis, a fellow freshman grad student who became Weinberg’s final student at
Harvard. Nick would also become my best friend, and a collaborator with whom I
have so far written 71 papers. Fall classes had not yet begun so we hiked over
to nearby MIT to hear Weinberg speak on spontaneous symmetry breaking. My
memory may have faded but I believe he was introduced by Viki Weisskopf, who
recalled Weinberg’s greatest contribution during his six years at MIT
(1967-1973). Of course that period witnessed the paper which would earn the
1979 Nobel Prize, but Weisskopf passed this over to instead commend Weinberg’s
insistence that one should be able to open the windows of MIT’s Center for
Theoretical Physics! The other thing I recall from that colloquium is that
Weinberg illustrated spontaneous symmetry breaking using the example of a
cigar: both ends are equivalent but one end is lighted and the other is
smoked.

Like many Harvard students, I took to calling Weinberg “Big Steve”. He taught
my first course on quantum field theory during the 1977-78 academic year and
it was one of the seminal experiences of my life. We students had a saying
that you must take QFT three times before really understanding it. In the end
I took it only twice, and if I have understood the subject, it is due to the
genius of the men who taught me: first, Steven Weinberg and then, Sidney
Coleman. Their styles were interestingly different. Whereas Sidney emphasized
mathematical elegance, Big Steve proceeded from universal principles and
erected a workmanlike framework of mutually reinforcing theory. His approach
reminded me of the enormous structural strength evident in bridges which
survive from the early industrial period of the United States.

Big Steve’s QFT course was a mind-expanding experience. After most lectures I
would stumble back to my room in Perkins Hall and lay there, staring at the
empty ceiling as my mind struggled to process the powerful concepts that were
being impressed upon it. The course also had humorous aspects. Harvard policy
required faculty to provide office hours, so Weinberg announced that his would
take place right after class on Tuesdays and Thursdays. However, he didn’t
really want to sit in his office for questions which might never come, so each
lecture ended with him apologizing that a dental appointment precluded holding
office hours that day. That same dental appointment stretched throughout the
entire 1977-78 academic year! The funny thing is, I don’t recall anyone
complaining: Big Steve’s lectures were so brilliantly clear that no office
hours were necessary.

History was important to Big Steve, and he made effective use of it in writing
and in teaching. His 1972 text on Gravitation and Cosmology prefaces its
historical review of mankind’s long struggle to perceive order in the cosmos
with a passage from Anna Comnena’s encomium to her father, who spent his life
restoring order from the chaos which followed Manzikert. Weinberg devoted the
first lectures of his QFT course to a historical review of quantum field
theory, with particular attention to the many attempts at supplanting it. His
message was that these alternatives always ended being recognized as either
incorrect, or else consequences of quantum field theory. Listening to Weinberg
speak made you share his conviction that QFT is the finest thing our species
ever created. I recall an Arthur C. Clarke novel, set more than a billion
years in the future, in which a character casually comments about needing to
consult a “field theory expert” to understand a particular device. You could
believe in that future after hearing Weinberg lecture.

Big Steve was full of historical anecdotes, such as Dirac’s explanation for
failing to quantize the photon field because he was afraid something “would go
wrong”. In fact, something did go wrong when the first loop computations
revealed ultraviolet divergences, and some physicists over-reacted by
rejecting quantum field theory. Then came the Shelter Island Conference, after
which the leading theorists of the day began to puzzle out renormalization.
Weinberg recounted [1] how students poked fun at the erroneous dismissal of
loop corrections with the quip, “Just because something is infinite does not
mean it is zero!” That story about divergences meant a lot to me when I later
encountered skepticism about large logarithmic loop corrections from
inflationary gravitons [2, 3, 4, 5, 6, 7, 8] because of the potential for
gauge dependence [9, 10, 11, 12]. A procedure for removing gauge dependence is
being developed [13, 14], and I intend to channel Big Steve when announcing
its success: “Just because something is gauge dependent does not mean it is
zero!”

Weinberg never questioned the reality of large logarithms when he discovered
them in loop corrections to the primordial power spectrum. Quoting from the
second of his famous papers on cosmological correlators [15]:

> In generic theories the $N$ integrals over time in $N$-th order perturbation
> theory will yield correlation functions at time $t$ that grow as
> $\ln^{N}[a(t)]$. Such a power series in $\ln[a(t)]$ can easily add up to a
> time dependence that grows like a power of $a(t)$, or even more
> dramatically. As everyone knows, the series of powers of the logarithm of
> energy encountered in various flat-space theories such as quantum
> chromodynamics can be summed by the method of the renormalization group. It
> will be interesting to see if the power series in $\ln[a(t)]$ encountered in
> calculating cosmological correlation functions at time $t$, though arising
> here in a very different way, can be summed by similar methods.

Some disagree [16], but I think Big Steve was right about the basic physics
[17]. And I hope he would appreciate the procedure being devised for
implementing the resummation whose potential he foresaw [18, 19].

Everyone who had the privilege of working with Weinberg knows the enthusiasm
he brought to research, and his determination to overcome all obstacles. I
well recall the time he asked me to visit the UT in 2005, in order to consult
about his work on loop corrections to the primordial power spectrum [20]. He
had just realized that the in-out amplitudes of conventional quantum field
theory are not the appropriate objects of study in cosmology because they are
dominated by assumptions about the yet-unknown future, and because even the
matrix elements of Hermitian operators are not generally real. I cannot count
the number of times I have grown irritated listening to intellectually
dishonest colleagues attempt to avoid these problems by devising tricks to
make the in-out formalism accomplish something for which it was never
intended, and for which it is not well suited. Not Big Steve. He understood
that true expectation values are the better object of study in cosmology and,
not knowing of the Schwinger-Keldysh formalism [21, 22, 23, 24, 25, 26, 27,
28], he devised a Hamiltonian technique for computing them [20], which is now
more commonly used than Schwinger’s method. Weinberg only learned of
Schwinger’s work from his then-student, Bua Chaicherdsakul, and he of course
gave full credit to Schwinger thereafter.

Aside from being blown away by Big Steve having invented his own version of
the Schwinger-Keldysh formalism, the incident I most recall from that visit
came during the morning of my final day. I had gotten up early and was waiting
in the hotel before departing for the airport. The phone rings and who should
it be but Steven Weinberg, apologizing for the early call and asking to
discuss a physics issue which had been troubling him. Another passage from
that Clarke novel comes to mind, about the men of the Dawn Ages never
permitting problems to hold them up for very long.

I’ll close by recalling part of the final e-mail notes we exchanged. I had
written to congratulate him on winning the Breakthrough Prize, and to share
some sad news about a mutual friend. That was September of 2020, when the
pandemic still dominated our lives, and I mentioned the travel bans which had
kept my wife and me apart for six months. Weinberg replied:

> I am glad that you and your wife are together again. My wife and I are
> together, fanatically isolated at home, but both of us getting a lot of work
> done and staying safe.

He was 87 at the time! I recall thinking how fine it would be, should I chance
to reach that age, to remain so active and so passionate about physics.111I
count seventeen books and papers written after he had turned 80 [29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]! Big Steve was a force
of nature; it was a privilege to have known him.

##  2 The True Origin of the $i\epsilon$

Weinberg believed that everything about quantum field theory should be
derived. He used the term “folk theorems” to describe commonly accepted
beliefs for which no derivation was currently available. I have chosen the
topic for this article to be a minor but irritating lacuna in the derivation
of propagators from the functional formalism. It was a point Big Steve derived
for us back in 1977-78, which is usually resolved by hand-waving.222A Russian
colleague pointed out a similar derivation by Slavnov and Faddeev [46]. I
refer to the famous “$i\epsilon$” part of the Feynman propagator. To make the
discussion transparent I will work in the context of a Simple Harmonic
Oscillator whose position is $q(t)$, and whose dynamics are controlled by the
Lagrangian,

$L=\frac{1}{2}m\dot{q}^{2}-\frac{1}{2}m\omega^{2}q^{2}\;.$ (1)

The propagator we will use the functional integral formalism to derive is,

$\displaystyle
i\Delta(t;t^{\prime})\equiv\Bigl{\langle}\Omega\Bigl{|}T\Bigl{[}q(t)q(t^{\prime})\Bigr{]}\Bigr{|}\Omega\Bigr{\rangle}=\frac{\hbar\,e^{-i\omega|t-t^{\prime}|}}{2m\omega}\;,$
(3) $\displaystyle=\\!\\!\lim_{\epsilon\rightarrow
0^{+}}\\!\frac{\hbar}{m}\\!\\!\int_{-\infty}^{\infty}\\!\\!\frac{dk^{0}}{2\pi}\frac{ie^{-ik^{0}(t-t^{\prime})}}{(k^{0}\\!-\\!\omega\\!+\\!i\epsilon)(k^{0}\\!+\\!\omega\\!-\\!i\epsilon)}=\\!\\!\lim_{\epsilon\rightarrow
0^{+}}\\!\frac{\hbar}{m}\\!\\!\int_{-\infty}^{\infty}\\!\\!\frac{dk^{0}}{2\pi}\frac{ie^{-ik^{0}(t-t^{\prime})}}{(k^{0})^{2}\\!-\\!\omega^{2}\\!+\\!i\epsilon}.\qquad$

Of course there is absolutely no ambiguity about the canonical derivation,
which proceeds from (3) to (3). The issue is how we would get (3) directly
from the functional integral formalism,

$\hbox
to0.0pt{$\Biggl{\rfloor}$\hss}\Biggl{\lceil}[dq]\,q(t)q(t^{\prime})\exp\Biggl{[}\frac{i}{\hbar}\int_{-\infty}^{\infty}\\!\\!\\!\\!\\!\\!ds\Bigl{\\{}\frac{1}{2}m\dot{q}^{2}(s)-\frac{1}{2}m\omega^{2}q^{2}(s)\Bigr{\\}}\Biggr{]}\;.$
(4)

The short answer is that it just isn’t possible. Although expression (4)
should result in some function which obeys the propagator equation,

$\frac{im}{\hbar}\Bigl{(}\frac{d^{2}}{dt^{2}}+\omega^{2}\Bigr{)}i\Delta(t;t^{\prime})=\delta(t\\!-\\!t^{\prime})\;,$
(5)

it is not at all clear how $i\Delta(t;t^{\prime})$ acquires the correct real
part from an exponent which is entirely imaginary. Some QFT texts argue that
the $i\epsilon$ follows from complex analysis, by the need to deform the
functional integration over $q(s)$ into the complex plane. However, this is
problematic when one considers the four different propagators of the
Schwinger-Keldysh formalism which each follow from a functional integral.
Other texts argue that the $i\epsilon$ has something to do with the temporal
integration running from $-\infty$ to $+\infty$, but this is also problematic
when one considers that the free Lagrangian (1) must produce the very same
propagator (3), no matter when the initial and final states are specified. Big
Steve had no patience with this sort of hand-waiving, and he discovered a
better explanation.

