## The Cosmic Universe

This page considers some of the key issues in cosmology, the science that attempts to grasp the biggest picture of all: how did the universe start, where is it going, and was there anything before?

### The Origin of the Universe

In 1970, Stephen Hawking and Roger Penrose showed that general relativity
predicted that the universe would have started with a *singularity*, a
single point with infinite density and infinite curvature of spacetime (see the
Penrose-Hawking
singularity theorems). This might sound like an elegant and compact solution
to how the universe started. However, this poses a problem for physics as our
current models break down at infinite densities (this is nicely explained here). For
example, if we attempt to calculate the initial density of the universe:

In this case we have complete freedom to choose any initial conditions for
the universe. So the situation can be summed-up as: **anything can happen
at a singularity**. Again, this might appear to be an elegant, compact
solution to how the universe started. But if the universe really did begin with
a singularity that would render the situation beyond mathematical analysis
(fortunately, we don't generally find infinities in physical reality, and the
presence of infinities in a theory is usually taken as a sign that the theory is
flawed). The result has been a succession of imaginative ideas by which the
singularity at the start of the universe might be avoided.

In 1973, Edward Tryon proposed the theory that the universe (as a small point
- a quantum particle) might have emerged out of nothing (*ex nihilo*
creation) due to a *quantum fluctuation*. According to the Heisenberg
uncertainty principle, there can be enough uncertainty about the energy at a
point, , to allow the creation of a particle/antiparticle pair, but only
for a very short time, :

In 1982, Alexander Vilenkin extended Tryon's proposal to consider the idea
that the universe could have inflated from one of these "virtual particles" (see
here).
Andre Linde realised that a succession of further quantum fluctuations within
the expanding universe would result in areas of secondary inflation. This
process can continue indefinitely in a process known as *eternal
inflation*. In an eternally inflating universe, "bubble" universes can
bubble-up out of an eternally inflating space like bubbles in a bottle of
champagne (see Andrei Linde's arXiv paper hep-th/0211048).

Inflation is now generally accepted as describing conditions in the earliest
moments **after** the origin of the universe. But inflationary
theories should not be considered as providing the "whole story" of the origin
of the universe: inflationary theories do not seem to avoid the initial
singularity, as Stephen Hawking point out on page 52 of his arXiv paper hep-th/9409195: *"Inflation
alone can not explain the present state of the universe. One can see this by
taking any state for the universe now and running it back in time. Providing it
contains enough matter, the singularity theorems will imply that there was a
singularity in the past."* (see also Neil Turok's article Inflation and the
Beginning of the Universe).

"Hidden Variables" Theories

(or "How to get something from nothing")Another problem with these inflationary theories is that it might be said that the quantum fluctuations do not really emerge out of "nothing". As J. Richard Gott says in his book

Time Travel in Einstein's Universe:"Making the universe out of literally 'nothing' seems difficult. How does 'nothing' know about the laws of physics? After all, any tunnelling-from-nothing model starts out with a quantum state obeying all the laws of physics - and that is not nothing. Indeed, trying to make the universe out of nothing may be considered odd, since 'nothing' is something that, by definition, would appear not to exist".This theme is continued in the section

"There is no such thing as nothing"towards the bottom of the Quantum Reality page. In that section it is explained how these "virtual" particles in fact appear out of a quantum vacuum. That vacuum has a quantum state (theground state) and can thus be described by a wavefunction. With this in mind, whether you think the particle actually appear out of "nothing" or not will depend on your opinion about the "ontological status" of the wavefunction: whether you think the wavefunction is just a mathematical tool for calculating probabilities or, in fact, describes something "real", an underlying structure.Diagram from the Quantum Reality page

This was all described in detail on the Quantum Reality page which proposed the idea of a "veiled reality" layer as a means of explaining counter-intuitive quantum behaviour such as non-locality (instantaneous connections over great distances - an effect which seems to be ignored in quantum cosmology). The veiled reality layer could also explain quantum particle production out of "apparently nothing". The veiled reality layer was proposed as describing a "world of wavefunctions", completely hidden from our eyes and our measuring apparatus which inhabit a "world of particles". This would explain how virtual particles appear "out of nothing": the appearance and disappearance of particles is controlled by the veiled reality layer. The veiled reality layer is certainly not "nothing", but the truth is hidden from our particle-oriented eyes.

(The "Pilot Wave" hidden variables theory of David Bohm is also described on the Quantum Reality page, though the pilot wave only controls the

positionof particles:"Bohmian mechanics does not account for phenomena such as particle creation and annihilation characteristic of quantum field theory"- see the last paragraph of Stanford Encyclopedia of Philosophy. However, a hidden variables theory could be generated which also controls the creation of particles in the manner described above.)