The key is incorporating the ground state wave function in the functional
integral expression (4). For the simple harmonic oscillator (1) this is,

$\Omega(q)=\Bigl{[}\frac{\pi
m\omega}{\hbar}\Bigr{]}^{\frac{1}{4}}\exp\Bigl{[}-\frac{m\omega}{2\hbar}q^{2}\Bigr{]}\;.$
(6)

Let us also make the range of temporal integration finite, and employ a
classical source $J(s)$,

$\displaystyle Z[J]\equiv\hbox
to0.0pt{$\Biggl{\rfloor}$\hss}\Biggl{\lceil}[dq]\,\Omega^{*}\Bigl{(}q(t_{f})\Bigr{)}$
(7) $\displaystyle\hskip
28.45274pt\times\exp\Biggl{[}\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}\\!\\!\\!\\!\\!\\!ds\Bigl{\\{}\frac{1}{2}m\dot{q}^{2}(s)-\frac{1}{2}m\omega^{2}q^{2}(s)+J(s)q(s)\Bigr{\\}}\Biggr{]}\Omega\Bigl{(}q(t_{i})\Bigr{)}\;.\qquad$

Note that the functional integration in expression (7) only extends over
functions $q(s)$ for $t_{i}\leq s\leq t_{f}$. We should be able to recover the
propagator (3) by functionally differentiating $Z[J]$,

$i\Delta(t;t^{\prime})=\frac{(-i\hbar)^{2}\delta^{2}Z[J]}{\delta J(t)\delta
J(t^{\prime})}\Biggl{|}_{J=0}\;.$ (8)

Expression (7) is the functional integral of $\exp[\frac{i}{\hbar}E[q,J]]$,
with exponent,

$E[q,J]=\frac{im\omega}{2}q^{2}(t_{f})+\int_{t_{i}}^{t_{f}}\\!\\!\\!\\!ds\Bigl{[}\frac{1}{2}m\dot{q}^{2}(s)-\frac{1}{2}m\omega^{2}q^{2}(s)+J(s)q(s)\Bigr{]}+\frac{im\omega}{2}q^{2}(t_{i})\;.$
(9)

The result of any Gaussian integral is the exponential evaluated at its
stationary point. To find this stationary point we vary (9), taking care to
include the surface variations,

$\displaystyle\frac{\delta E[q,J]}{\delta
q(t)}=m\delta(t\\!-\\!t_{f})\Bigl{[}\dot{q}(t_{f})\\!+\\!i\omega
q(t_{f})\Bigr{]}$ (10) $\displaystyle\hskip
71.13188pt-m\Bigl{[}\ddot{q}(t)\\!+\\!\omega^{2}q(t)\\!-\\!\frac{J(t)}{m}\Bigr{]}-m\delta(t\\!-\\!t_{i})\Bigl{[}\dot{q}(t_{i})\\!-\\!i\omega
q(t_{i})\Bigr{]}\;.\qquad$

Note that the surface variations enforce Feynman boundary conditions, which is
the part one couldn’t get without including the initial and final states.
Setting (10) to zero has the unique solution,

$q[J](t)=\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}\\!\\!\\!\\!dt^{\prime}\,i\Delta(t;t^{\prime})J(t^{\prime})\;.$
(11)

Partial integration makes it simple to evaluate the exponent (9) at its
stationary point,

$\displaystyle
E\Bigl{[}q[J],J\Bigr{]}=\frac{im}{2}q(t_{f})\Bigl{[}\dot{q}(t_{f})\\!+\\!i\omega
q(t_{f})\Bigr{]}+\int_{t_{i}}^{t_{f}}\\!\\!\\!\\!dt\Biggl{\\{}-\frac{1}{2}mq(t)\Bigl{[}\ddot{q}\\!+\\!\omega^{2}q(t)\\!-\\!\frac{J(t)}{m}\Bigr{]}$
(12) $\displaystyle\hskip
128.0374pt+\frac{1}{2}q(t)J(t)\Biggr{\\}}+\frac{im}{2}q(t_{i})\Bigl{[}\dot{q}(t_{i})\\!-\\!i\omega
q(t_{i})\Bigr{]}\;,\qquad$ $\displaystyle=$
$\displaystyle\frac{i}{2\hbar}\int_{t_{i}}^{t_{f}}\\!\\!\\!\\!dtJ(t)\\!\\!\int_{t_{i}}^{t_{f}}\\!\\!\\!\\!dt^{\prime}J(t^{\prime})\,i\Delta(t;t^{\prime})\;.$
(13)

Hence we conclude,

$\ln\Bigl{[}Z[J]\Bigr{]}=-\frac{1}{2\hbar^{2}}\int_{t_{i}}^{t_{f}}\\!\\!\\!\\!dtJ(t)\\!\\!\int_{t_{i}}^{t_{f}}\\!\\!\\!\\!dt^{\prime}J(t^{\prime})\,i\Delta(t;t^{\prime})\;,$
(14)

which obviously gives the Feynman propagator in expression (8).

The generalization to quantum field theory is straightforward; in making it I
shall adopt the usual convention of setting $\hbar=c=1$. The Lagrangian
density of a free scalar field on flat space with spacelike signature is,

$\mathcal{L}=-\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi-\frac{1}{2}m^{2}\varphi^{2}\;.$
(15)

It seems that application of quantum field theory brings me back to something
else Big Steve taught us: that all free theories become simple harmonic
oscillators in spatial Fourier coordinates,

$\widetilde{\varphi}(t,\vec{k})\equiv\int\\!\\!d^{3}x\,e^{i\vec{k}\cdot\vec{x}}\varphi(t,\vec{x})\;.$
(16)

We can identify each modes’ mass and frequency by using Parseval’s theorem on
the Lagrangian,

$L\equiv\int\\!\\!d^{3}x\mathcal{L}=\int\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\Bigl{\\{}\frac{1}{2}\dot{\widetilde{\varphi}}(t,\vec{k})\dot{\widetilde{\varphi}}^{*}(t,\vec{k})+\frac{1}{2}(k^{2}\\!+\\!m^{2})\widetilde{\varphi}(t,\vec{k})\widetilde{\varphi}^{*}(t,\vec{k})\Bigr{\\}}\;.$
(17)

Comparison with (1) implies the identifications,

$\omega\rightarrow\sqrt{k^{2}+m^{2}}\qquad,\qquad
m\longrightarrow\frac{d^{3}k}{(2\pi)^{3}}\;.$ (18)

Hence the ground state wave functional is,

$\displaystyle\Omega\Bigl{[}\varphi(t)\Bigr{]}$
$\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N\exp\Biggl{[}\frac{1}{2}\sum_{\vec{k}}m\omega\,\widetilde{\varphi}(t,\vec{k})\widetilde{\varphi}^{*}(t,\vec{k})\Biggr{]}\;,$
(19) $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N\exp\Biggl{[}\frac{1}{2}\int\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\,\widetilde{\varphi}(t,\vec{k})\sqrt{k^{2}+m^{2}}\,\widetilde{\varphi}^{*}(t,\vec{k})\Biggr{]}\;,$
(20) $\displaystyle\\!\\!\\!=\\!\\!\\!$ $\displaystyle
N\exp\Biggl{[}\frac{1}{2}\int\\!\\!d^{3}x\,\varphi(t,\vec{x})\sqrt{-\nabla^{2}+m^{2}}\,\varphi(t,\vec{x})\Biggr{]}\;.$
(21)

##  3 State Wave Functionals in Cosmology

I hope people will agree that Big Steve’s functional derivation of the
$i\epsilon$ in the propagator is far superior to the usual hand-waiving.
However, we already knew the answer from the canonical formalism. The real
efficacy of state wave functionals becomes evident when applying the
Schwinger-Keldysh formalism to study the time evolution of expectation values
in cosmology.

###  3.1 Cosmological Particle Production

Quantum field theory in cosmology is much more interesting than its flat space
cousin owing to what Schrödinger termed “the alarming phenomena” of particle
production “caused by accelerated expansion.” [47, 48, 49, 50]. To understand
this, consider a homogeneous, isotropic and spatially flat geometry with scale
factor $a(t)$, Hubble parameter $H(t)$ and first slow roll parameter
$\epsilon(t)$,

$ds^{2}=-dt^{2}+a^{2}(t)d\vec{x}\\!\cdot\\!d\vec{x}\qquad\Longrightarrow\qquad
H(t)\equiv\frac{\dot{a}}{a}\quad,\quad\epsilon(t)\equiv-\frac{H}{H^{2}}\;.$
(22)

Students of QFT are familiar with how the Energy-Time Uncertainty Principle
permits a virtual particle of energy $E=\sqrt{k^{2}+m^{2}}$ to exist for a
time $\Delta t\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}1/E$. The simplest way of understanding many QFT
effects is by positing the existence of these virtual particles and then using
classical field theory, from which it follows that the strongest effects come
from virtual particles with the longest persistence times $\Delta t$. An
expanding universe ($H>0$) strengthens QFT effects because the momentum $k$
redshifts to increase the persistence time, which is governed by the integral,

$\int_{t}^{t+\Delta
t}\\!\\!\\!\\!\\!\\!\\!\\!\\!dt^{\prime}\sqrt{\frac{k^{2}}{a^{2}(t^{\prime})}+m^{2}}\qquad\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}\qquad 1\;.$ (23)

Just as in flat space, the longest-lived virtual particles are the lightest.
Taking the massless limit gives,

$\int_{t}^{t+\Delta
t}\\!\\!\\!\\!\\!\\!\\!\\!\\!dt^{\prime}\frac{k}{a(t^{\prime})}\qquad\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}\qquad 1\;.$ (24)

For accelerated expansion ($\epsilon<1$) the integral converges even as
$\Delta t\rightarrow\infty$. Hence a sufficiently long wave length virtual
particle can live forever.

Although correct, the preceding discussion fails to account for the rate
$dN/dt$ at which virtual particles emerge from the vacuum. This is very
important because almost all massless particles possess a killer symmetry
known as conformal invariance which means they cannot tell the difference
between a metric $g_{\mu\nu}(x)$ and another $\Omega^{2}(x)\times
g_{\mu\nu}(x)$. Changing the time coordinate $t$ of the cosmological geometry
(22) to “conformal time” $\eta$ with $d\eta\equiv dt/a(t)$ makes the geometry
conformal to flat space,

$ds^{2}=-dt^{2}+a^{2}d\vec{x}\\!\cdot\\!d\vec{x}=a^{2}\Bigl{[}-d\eta^{2}+d\vec{x}\\!\times\\!d\vec{x}\Bigr{]}\;.$
(25)

This means that the emergence rate in conformal coordinates is the same as in
flat space,

$\frac{dN}{d\eta}=\Gamma_{\rm
flat}\qquad\Longrightarrow\qquad\frac{dN}{dt}=\frac{dN}{d\eta}\\!\cdot\\!\frac{d\eta}{dt}=\frac{1}{a(t)}\\!\cdot\\!\Gamma_{\rm
flat}\;.$ (26)

Therefore, any sufficiently long wave length, massless particle which emerges
during accelerated expansion can exist forever, but very few emerge.

Massless fermions and vector gauge bosons are conformally invariant, and so
are conformally coupled scalars. These particles do not, by themselves, give
rise to interesting effects in cosmology. However, massless, minimally coupled
scalars are not conformally invariant, nor are gravitons, which obey the same
linearized equation of motion [51]. On de Sitter background (which means
$\epsilon=0$, $H$ is constant and $a(t)=e^{Ht}$) the occupation number for a
scalar of wave vector $\vec{k}$ (or each polarization of graviton) is
staggering,

$N(t,\vec{k})=\Bigl{[}\frac{Ha(t)}{2\|\vec{k}\|}\Bigr{]}^{2}\;.$ (27)

In addition to the obvious exponential growth, note also that the occupation
number only becomes of order one at horizon crossing, $\|\vec{k}\|\equiv
k=Ha(t)$. This is very important because it guarantees that cosmological
particle production is an infrared effect which can be studied reliably using
general relativity as a low energy effective field theory, without needing its
unknown ultraviolet completion.

The explosive growth evident in expression (27) is the origin of two
observables from the epoch of primordial inflation: the power spectra of
gravitons [55] and scalars [56]. In fact, the occupation number (27) can be
expressed in terms of the tensor power spectrum $\Delta^{2}_{h}(k)$, not just
for de Sitter ($\epsilon=0$) but for general $\epsilon<1$,

$N(t,\vec{k})\longrightarrow\frac{\pi\Delta^{2}_{h}(k)}{64Gk^{2}}\\!\times\\!a^{2}(t)\;.$
(28)

The power spectra are tree order quantum effects but the quanta from which
they derive must interact with themselves and with other particles. In certain
cases these interactions lead to time dependent effects which grow as more and
more particles are ripped out of the vacuum.

The choice of vacuum is so contentious in cosmological QFT that I had better
explain how expression (27) was derived. The spatial integral of the
Lagrangian density of a massless, minimally coupled scalar
$\varphi(t,\vec{x})$ can be written, using Parseval’s theorem, as a Fourier
mode sum of independent harmonic oscillators $\widetilde{\varphi}(t,\vec{k})$,

$\int\\!\\!d^{3}x\,a^{3}\Bigl{[}\frac{1}{2}\dot{\varphi}^{2}-\frac{1}{2a^{2}}\|\vec{\nabla}\varphi\|^{2}\Bigr{]}=\int\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\,a^{3}\Bigl{[}\frac{1}{2}|\widetilde{\varphi}|^{2}-\frac{k^{2}}{2a^{2}}|\widetilde{\varphi}|^{2}\Bigr{]}\;.$
(29)

Each wave vector $\vec{k}$ corresponds to an independent harmonic oscillator
with time-dependent mass and frequency,

$m(t)\longrightarrow\frac{d^{3}k}{(2\pi)^{3}}\\!\times\\!a^{3}(t)\qquad,\qquad\omega(t,k)\longrightarrow\frac{k}{a(t)}\;.$
(30)

At any instant this system is a harmonic oscillator, so we can define the
instantaneous occupation number $N$ by writing the expectation value of its
energy as $E=\omega(\frac{1}{2}+N)$. For the de Sitter geometry the mode
function $u(t,k)$ is simple and we find,

$\displaystyle
u(t,k)=\frac{H}{\sqrt{2k^{3}}}\Bigl{[}1-\frac{ik}{aH}\Bigr{]}\exp\Bigl{[}\frac{ik}{aH}\Bigr{]}=\frac{H}{\sqrt{2k^{3}}}\Bigl{[}1+ik\eta\Bigr{]}e^{-ik\eta}$
(31) $\displaystyle\hskip
113.81102pt\Longrightarrow\frac{1}{2}a^{3}\Bigl{[}|\dot{u}|^{2}+\frac{k^{2}}{a^{2}}|u|^{2}\Bigr{]}=\frac{k}{a}\Bigl{[}\frac{1}{2}+\Bigl{(}\frac{aH}{2k}\Bigr{)}^{2}\Bigr{]}\;.\qquad$

The mode function (31) is known as Bunch-Davies vacuum [52, 53, 54], and one
can see from the occupation number (27) that it corresponds to a state which
was empty in the distant past when $a(t)\rightarrow 0$.

###  3.2 Time Dependence in Cosmological QFT

Accelerated expansion changes cosmological quantum field theory in profound
ways. First, it is nonsense to base the theory on asymptotic scattering
experiments. At least classically, the universe began with a singularity at
some finite time, and no one knows how it will end. Fixtures of flat space QFT
such as Euclideanization and defining “the vacuum” as “the unique,
normalizable energy eigenstate” come to seem quaint. Instead of in-out matrix
elements between states which were free at asymptotically early and late
times, we must become accustomed to computing true expectation values in the
presence of states which are defined at some finite time. Finally, the
explosive production of massless, minimally coupled scalars and gravitons
evident in expression (27) implies that we must expect time-dependent effects
as more and more virtual particles emerge from the vacuum.

Consider a self-interacting scalar field on de Sitter background,

$\mathcal{L}=-\frac{1}{2}\partial_{\mu}\varphi\partial_{\nu}\varphi
g^{\mu\nu}\sqrt{-g}-\frac{1}{4}\lambda\varphi^{4}\sqrt{-g}\;.$ (32)

The first dimensionally regulated computation I ever did on de Sitter
background was the expectation value of this model’s stress-energy tensor at 1
and 2-loop order [57, 58],

$T_{\mu\nu}=\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\partial_{\rho}\varphi\partial_{\sigma}\varphi-\frac{1}{4}g_{\mu\nu}\lambda\varphi^{4}\;.$
(33)

Although the vacuum is not unique, most people work on Bunch-Davies vacuum,
which we saw from (31) was empty in the distant past. However, cosmological
particle production means that we must release the state at some finite time
which can be taken to be $t=0$, at which point the de Sitter scale factor
$a(t)=e^{Ht}$ is unity.

Because the state is homogeneous and isotropic, the expectation value takes
the perfect fluid form,

$\Bigl{\langle}\Omega\Bigl{|}T_{\mu\nu}(t,\vec{x})\Bigr{|}\Omega\Bigr{\rangle}=u_{\mu}u_{\nu}\\!\times\\!\Bigl{[}\rho(t)\\!+\\!p(t)\Bigr{]}+g_{\mu\nu}\\!\times\\!p(t)\qquad,\qquad
u_{\mu}\equiv\delta^{0}_{~{}\mu}\;.$ (34)

The renormalized result consists of a “simple” part which grows with time, or
remains constant, plus a “complicated” part that falls off exponentially [59],

$\displaystyle\rho(t)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{3H^{2}}{8\pi G}+\frac{\lambda
H^{4}}{(2\pi)^{4}}\Biggl{\\{}2\ln^{2}(a)+\frac{13}{6}\ln(a)-\frac{43}{18}+\frac{\pi^{2}}{3}$
(35) $\displaystyle\hskip
153.6447pt-\frac{3}{2a^{2}}-2\sum_{n=4}^{\infty}\frac{(n\\!+\\!1)}{n^{2}a^{n}}\Biggr{\\}}+O(\lambda^{2})\;,\qquad$
$\displaystyle p(t)$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{3H^{2}}{8\pi G}+\frac{\lambda
H^{4}}{(2\pi)^{4}}\Biggl{\\{}-2\ln^{2}(a)-\frac{7}{2}\ln(a)+\frac{5}{3}-\frac{\pi^{2}}{3}$
(36) $\displaystyle\hskip
125.19194pt+\frac{1}{2a^{2}}-\frac{2}{3}\sum_{n=4}^{\infty}\frac{(n\\!-\\!3)(n\\!+\\!1)}{n^{2}a^{n}}\Biggr{\\}}+O(\lambda^{2})\;.\qquad$

A nice check on accuracy is that (35-36) obey conservation,
$\dot{\rho}=-3(\rho+p)$.