#### Causal Loop Theories

Let's imagine what sort of objects could appear out of nothingness. It would
appear that only those objects **which could create themselves**
could come into existence. But what sort of objects can create themselves? And,
most importantly, could our very universe have created itself?

Theories which seem most successful at avoiding an initial singularity seem to those which involve some sort of looping in time, or "curving" of spacetime. These theories seem to have the advantage of theories involving linear time which inevitably lead back to an initial singularity. Also, some of these "curvy" theories seem to propose a universe which can, indeed, "create itself", thus providing an all-important motive for our existence.

John Wheeler has proposed a model for a *participatory universe* which
would appear to belong to this category of "universes which could create
themselves". The principle of the participatory universe is summed-up by Martin
Rees: *"The universe could only come into existence if someone observed it.
It does not matter that the observers turned up several billion years later. The
universe exists because we are aware of it."* In this sense, the
participatory universe borrows strongly from the quantum mechanics
interpretation that "observation creates reality" - by observing the universe,
we somehow bring it into existence. However, I (in common with the majority of
physicists) do not subscribe to the controversial "conscious observation creates
reality" interpretation of quantum mechanics (see the page on Quantum
Reality for my reasoning).

What other possibilities are on the table? I prefer examples which create
themselves through some sort of *causal loop*. For example, in the film
*The Terminator* a robotic killing machine (Arnold Schwarzenegger) is
sent back in time to kill a rebel's mother before the rebel is born. The
terminator is defeated, but some of his advanced technological body parts are
retained, and it is the technological advances which result from examining those
body parts which allow future scientists to construct the Terminator. So nobody
actually invented the technology used to construct the Terminator! (I'm
convinced this is the technique Apple used to develop the amazing new
touch-screen iPods!). An object such as this which is "invented out of nothing"
by a causal loop is called a *jinn*.

As strange as this may sound, there is no logical inconsistency here. There is nothing in physics to say this is not allowed: nothing both happens and doesn't happen. And if the technique can produce something as complex as the Terminator from nothing then maybe it could also produce a universe from nothing.

J. Richard Gott has suggested a method for a universe to create itself if it
formed a causal loop in spacetime (see here).
His principle imagines an extreme curvature of spacetime forming a
*wormhole*, allowing time travel to the past. The diagram below shows the
loop in time:

There is another possible causal loop solution which would also explain anthropic fine-tuning. Maybe an advanced (human?) civilisation in the future could travel back in time to create the universe which they knew would evolve themselves into existence. Such a process would explain fine-tuning because they would have to set the parameters very precisely to ensure their particular civilisation and its inhabitants would result.

This idea is considered by J. Richard Gott on page 191 of his book *Time
Travel in Einstein's Universe*: *"With a time-travel loop, an intelligent
civilisation could produce the trunk as well"*. This ties in with the work
of Farhi, Guth, and Guven who proposed that baby universes could be created in
the lab by an advanced civilisation (see the page on The Big Brother
Universe).

#### The "No Boundary" Proposal

Another "curvy time" solution is the "No Boundary" proposal of Stephen Hawking and James Hartle.

As we trace the universe back in time to the singularity we not only find our laws of physics breaking down, but we are also left with the apparently unanswerable question of "what happened before the Big Bang"?

In 1981, Stephen Hawking and James Hartle came up with an imaginative proposal which promised to avoid the singularity at the origin of the universe, and also gave a answer to the question of why there was no time before the Big Bang. But before we can consider the theory, we need to introduce a couple of concepts.

Firstly, we need to introduce the idea of a *metric*, which is a way
of defining distance. In our *(x, y, z)* three-dimensional space, the
formula for distance is provided by Pythagoras's theorem:

When we extend this notion to 4-dimensional spacetime, it might be imagined
that the time axis is treated the same way (creating the *Euclidean* or
*Riemannian* metric):

However, Einstein's theory of special relativity says that the clock which
travels the furthest actually shows the **smallest** time
measurement, not the largest. So we need to use the *Lorentzian* metric
(the "time" element becomes negative):

Hartle and Hawking's proposal was to employ a mathematical transformation
called a *Wick rotation* to modify the time axis to avoid the
singularity. In the Wick rotation, the time axis is multiplied by the imaginary
number *i* (the square root of minus one), in which case the Lorentzian
metric is converted to a Euclidean metric:

As a result of the Wick rotation, the time axis is converted to a complex
number. The time axis is rotated 90° anticlockwise from the original time axis
to become the *imaginary time* axis:

The following diagram shows the resultant graph rotated by a further 90° (but
this time **clockwise**) so that the **imaginary
time** axis now points in the vertical direction - taking the place of
the old **time** axis:

It is as though when we travel back in time we find time itself curving round so that spacetime forms a smooth surface, instead of coming to a point singularity:

So how can the No Boundary proposal answer the question: "How did the
universe appear out of nothing?" Well, it would say that the question is flawed.
According to the No Boundary proposal, there is no time before the start of the
universe, time is self-contained within the universe. So nothing could have
created the universe: it just **is**. It just "exists", essentially
for no reason.