###  3.3 Eliminating the Divergence with the Initial State

In addition to being complicated, the infinite sums actually cause expressions
(35-36) to diverge at $t=0$, for which $a=1$. These divergences have nothing
to do with inflationary particle production. They derive instead from the
initial state having been free, in spite of the interaction. Similar
divergences would occur in flat space QFT if we had specified the state to be
free vacuum at some finite time. The cure for these divergences is to correct
the initial state. Note that we do not need the full state — no one is ever
going to find the full state wave functional for an interacting quantum field
theory in $3+1$ dimensions. All we need is the order $\lambda$ correction;
only this part of the initial state can affect our perturbative results
(35-36).

It is well to recall how a quartic interaction would change the ground state
of a simple harmonic oscillator (1) in quantum mechanics,

$H=\frac{p^{2}}{2m}+\frac{m\omega^{2}q^{2}}{2}+\frac{\lambda
q^{4}}{4}\;,\;|\Omega\rangle=|0\rangle+\sum_{n=1}^{\infty}a_{n}|n\rangle\Rightarrow
a_{n}=-\frac{\lambda\langle n|q^{4}|0\rangle}{4n\omega}+O(\lambda^{2})\;.$
(37)

If we release the system at $t=0$ in a functional integral over configurations
$q(t)$ then its wave function depends upon the position $q_{0}\equiv q(t=0)$.
The first order correction is,

$\Omega_{1}(q_{0})=-\frac{\lambda}{16m^{2}\omega^{3}}\Bigl{[}1-2m\omega
q_{0}^{2}\Bigr{]}^{2}\times\Omega_{0}(q_{0})\;,$ (38)

where $\Omega_{0}(q_{0})$ is the harmonic oscillator ground state (6).

If we release the QFT state at $t=0$ in the functional formalism then the
state wave functional will depend on
$\varphi_{0}(\vec{x})\equiv\varphi(t=0,\vec{x})$. From the preceding quantum
mechanical discussion we expect that the order $\lambda$ correction to Bunch-
Davies vacuum $\Omega_{0}[\varphi_{0}]$ will involve zero, two and four powers
of $\varphi_{0}$. Because the stress tensor operator (33) has the form
$\partial\varphi\partial\varphi+\lambda\varphi^{4}$, only the
$\partial\varphi\partial\varphi$ part can couple with the order $\lambda$
state correction to contribute to the expectation value at order $\lambda$. If
this contribution is to depend upon the time $t$ at which the stress tensor is
evaluated, it must couple with the order $\lambda$ state correction involving
two powers of $\varphi_{0}$. Homogeneity and isotropy then imply a correction
of the form,

$\Omega_{1}[\varphi_{0}]=\frac{\lambda
H}{2}\\!\int\\!\\!\frac{d^{3}k}{(2\pi)^{3}}\,F\Bigl{(}\frac{k}{H}\Bigr{)}\Bigl{|}\widetilde{\varphi}_{0}(\vec{k})\Bigr{|}^{2}\times\Omega_{0}[\varphi_{0}]\;.$
(39)

A straightforward calculation reveals that the kernel is [59],

$F(x)=\frac{ie^{2ix}}{32\pi^{2}(1\\!+\\!ix)^{2}}\Biggl{\\{}e^{-2ix}-x^{3}\\!\int\\!\\!\frac{dz}{z^{4}}\,e^{-2iz}\Bigl{[}1+2\ln\Bigl{(}\frac{z}{x}\Bigr{)}\Bigr{]}\Biggr{\\}}\;.$
(40)

And the corrected energy density and pressure are free of both divergences and
exponentially falling contributions,

$\displaystyle\rho_{\rm new}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{3H^{2}}{8\pi G}+\frac{\lambda
H^{4}}{(2\pi)^{4}}\Biggl{\\{}2\ln^{2}(a)+\frac{13}{6}\ln(a)-\frac{43}{18}+\frac{\pi^{2}}{3}\Biggr{\\}}+O(\lambda^{2})\;,$
(41) $\displaystyle p_{\rm new}$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle-\frac{3H^{2}}{8\pi G}+\frac{\lambda
H^{4}}{(2\pi)^{4}}\Biggl{\\{}-2\ln^{2}(a)-\frac{7}{2}\ln(a)+\frac{5}{3}-\frac{\pi^{2}}{3}\Biggr{\\}}+O(\lambda^{2})\;.$
(42)

###  3.4 Can We Absorb the Logarithms?

The factors of $\ln(a)=Ht$ in expressions (41-42) derive from inflationary
particle production. There are two such factors because the expectation value
of the $\varphi^{4}$ part of the stress tensor (33) involves two coincident
propagators, each of which contributes a $\ln(a)$ [60, 61, 62],

$\Bigl{\langle}\Omega_{0}\Bigl{|}\varphi^{2}(t,\vec{x})\Bigr{|}\Omega_{0}\Bigr{\rangle}={\rm
UV\ Divergence}+\frac{H^{2}}{4\pi^{2}}\ln(a)\;.$ (43)

The physical way of understanding this is that inflationary particle
production forces the scalar further and further up its potential.

Higher loop corrections involve even more factors of $\ln(a)$. At order
$\lambda^{N}$ there can be up to $2N$ factors of $\ln(a)$ [63]. Starobinsky
has inferred a stochastic formalism [64] which can be proven to reproduce the
leading logarithms at each order in perturbation theory [63]. In cases, such
as this model (32), where a time-independent limit is approached,
Starobinsky’s technique can even be used to find this limit [65],

$\Bigl{\langle}\Omega\Bigl{|}T_{\mu\nu}(t,\vec{x})\Bigr{|}\Omega\Bigr{\rangle}\longrightarrow\frac{3H^{4}}{32\pi^{2}}\,g_{\mu\nu}\;.$
(44)

The physical picture is that inflationary particle production pushes the
scalar up its potential until an equilibrium is reached with the classical
downward force. Of course that equilibrium corresponds to a particular state
wave functional and, were one to release the system in this state, there would
be no time dependence.

Some people invoke the existence of an equilibrium state to argue that
inflationary particle production is somehow not real, or else does not induce
time dependence in QFT. This shows the same level of disingenuity as denying
the inflationary origin of primordial perturbations because one could obtain
identical results by starting with a state which had exactly the desired
pattern of correlations. However, the anti-time dependence position becomes
completely untenable when one considers models for which no static limit is
approached. Although scalar quantum electrodynamics approaches a static limit
[66], a Yukawa-coupled massless, minimally coupled scalar does not [67].
Nonlinear sigma models also show large logarithms [63, 68, 69, 70], and some
of these models do not approach a static limit [18, 19].

Nonlinear sigma models are interesting in that capturing their large
logarithms requires a variant of the renormalization group in addition to a
variant of Starobinsky’s stochastic formalism [18, 19]. This is because their
interactions involve derivatives, unlike the $\varphi^{4}$ interaction of the
scalar model (32). Quantum gravity also involves derivative interactions and
its large inflationary logarithms [3, 4, 5, 6, 71, 7, 8] are not completely
captured by Starobinsky’s stochastic formalism [72]. I suspect that combining
Starobinsky’s formalism with a variant of the renormalization group will
suffice [6], as it did for nonlinear sigma models, but that remains to be
proven. It is not known if the large logarithms of quantum gravity add up to
approach a static limit.

##  4 Conclusions

Steven Weinberg dominated high energy particle theory for decades and inspired
generations of physicists. I was fortunate to be among them. Two of the things
I most admired about Weinberg were his refusal to accept dogma and his
willingness to take risks. In addition to sharing some reminiscences, I have
here discussed a topic which illustrates both of those characteristics: the
role of state wave functionals in quantum field theory.

State wave functionals are the correct way to derive the $i\epsilon$ of the
Feynman propagator from the flat space functional formalism. They assume a
larger role in cosmological QFT because one must specify states at finite
time, and because inflationary particle production sometimes injects secular
growth into expectation values, the potential for which Weinberg discovered in
the scalar power spectrum [15]. Perturbative corrections to free vacuum are
necessary to remove divergences on the initial value surface [59], which
leaves large logarithms from inflationary particle production. Weinberg
realized the importance of resumming these logarithms [15]. In some cases,
they can be summed to produce a static limit which can be subsumed into a
highly nonlinear state wave functional [65]. For other models, time dependence
persists forever [18, 19] and one is tempted to wonder if some otherwise
curious features of late time cosmology might be explained as residual QFT
effects [73, 74]. Wherever he is now, Big Steve would love that.

Acknowledgements

I am grateful for conversations on this subject with S. Deser, D. Glavan, E.
O. Kahya, S. P. Miao, V. K. Onemli, T. Prokopec and N. C. Tsamis. This work
was supported by NSF grant PHY-2207514 and by the Institute for Fundamental
Theory at the University of Florida.

## References

  * [1] S. Weinberg, The Quantum Theory of Fields, Volume 1 (Cambridge University Press), p. 35.
  * [2] N. C. Tsamis and R. P. Woodard, Annals Phys. 253, 1-54 (1997) doi:10.1006/aphy.1997.5613 [arXiv:hep-ph/9602316 [hep-ph]].
  * [3] S. P. Miao and R. P. Woodard, Phys. Rev. D 74, 024021 (2006) doi:10.1103/PhysRevD.74.024021 [arXiv:gr-qc/0603135 [gr-qc]].
  * [4] D. Glavan, S. P. Miao, T. Prokopec and R. P. Woodard, Class. Quant. Grav. 31, 175002 (2014) doi:10.1088/0264-9381/31/17/175002 [arXiv:1308.3453 [gr-qc]].
  * [5] C. L. Wang and R. P. Woodard, Phys. Rev. D 91, no.12, 124054 (2015) doi:10.1103/PhysRevD.91.124054 [arXiv:1408.1448 [gr-qc]].
  * [6] D. Glavan, S. P. Miao, T. Prokopec and R. P. Woodard, JHEP 03, 088 (2022) doi:10.1007/JHEP03(2022)088 [arXiv:2112.00959 [gr-qc]].
  * [7] L. Tan, N. C. Tsamis and R. P. Woodard, [arXiv:2107.13905 [gr-qc]].
  * [8] L. Tan, N. C. Tsamis and R. P. Woodard, [arXiv:2206.11467 [gr-qc]].
  * [9] J. Garriga and T. Tanaka, Phys. Rev. D 77, 024021 (2008) doi:10.1103/PhysRevD.77.024021 [arXiv:0706.0295 [hep-th]].
  * [10] N. C. Tsamis and R. P. Woodard, Phys. Rev. D 78, 028501 (2008) doi:10.1103/PhysRevD.78.028501 [arXiv:0708.2004 [hep-th]].
  * [11] A. Higuchi, D. Marolf and I. A. Morrison, Class. Quant. Grav. 28, 245012 (2011) doi:10.1088/0264-9381/28/24/245012 [arXiv:1107.2712 [hep-th]].
  * [12] S. P. Miao, N. C. Tsamis and R. P. Woodard, Class. Quant. Grav. 28, 245013 (2011) doi:10.1088/0264-9381/28/24/245013 [arXiv:1107.4733 [gr-qc]].
  * [13] S. P. Miao, T. Prokopec and R. P. Woodard, Phys. Rev. D 96, no.10, 104029 (2017) doi:10.1103/PhysRevD.96.104029 [arXiv:1708.06239 [gr-qc]].
  * [14] S. Katuwal and R. P. Woodard, JHEP 21, 029 (2020) doi:10.1007/JHEP10(2021)029 [arXiv:2107.13341 [gr-qc]].
  * [15] S. Weinberg, Phys. Rev. D 74, 023508 (2006) doi:10.1103/PhysRevD.74.023508 [arXiv:hep-th/0605244 [hep-th]].
  * [16] L. Senatore and M. Zaldarriaga, JHEP 12, 008 (2010) doi:10.1007/JHEP12(2010)008 [arXiv:0912.2734 [hep-th]].
  * [17] E. O. Kahya, V. K. Onemli and R. P. Woodard, Phys. Lett. B 694, 101-107 (2011) doi:10.1016/j.physletb.2010.09.050 [arXiv:1006.3999 [astro-ph.CO]].
  * [18] S. P. Miao, N. C. Tsamis and R. P. Woodard, JHEP 03, 069 (2022) doi:10.1007/JHEP03(2022)069 [arXiv:2110.08715 [gr-qc]].
  * [19] R. P. Woodard and B. Yesilyurt, [arXiv:2302.11528 [gr-qc]].
  * [20] S. Weinberg, Phys. Rev. D 72, 043514 (2005) doi:10.1103/PhysRevD.72.043514 [arXiv:hep-th/0506236 [hep-th]].
  * [21] J. S. Schwinger, J. Math. Phys. 2, 407-432 (1961) doi:10.1063/1.1703727
  * [22] K. T. Mahanthappa, Phys. Rev. 126, 329-340 (1962) doi:10.1103/PhysRev.126.329
  * [23] P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4, 1-11 (1963) doi:10.1063/1.1703883
  * [24] P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4, 12-16 (1963) doi:10.1063/1.1703879
  * [25] L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515-1527 (1964)
  * [26] K. c. Chou, Z. b. Su, B. l. Hao and L. Yu, Phys. Rept. 118, 1-131 (1985) doi:10.1016/0370-1573(85)90136-X
  * [27] R. D. Jordan, Phys. Rev. D 33, 444-454 (1986) doi:10.1103/PhysRevD.33.444
  * [28] E. Calzetta and B. L. Hu, Phys. Rev. D 35, 495 (1987) doi:10.1103/PhysRevD.35.495
  * [29] S. Weinberg, Int. J. Mod. Phys. A 28, 1330055 (2013) doi:10.1142/S0217751X1330055X
  * [30] S. Weinberg, Phys. Rev. Lett. 110, 261601 (2013) doi:10.1103/PhysRevLett.110.261601 [arXiv:1303.0342 [hep-ph]].
  * [31] S. Weinberg, Phys. Rev. Lett. 110, no.24, 241301 (2013) doi:10.1103/PhysRevLett.110.241301 [arXiv:1305.1971 [astro-ph.CO]].
  * [32] S. Weinberg, Int. J. Mod. Phys. A 29, 1430004 (2014) doi:10.1142/S0217751X1430004X
  * [33] S. Weinberg, Phys. Rev. A 90, no.4, 042102 (2014) doi:10.1103/PhysRevA.90.042102 [arXiv:1405.3483 [quant-ph]].
  * [34] S. Weinberg, Int. J. Mod. Phys. A 31, no.06, 1630007 (2016) doi:10.1142/S0217751X16300076
  * [35] S. Weinberg, Phys. Rev. A 93, 032124 (2016) doi:10.1103/PhysRevA.93.032124 [arXiv:1603.06008 [quant-ph]].
  * [36] S. Weinberg, Phys. Rev. A 94, no.4, 042117 (2016) doi:10.1103/PhysRevA.94.042117 [arXiv:1610.02537 [quant-ph]].
  * [37] R. Flauger and S. Weinberg, Phys. Rev. D 97, no.12, 123506 (2018) doi:10.1103/PhysRevD.97.123506 [arXiv:1801.00386 [astro-ph.CO]].
  * [38] S. Weinberg, Phys. Rev. Lett. 121, no.22, 220001 (2018) doi:10.1103/PhysRevLett.121.220001
  * [39] S. Weinberg, Phys. Rev. D 99, no.7, 076018 (2019) doi:10.1103/PhysRevD.99.076018 [arXiv:1903.11168 [astro-ph.GA]].
  * [40] R. Flauger and S. Weinberg, Phys. Rev. D 99, no.12, 123030 (2019) doi:10.1103/PhysRevD.99.123030 [arXiv:1906.04853 [hep-th]].
  * [41] S. Weinberg, Phys. Rev. D 101, no.3, 035020 (2020) doi:10.1103/PhysRevD.101.035020 [arXiv:2001.06582 [hep-th]].
  * [42] R. Bousso, F. Quevedo and S. Weinberg, [arXiv:2002.02371 [physics.hist-ph]].
  * [43] S. Weinberg, Phys. Rev. D 102, no.9, 095022 (2020) doi:10.1103/PhysRevD.102.095022 [arXiv:2010.05823 [hep-th]].
  * [44] S. Weinberg, Eur. Phys. J. H 46, no.1, 6 (2021) doi:10.1140/epjh/s13129-021-00004-x [arXiv:2101.04241 [hep-th]].
  * [45] S. Weinberg, Cambridge University Press, 2021, ISBN 978-1-108-89484-5, 978-1-108-84176-4 doi:10.1017/9781108894845
  * [46] L. D. Faddeev and A. A. Slavnov, Front. Phys. 50, 1-232 (1980)
  * [47] E. Schrödinger, Physica 6, 899 (1939) doi.org/10.1016/S0031-8914(39)90091-1
  * [48] L. Parker, Phys. Rev. Lett. 21, 562-564 (1968) doi:10.1103/PhysRevLett.21.562
  * [49] L. Parker, Phys. Rev. 183, 1057-1068 (1969) doi:10.1103/PhysRev.183.1057
  * [50] L. Parker, Phys. Rev. D 3, 346-356 (1971) [erratum: Phys. Rev. D 3, 2546-2546 (1971)] doi:10.1103/PhysRevD.3.346
  * [51] E. Lifshitz, J. Phys. (USSR) 10, no.2, 116 (1946) doi:10.1007/s10714-016-2165-8
  * [52] N. A. Chernikov and E. A. Tagirov, Ann. Inst. H. Poincare Phys. Theor. A 9, 109 (1968)
  * [53] C. Schomblond and P. Spindel, Ann. Inst. H. Poincare Phys. Theor. 25, 67-78 (1976)
  * [54] T. S. Bunch and P. C. W. Davies, Proc. Roy. Soc. Lond. A 360, 117-134 (1978) doi:10.1098/rspa.1978.0060
  * [55] A. A. Starobinsky, JETP Lett. 30, 682-685 (1979)
  * [56] V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33, 532-535 (1981)
  * [57] V. K. Onemli and R. P. Woodard, Class. Quant. Grav. 19, 4607 (2002) doi:10.1088/0264-9381/19/17/311 [arXiv:gr-qc/0204065 [gr-qc]].
  * [58] V. K. Onemli and R. P. Woodard, Phys. Rev. D 70, 107301 (2004) doi:10.1103/PhysRevD.70.107301 [arXiv:gr-qc/0406098 [gr-qc]].
  * [59] E. O. Kahya, V. K. Onemli and R. P. Woodard, Phys. Rev. D 81, 023508 (2010) doi:10.1103/PhysRevD.81.023508 [arXiv:0904.4811 [gr-qc]].
  * [60] A. Vilenkin and L. H. Ford, Phys. Rev. D 26, 1231 (1982) doi:10.1103/PhysRevD.26.1231
  * [61] A. D. Linde, Phys. Lett. B 116, 335-339 (1982) doi:10.1016/0370-2693(82)90293-3
  * [62] A. A. Starobinsky, Phys. Lett. B 117, 175-178 (1982) doi:10.1016/0370-2693(82)90541-X
  * [63] N. C. Tsamis and R. P. Woodard, Nucl. Phys. B 724, 295-328 (2005) doi:10.1016/j.nuclphysb.2005.06.031 [arXiv:gr-qc/0505115 [gr-qc]].
  * [64] A. A. Starobinsky, Lect. Notes Phys. 246, 107-126 (1986) doi:10.1007/3-540-16452-9_6
  * [65] A. A. Starobinsky and J. Yokoyama, Phys. Rev. D 50, 6357-6368 (1994) doi:10.1103/PhysRevD.50.6357 [arXiv:astro-ph/9407016 [astro-ph]].
  * [66] T. Prokopec, N. C. Tsamis and R. P. Woodard, Annals Phys. 323, 1324-1360 (2008) doi:10.1016/j.aop.2007.08.008 [arXiv:0707.0847 [gr-qc]].
  * [67] S. P. Miao and R. P. Woodard, Phys. Rev. D 74, 044019 (2006) doi:10.1103/PhysRevD.74.044019 [arXiv:gr-qc/0602110 [gr-qc]].
  * [68] H. Kitamoto and Y. Kitazawa, Phys. Rev. D 83, 104043 (2011) doi:10.1103/PhysRevD.83.104043 [arXiv:1012.5930 [hep-th]].
  * [69] H. Kitamoto and Y. Kitazawa, Phys. Rev. D 85, 044062 (2012) doi:10.1103/PhysRevD.85.044062 [arXiv:1109.4892 [hep-th]].
  * [70] H. Kitamoto, Phys. Rev. D 100, no.2, 025020 (2019) doi:10.1103/PhysRevD.100.025020 [arXiv:1811.01830 [hep-th]].
  * [71] S. Park, T. Prokopec and R. P. Woodard, JHEP 01, 074 (2016) doi:10.1007/JHEP01(2016)074 [arXiv:1510.03352 [gr-qc]].
  * [72] S. P. Miao and R. P. Woodard, Class. Quant. Grav. 25, 145009 (2008) doi:10.1088/0264-9381/25/14/145009 [arXiv:0803.2377 [gr-qc]].
  * [73] N. C. Tsamis and R. P. Woodard, Int. J. Mod. Phys. D 20, 2847-2851 (2011) doi:10.1142/S0218271811020652 [arXiv:1103.5134 [gr-qc]].
  * [74] R. P. Woodard, Universe 4, no.8, 88 (2018) doi:10.3390/universe4080088 [arXiv:1807.01791 [gr-qc]].


# Lowest-cost virus suppression

Jacob L.L.C.C. Janssen1 Corresponding author. Email<EMAIL_ADDRESS>Yaneer Bar-Yam2 1 Amsterdam, TNO, the Netherlands  2New England Complex
Systems Institute, Cambridge, MA

(August 27, 2024)

###### Abstract

Analysis of policies for managing epidemics require simultaneously an economic
and epidemiological perspective. We adopt a cost-of-policy framework to model
both the virus spread and the cost of handling the pandemic. Because it is
harder and more costly to fight the pandemic when the circulation is higher,
we find that the optimal policy is to go to zero or near-zero case numbers.
Without imported cases, if a region is willing to implement measures to
prevent spread at one level in number of cases, it must also be willing to
prevent the spread with at a lower level, since it will be cheaper to do so
and has only positive other effects. With imported cases, if a region is not
coordinating with other regions, we show the cheapest policy is continually
low but nonzero cases due to decreasing cost of halting imported cases. When
it is coordinating, zero is cost-optimal. Our analysis indicates that within
Europe cooperation targeting a reduction of both within country transmission,
and between country importation risk, should help achieve lower transmission
and reduced costs.

COVID-19, caused by the novel coronavirus SARS-CoV-2 has become a severe
pandemic WHO (2020). Understanding how the cost of suppressing the virus
interacts with the dynamics of disease transmission, both within and between
regions provides insight into the best course of action Acemoglu et al.
(2020); Bognanni M and K (2020, 2020); Kissler et al. (2020); Siegenfeld and
Bar-Yam (2020); Norman et al. (2020). In this article, we show that policies
that allow for a continually low virus circulation are optimal in situations
both with and without inter-regional travel. Not considering the cost of
limiting travel at first, we find the cost of suppressing the virus can be
described as an increasing function of case numbers. Conversely, the cost of
limiting imported cases is decreasing in the number of cases. When we abstract
from dealing with inter-regional travel, the increasing nature of the cost
function implies the cost of suppressing the virus is lowest at zero cases.
Since there are only positive side effects to having less virus, the optimal
policy is to go to zero.

Due to globalization and inter-regional travel the world has become more
vulnerable to pandemics and suppressing the virus has become more difficult
Bar-Yam (2016); Petersen et al. (2018) and more costly Antràs et al. (2020),
sometimes even to the point of being used as an argument for giving up
preventing the virus from becoming widespread entirely Paterlini (2020);
Kulldorf M (2020); Rijksoverheid (2020). The cost of limiting imported cases
when analysing multiple regions is based on the simultaneous cost of
suppressing the virus within a region and the cost of limiting imported cases
from outside the region. As the cost of limiting imported cases is decreasing
in the number of cases, and the cost of suppressing the virus is increasing in
the number of cases, a natural question is at what number of cases the cost of
suppressing the virus is lowest. We show that in spite of the cost of limiting
inter-regional travel, the impact of imported cases on costly society-wide
restrictions implies the best course of action is to go to very low number of
cases that can be controlled through testing, tracing and isolating (TTI) of
individual cases.

##  I Costs of policy: Increasing in case numbers

Naturally, when there is more virus, one must do more to stop the spread. This
means the cost of suppressing the virus is increasing in the number of cases.
Measures to halt the spread of the virus each have their own financial and
societal cost, and their own efficacy; shutting down large congregations,
closing bars, schools, or other venues, and aggressive social distancing.
Choosing measures with the highest efficacy and lowest societal cost should
have the highest priority, and there is a natural incentive to pick those
measures. Moreover, when there are only a few new cases per day, it is
possible to prevent further spreading by TTI of individual cases WHO (2020).
This is a relatively cheap way to stop the virus from spreading. At higher
levels of spread, one must rely on other, more expensive, region-wide
measures, like closing down venues and stricter social distancing. Therefore,
the cost per case is also lower when you can do TTI and contain the virus
entirely.

We stress the point where a transition is made from ‘firefighting’ (TTI) to
society wide restrictions, which happens at a few cases. A few cases means all
contacts can be traced, quarantined, and further spread can effectively be
prevented with other people having to make only relatively small changes to
their daily lives. After this point, more drastic society-wide measures will
be required. However, a similar mechanism occurs at higher levels of
circulation. If some number of measures together with TTI is sufficient to
prevent further spreading, there is a similar transition point where TTI
breaks down. Here again, when the limits of TTI are reached and the system
loses some of its efficacy, harsher measures are needed to stop the virus from
spreading further. This means that around points where TTI breaks down, the
costs are locally convex, and it is important to protect the coverage of TTI
at any level of circulation.

A note on convexity of costs
It immediately follows from a linear impact analysis that the larger the
number of cases the larger the cost to society from an emergent disease. As
TTI is a relatively cheap way to suppress the virus, the cost per case of
suppressing the virus is lower when doing TTI than when not doing TTI. This
means that around points where TTI breaks down and society must rely on
additional measures, independent of what additional measures were already in
place, the cost of suppressing the virus is locally convex. To further
illustrate this point, consider two regions with small outbreaks that can both
only just be effectively controlled through TTI. This requires a serious
effort to prevent the outbreaks from creating community transmission and
requireing more extensive measures. However, if the same number of cases would
appear in a single outbreak the contact tracing system may pass the transition
point and lose the completeness of its coverage. This is much more serious and
costly for the affected region. Though the cost is increasing in the number of
cases and has pockets of convexity around points where TTI breaks down, the
cost as a function of the number of cases is not convex overall. The
consequences for the healthcare system, society and the wider economy may
still have convex costs associated with them that are not considered here.
Indeed, in general we see that as the shock to a complex system is larger, it
is more costly or difficult to adapt to it. While a linear approximation may
hold for a low level of outbreak, those costs may scale in a nonlinear way.

##  II One Region, only domestic transmissions

Here we abstract from considering inter-regional travel. Considering only the
cost of halting local transmission, we find the optimal solution is to go to
zero cases.

1\. If a region is willing to implement measures to prevent spread at one
level in number of cases, it must also be willing to prevent the spread at a
lower level, since it will be cheaper to do so and has only positive other
effects.

Relaxing measures to save costs while increasing case numbers, only to
implement more stringent measures later to reduce the virus presence, is more
costly than keeping measures at such a level that case numbers always keep
going down. This follows from the fact that costs are an increasing function
of case numbers. As an example, recent increases in numbers of cases after
relaxing containment measures and travel restrictions in Europe, followed by
new travel restrictions and stronger containment measures within countries
indicates policy choices are non-optimal. From the first proposition quickly
follows a second one:

2\. Without imported cases, the cheapest policy is to have zero cases.

When there are no imported cases, the increasing nature of the cost function
tell us that the total cost over time is lowest when there are no new cases.
This implies that the best policy is at either 0 cases or by having no
policies at all. For COVID, having no policies at all would mean that a
majority would be infected, suffering damages and economic costs that should
certainly be avoided. Some of the economic harm occurs directly because of the
health harm, but also from risk reduction by individuals regardless of
government policy. This spontaneous adoption of a social policy as the number
of transmission events increases leads to high economic costs.

We note that some argue that it would impossible to go to zero cases Paterlini
(2020), but despite our best efforts we found no reason for this to be the
case. Instead, empirical evidence suggests not only that it is possible to go
to zero cases, but also that it is less costly than controlling the spread at
a higher level. The intuitive result that there is no trade off between the
economy and fighting the virus also follows from empirical data, as is shown
in in Data (2020c). Comparing the COVID-19 death rate with the latest GDP
data, we in fact see the opposite: countries that have managed to protect
their population’s health in the pandemic have generally also protected their
economy too.

##  III Policy for one region, including imported cases

Here we consider the costs of suppressing local transmission and the cost of
limiting imported cases from the perspective of an individual region without
policy cooperation with the rest of the world. We find that the optimal
situation is with a very low number of cases.

From the perspective of any individual region, the costs of border controls
that are needed to limit imported cases are decreasing in the number of newly
imported cases as long as the virus is present in the world. Costs include the
loss of economic benefits of the travel in tourism and other business
activity. These border control and other economic costs are offset by the cost
of the outbreak response (TTI) or wider social restrictions that are needed to
control transmission due to imported cases when travel is present. In this
paper we deal only with the following costs; the cost of limiting local
transmissions $c_{T}(x)$ to suppress the virus when there are $x$ new
infections each day, the cost of border controls $c_{b}(I)$ with $I$ the
average number of infected cases per day, and the cost that the outbreak
brings to society $c_{O}(x)$:

$cost=c_{T}+c_{b}-c_{O}.$

The cost of the outbreak $c_{O}$ is written down with a minus as we consider
the holistic cost of the policies to prevent spread. This means that while the
cost of implementing the policy increases as we put in more effort to control
this disease, the reduction in the effects of the outbreak and its associated
costs will mean the total cost will be substantially reduced. $c_{T}(x)$ and
$c_{O}(x)$ are both increasing in the number of cases as we shall see next,
and $c_{b}(I)$ is decreasing in the average number of imported cases (thus
increasing with stronger border control). The cost of the outbreak $c_{O}$
pertains to health effects, indirect economic effects, restricted liberties,
and uncertainties to arise from a virus outbreak.

Suppose a traveler coming from any other region $R$ at a certain moment in
time is infectious with probability $p_{R}$. The expected value of the number
of imported cases is then $\sum_{R}p_{R}k_{R}=I$, where $k_{R}$ is the number
of travelers from region $R$. Since the probability $p_{R}$ is different for
each region, it is more effective to prevent imported cases from regions with
high probability of importing infectious travelers. The cost of accepting a
low number of cases, for instance by allowing only travel from areas with very
low infection rates is thus lower per avoided case by limiting travel from
high-risk areas. Similarly, screening arriving passengers with temperature
checks, testing and/or imposing quarantines can be relatively cheap ways to
limit the amount of imported virus.

3\. The cost of controlling the virus is minimal for some constant, possibly
fractional, positive case number.

In this case the cost function for limiting the virus spread changes from when
there is only domestic transmission. The number of new domestic infections $x$
with associated policy cost $c_{T}(x)$ is insufficient to suppress the virus
when there are also imported cases. Instead cost of policy to suppress the
virus $c_{T}(x^{\prime})$ must be made with $x^{\prime}=x+\alpha I$, where
$\alpha I$ is the number of imported cases plus additional cases of
transmission resulting from those imported cases, which we define to be given
by a constant multiple of $I$. Additionally, there are costs of border control
$c_{b}(I)$. The policymaker may then solve the sequential minimization problem

$\min_{I}\\{\ \min_{x}[c_{T}(x+\alpha I)+c_{b}(I)]\ \\}=\min_{I}[c_{T}(\alpha
I)+c_{b}(I)]$ $\equiv\min_{I}C(I),$

where the cost aggregate $C(I)$ is the minimal cost of policy to suppress the
virus given any level of domestic transmissions $x$ each day including
imported cases. The minimization is at $x=0$ as $c_{T}(x)$ is an increasing
function of $x$. Therefore, assuming the cost of total and complete travel
restriction is high enough, $C(I)$ is decreasing at $I=0$ when $c_{b}$
dominates, and increasing as $I$ is large enough (for sufficiently large $I$,
$c_{T}$ dominates $c_{b}$). $c_{b}(I)$ is not just decreasing in the number of
imported cases, it is also true that the most expensive travel restrictions
are for the most essential travelers. So, it is easier to prevent $\it{some}$
imported cases (compared to no border controls at all), than to have no
imported cases at all (compared to allowing a low number of travelers with
risk of importing $\it{some}$ cases).

The assumptions on costs associated with restrictions limiting travel leads to
a conclusion that there is an optimal number of imported cases. However, we
note that the minimum may occur when the number of imported cases $I$ is so
low that it is fractional, i.e. for a rate of importing cases that is less
than one (so, in practice intermittently periods of nonzero cases). Importing
new cases brings the risk of causing a significant initially undetected
outbreak requiring both extensive TTI and social restrictions, as in recent
experience in New Zealand Lewis (2020). Since these costs are large compared
to conventional tourism and business opportunities, this is very well
possible. Careful scrutiny of the associated costs and benefits of travelers
in terms of the overall costs is well justified.

We note also that the cost of allowing imported cases varies between
countries. For example, in island nations, controlling borders is easier.
Conversely, for economies that heavily rely on tourism the benefit of tourism
and thus the cost of closing borders is higher relative to the domestic
economy OECD (2020). Where cross-border travel plays a larger role in the
economy, it becomes more expensive and difficult to implement border
restrictions. This is also related to the size of a region. The effect is
reminiscent to how in physics large objects have a low surface-area to volume
ratio. Similarly, large countries have a lower border cost to economy ratio
for restricting cross-border travel compared to small countries.

Thus, for example, it would be expected that Luxembourg would be more
dependent on allowing cross border travel than Germany. However, we note that
Luxembourg imposed travel restrictions that allowed cross border workers while
not allowing tourism. For this and other reasons found in for example Shen et
al. (2020), we can infer that restricting cross border workers would be a
higher cost, and that workers would also represent a lower transmission risk
either due to inherent social, psychological, and behavioral differences from
tourists and/or due to enhanced precautions that they would take due to the
conditions present. Moreover, it is well known that the smallest European
countries, including Andorra, Luxembourg, Liechtenstein and Monaco, were
initially among the best performers in reducing COVID cases in Europe in Data
(2020a) (however, when European travel restrictions were collectively removed
their situation worsened). The reason is not clear from this economic analysis
but may be due to the possibility that smaller countries are more nimble, in
particular having a greater ability to act collectively in response to the
outbreak and optimize policy decisions. Some support for this observation
arises from the recognition in larger countries such as Germany that
differentiated policy decisions in localities allows better optimization of
the response efforts in Data (2020b).

It is important to recognize that using a reference for the benefits of travel
that is in a COVID-free (pre-COVID) world is the wrong starting point for
calculating the cost impacts of travel restrictions. Instead we need to
consider the overall costs of travel limitations. Thus, for example, when a
country calculates the benefit of allowing travel because of a per traveler
tourism income in the pre-COVID world, this is not an estimate of the benefit
in a world with COVID. Without travelers a country would have a different
level of outbreak response (TTI) and, significantly, a different need for
widespread social restrictions affecting economic activity Times (2020);
McKinsey (2020). Due to the high costs of outbreak control measures, the
number of travelers that provides a benefit is much lower than those in a
world without COVID, and the desirable/optimal number decreases with
increasing prevalence of COVID in other locations. This is in line with other
analyses Green et al. (2020); Norman et al. (2020).

##  IV Policies for two regions

Here we consider both local transmissions and imported cases, in a situation
where two regions consider each other’s policies and disease dynamics. We find
that cooperation and coordination can play an important role in getting to low
circulation.

Suppose there are only two regions, $i$ and $j$, both in which a percentage of
$L_{\nu}$ people are infected, $\nu\in\\{i,j\\}$. We assume that each day $k$
randomly selected***In reality, those who travel more often may well not be a
random selection. We abstract from that for simplicity, and note this is an
underestimation of the number of imported cases. people travel from $i$ to
$j$, and $N_{\nu}$ is the population of each region. The probability of having
at least $n$ cases in those $k$ people is given by

$P(\\#imported\ cases\leq
n)=\sum_{\nu=1}^{n}\frac{\binom{L_{i}N_{i}}{\nu}\binom{N_{i}-L_{i}N_{i}}{k-\nu}}{\binom{N}{k}},$

which approaches

$\sum_{\nu=1}^{n}\binom{k}{\nu}L_{i}^{\nu}$

when $k\ll N_{i}L_{i}\ll N_{i}$.††† Notice that
$\frac{\binom{L_{i}N_{i}}{\nu}\binom{N_{i}-L_{i}N_{i}}{k-\nu}}{\binom{N}{k}}=\frac{\binom{k}{\nu}\binom{N_{i}-k}{L_{i}N_{i}-\nu}}{\binom{N_{i}}{N_{i}L_{i}}}=$
$=\binom{k}{\nu}\frac{(N_{i}-L_{i}N_{i})...(N_{i}-L_{i}N_{i}-(k-\nu))\times(L_{i}N_{i}...(L_{i}N_{i}-\nu))}{N_{i}...(N_{i}-k)}\rightarrow$
$\binom{k}{\nu}L_{i}^{\nu}.$ The expected number of imported cases from region
$i$ to region $j$ given $k$ travelers is given by

$\sum_{\nu=1}^{k}\binom{k}{\nu}L_{i}^{\nu}\nu\equiv I_{ik}$

Region $i$ can take action and through screening or reducing travel get
$I_{ik}$ down to $I_{ik}F$. When region $j$ assumes that there are cases in
region $i$ and there will be cases indefinitely, it will weigh the cost of
limiting travel from region $i$ against the cost of accepting a number of
imported cases from region $i$. However, when the two regions consider their
strategy holistically, they would consider themselves one region and would go
for zero cases, as discussed above. This would require stronger measures to
fight the epidemic in the region with more active transmission as well as the
region with less transmission, and/or stricter limitations on the number of
people traveling from the region with more active transmission. The subsequent
result would be lower costs for both.

We can formalize the cost for region $j$ similar to the one-region case. The
cost of limiting domestic cases $c_{T}(x)$ to a level of $x$ cases per day is
represented by $c_{T}(0)=c_{0}>0,c_{T}^{\prime}(x)>0$ (increasing). The cost
of limiting imported cases to a level of $y$ cases, represented by $c_{b}(x)$,
per day is characterized by $c_{b}(0)>0,c_{b}^{\prime}(y)<0$, (decreasing),
$c_{b}(I_{ik})=0$ (no border costs when fully open). Region $j$ does not
consider the impact of its own number of cases on the strategy in region $i$.
In this scenario we identify the minimization problem for region $j$ as

$\min_{F}c_{b}+c_{T}$ (1)

with interior solution

$\frac{dc_{T}(x+\alpha I_{ik}F)}{dF}=-\frac{dc_{b}(I_{ik}F)}{dF}.$

The minimization problem would have boundary solution $F=1$ (no border control
at all) when $\frac{dc_{T}(x+\alpha
I_{ik}F)}{dF}|_{F=1}<-\frac{dc_{b}(I_{ik}F)}{dF}|_{F=1}$. This is only a
hypothetical scenario, as it occurs when the cost of limiting imported cases
is so expensive that it is always cheaper to just accept the cases and limit
the spread locally. Additionally, the boundary solution for $F=0$ is found
when the condition holds that the cost of holding cases at zero, plus the cost
of holding cases at near-zero is when a new case is imported, is larger than
the marginal cost savings of allowing even a little bit of travel between
regions $i$ and $j$: $\frac{dc_{T}(\alpha
I_{ik}F)}{dF}|_{F=0}>-\frac{dc_{b}(I_{ik}F)}{dF}|_{F=0}$.‡‡‡When all borders
are completely closed, the pandemic preparedness at 0 zero cases can also be
reduced significantly, giving rise to the new assumptions $\lim_{x\downarrow
0}c_{T}(x)=c_{0}$ and $c_{T}(0)\approx 0$, so that we alternatively find $F=0$
when $c_{0}+\frac{dc_{T}(\alpha
I_{ik}F)}{dF}|_{F=0}>-\frac{dc_{b}(I_{ik}F)}{dF}|_{F=0}.$

The above analysis gives rise to somewhat of a paradox: If region $i$ acts
assuming region $j$ does not eradicate the virus, and region $j$ assumes $i$
will not eradicate the virus, then for both regions the lowest-cost approach
is to accept some infected individuals (perhaps fractional per day, i.e. some
risk of an infected individual arriving). When both region $i$ and $j$
determine their strategy together, they are back in the situation of only
domestic transmission described above, and they should eradicate the virus.
The maximization problem for a single region is the same as in section B above
with one region without imported cases. The cost here reduces back to

$cost=c_{T}(x)-c_{O}(x)$

and since $c_{T}$ is increasing in $x$ and $-c_{O}$ is decreasing in $x$, the
cost is minimal when the number of cases is zero. This occurs when both
regions act under one government. This is naturally also the case when both
regions coordinate. When weaker assumptions are made, zero can still be the
optimal strategy. If one region assumes rationality on the part of the other
region, then its own rational solution is to aim for zero cases. In the case
where one or both regions assume, wrongly or not, that the other region will
not contain its virus presence, it faces the choice of limiting transport
between regions indefinitely and at high cost, and may as a consequence accept
some transmission.

So, while even without coordination each region should aim for zero cases when
assuming rationality on the part of the other region, this still highlights
the utility of inter-regional cooperation. A practical example of where such a
coordinated effort is possible, is found in the European Union, where
countries have recently started a shared response Union (2020). As an
extension of this analysis, it has been shown that due to the fact that Europe
did its lockdowns in its first wave at the same time, they were much more
effective Ruktanonchai et al. (2020).

One reason a region may not go to zero is because it makes the assumption that
it’s impossible to do so. Although some scientists Paterlini (2020) suggest
that it is impossible to go to zero cases, there is no reason for that
assumption. In fact, several countries have already shown it can be done.
Finally, we note that at the heart of this there is a collective action
problem regarding the cost of limiting transportation; if both regions go to
zero cases, travel between the regions could be freed up and corona-free
without any cost, and that would be the optimal (lowest-cost) strategy for
each region. If either region decides not to, it is harder for the other
region to likewise go to zero because it has to maintain travel restrictions
indefinitely. The solution to the collective action problem in this case is
simple, as it is in each region’s individual interest to have a low number of
cases. There is a tragedy of the commons dynamics due to the fact that a
region maintaining a large number of cases has the externality of infecting
other regions. What is unusual in comparison to typical analyses, is that what
seems to be an exploitative action by one of the players is actually a cost
increasing (irrational) action.

Note that we have not used $c_{O}$ much in this analysis. In addition to the
direct cost of policy discussed here, there are many more arguments to contain
outbreaks. These pertain to the costs associated with $c_{O}$; health effects
(many people get sick, and even die), indirect economic effects (even without
strong anti-transmission measures, the economy would respond strongly to the
virus presence Koolman (2020)), ongoing restricted liberties, ethics, possible
mutations and uncertainties Norman et al. (2020) that are exacerbated by not
going to zero cases. The costs included in $c_{O}$ could be absorbed into the
economic costs of suppressing the virus $c_{T}$, and the analysis would not
change qualitatively.

In conclusion, we identified a reason for a nonzero COVID outcome as the
individual country solution to a cost-of-policy optimization where it is
assumed that other countries themselves irrationally adopt a costly nonzero
COVID policy. This response is one that some European countries seem to be
aiming for given their policies are sufficient to keep outbreaks in check, but
insufficient to eliminate the virus from their countries entirely. We note,
however, that optimization indicates that if a region is willing to implement
measures to prevent spread at one level in number of cases, it must also be
willing to prevent the spread at a lower level, since it will be cheaper to do
so and has only positive other effects. Therefore the strategy of keeping the
virus in check is at higher levels of circulation is not only impractical due
to the many uncertainties surrounding the further development of the virus and
its spread, but also not optimal from a dynamic optimization perspective.
Moreover, we note that travel induced cases should be limited to very low
levels due to the very high costs of country wide actions to contain outbreaks
in the presence of COVID. Lastly, we note that cooperation (and/or rationality
assumptions) should be promoted to give rise to joint zero COVID strategies.

## References

  * Acemoglu et al. [2020] Daron Acemoglu, Victor Chernozhukov, Iván Werning, and Michael D Whinston.  A multi-risk sir model with optimally targeted lockdown.  Technical report, National Bureau of Economic Research, 2020.
  * Antràs et al. [2020] Pol Antràs, Stephen J Redding, and Esteban Rossi-Hansberg.  Globalization and pandemics.  Technical report, Harvard University Working Paper, 2020.
  * Bar-Yam [2016] Yaneer Bar-Yam.  Transition to extinction: Pandemics in a connected world, 2016.
  * Bognanni M and K [2020] Kolliner D Bognanni M, Hanley D and Mitman K.  Economics and epidemics: Evidence from an estimated spatial econ-sir model, 2020.
  * Green et al. [2020] Aaron Green, Chen Shen, and Yaneer Bar-Yam.  Case studies of covid-19 travel restrictions.  _New England Complex Systems Institute_ , 2020.
  * in Data [2020a] Our World in Data.  Luxembourg: Coronavirus pandemic country profile.  https://ourworldindata.org/coronavirus/country/luxembourg, 2020a.  Accessed: 2020-09-11.
  * in Data [2020b] Our World in Data.  Emerging covid-19 success story: Germany’s strong enabling environment.  https://ourworldindata.org/covid-exemplar-germany, 2020b.  Accessed:2020-09-11.
  * in Data [2020c] Our World in Data.  Which countries have protected both health and the economy in the pandemic?  https://ourworldindata.org/covid-health-economy, 2020c.  Accessed:2020-09-11.
  * Kissler et al. [2020] Stephen M Kissler, Christine Tedijanto, Edward Goldstein, Yonatan H Grad, and Marc Lipsitch.  Projecting the transmission dynamics of sars-cov-2 through the postpandemic period.  _Science_ , 368(6493):860–868, 2020.
  * Koolman [2020] Xander Koolman.  It’s the virus stupid!  _Academic Health Economists’ Blog_ , 2020.
  * Kulldorf M [2020] et al. Kulldorf M.  The great barrington declaration.  https://gbdeclaration.org/, 2020.
  * Lewis [2020] Dyani Lewis.  ”we felt we had beaten it”: New zealand’s race to eliminate the coronavirus again.  _Nature_ , 584(7821):336, 2020.
  * McKinsey [2020] McKinsey.  Covid-19: Implications for business.  https://www.mckinsey.com/business-functions/risk/our-insights/covid-19-implications-for-business, 2020\.
  * Norman et al. [2020] Joseph Norman, Yaneer Bar-Yam, and Nassim Nicholas Taleb.  Systemic risk of pandemic via novel pathogens—coronavirus: A note.  _New England Complex Systems Institute (January 26)_ , 2020.
  * OECD [2020] OECD.  Tourism policy responses to the coronavirus (covid-19).  https://www.oecd.org/coronavirus/, 2020.  Accessed:2020-09-11.
  * Paterlini [2020] Marta Paterlini.  ”closing borders is ridiculous”: the epidemiologist behind sweden’s controversial coronavirus strategy.  https://www.nature.com/articles/d41586-020-01098-x, 2020.
  * Petersen et al. [2018] Eskild Petersen, Nicola Petrosillo, Marion Koopmans, N Beeching, A Di Caro, E Gkrania-Klotsas, A Kantele, R Kohlmann, P-L Lim, A Markotic, et al.  Emerging infections—an increasingly important topic: review by the emerging infections task force.  _Clinical Microbiology and Infection_ , 24(4):369–375, 2018.
  * Rijksoverheid [2020] Rijksoverheid.  Tv-toespraak van minister-president mark rutte.  www.rijksoverheid.nl/documenten/toespraken/2020/03/16/tv-toespraak-van-minister-president-mark-rutte, 2020\.
  * Ruktanonchai et al. [2020] Nick Warren Ruktanonchai, JR Floyd, Shengjie Lai, Corrine Warren Ruktanonchai, Adam Sadilek, Pedro Rente-Lourenco, Xue Ben, Alessandra Carioli, Joshua Gwinn, JE Steele, et al.  Assessing the impact of coordinated covid-19 exit strategies across europe.  _Science_ , 369(6510):1465–1470, 2020.
  * Shen et al. [2020] Chen Shen, Gerry F Killeen, Anthony Staines, and Yaneer Bar-Yam.  A green zone strategy for ireland, 2020.
  * Siegenfeld and Bar-Yam [2020] Alexander F Siegenfeld and Yaneer Bar-Yam.  Eliminating covid-19: The impact of travel and timing.  _arXiv preprint arXiv:2003.10086_ , 2020.
  * Times [2020] Financial Times.  Real-time data show virus hit to global economic activity.  https://www.ft.com/content/d184fa0a-6904-11ea-800d-da70cff6e4d3, 2020.
  * Union [2020] European Union.  The common eu response to covid-19.  https://europa.eu/european-union/coronavirus-response_en, 2020.
  * WHO [2020] WHO.  Director-general’s opening remarks at the media briefing on covid-19 \- 16 march 2020.  https://www.who.int/, 2020.  Accessed: 2020-09-11.