For me, this doesn't answer the question "Why?": "Why should the universe exist at all instead of not existing?". I feel this rather passive universe, lacking any form of motive its existence means the No Boundary proposal is not as compelling as theories in which the universe is an object capable of creating itself (such as the causal loop theories presented above). At least those theories answer the question "Why does the universe exist instead of not existing?".

For more details on the No Boundary proposal, see
chapter 3 of Stephen Hawking's arXiv paper hep-th/9409195. There is also a
series of video lectures on this subject by Stephen Hawking on
*YouTube*:

- Origin of the Universe - Stephen Hawking (1 of 5)
- Origin of the Universe - Stephen Hawking (2 of 5)
- Origin of the Universe - Stephen Hawking (3 of 5)
- Origin of the Universe - Stephen Hawking (4 of 5)
- Origin of the Universe - Stephen Hawking (5 of 5)

### Quantum Gravity: The Wheeler-DeWitt Equation

*Quantum cosmology* is the attempt to find a fully quantum mechanical
description of the large-scale universe and to explain the origin of the
universe. In order to study the universe in this way, we need to examine
spacetime - the underlying substrate which seems to compose the universe.
Because gravity is treated as a curvature of spacetime itself (in Einstein's
theory of general relativity), most quantum cosmology theories are based around
trying to find a quantum theory of gravity: *quantum gravity*.

A great puzzle in physics has been how to reconcile Einstein's theory of general relativity with quantum mechanics. General relativity remains our main theory for describing gravity, and is extremely accurate for with large objects (stars and planets, etc.). Quantum mechanics, on the other hand, is our main theory for dealing with microscopic objects, and the other three fundamental forces which act at the atomic scale (see the It's a Small World page for a description of those other three fundamental forces). General relativity describes space as being a smooth surface, but quantum mechanics reveals a discontinuous microscopic world with constant fluctuations and activity. So, each of these theories is accurate in its own right but they describe the nature of space and matter so differently that it has proven highly problematic to combine the theories into a single unified theory.

But perhaps there is an even more fundamental difference between the two
theories. Quantum mechanics attempts to describe the behaviour of particles as
they move **within** a universe composed of spacetime. However, a
theory of quantum gravity would seek to describe the behaviour of that
background spacetime itself. They seem to be two different entities, and we, as
yet, have no evidence that it is valid to apply the principles of quantization
to spacetime. This question will be considered later on this page.

Before we consider quantum gravity, we need to introduce the concept of the
*Lagrangian*. The Lagrangian is a reformulation of classical mechanics,
i.e., it is a different approach to describing the motion of a system. The
Lagrangian allows us to describe the complete motion of a system - however
complex, however many objects it contains - by a single equation: the
*Lagrangian*. The Lagrangian is defined as the kinetic energy of a system
minus its potential energy. The Lagrangian is defined for all possible
configurations of all elements of the system, so it is possible to plot the
Lagrangian as a surface in space (more accurately, an N-dimensional manifold):

It can be shown that the state of the system as it moves from point A to
point B along the Lagrangian follows the *path of least action* (in other
words, the shortest distance between the two points - see here). This
essentially means that nature wants to convert potential energy to kinetic
energy (or vice versa) by the most efficient means possible with respect to
time. In the diagram above, it can be seen that the state of the system moves
along the base of the trough between the start and end points - the shortest
distance between the two points. The Lagrangian is such a powerful tool as once
you have managed to find the Lagrangian for a system you know precisely how it
will behave: it will follow the path of least action. The Lagrangian alone is
enough to describe the system. For this reason, in modern fundamental physics,
when some new theory is proposed, it is almost invariably given in the form of a
Lagrangian (*"For reasons as yet utterly mysterious, this quantity stays as
small as possible under all circumstances. Theorists are convinced that action
must be incredibly important - so much so that the discovery of any new
fundamental law prompts a race to work out the particular action needed to
produce it. The trouble is that no one understands the principles behind
nature's infatuation with action."* - quoted from this *New
Scientist* article).

The *Hamiltonian* is closely related to the Lagrangian, but instead of
the difference between the kinetic and potential energy of a system, we now
consider the **sum** of the kinetic and potential energy. To be
precise, the Hamiltonian is the sum of the kinetic and potential energy of a
closed system expressed in terms of momentum, position, and time (see here).

Now let's see how the Hamiltonian was used to produce the very first results
of quantum mechanics. Consider the case of a single particle of mass *m*
moving in some external field given by a potential energy function, *V*,
which can depend on position: *V=V(x,y,z)*.

The (classical) Hamiltonian is the sum of kinetic plus potential energy. Let's derive it:

where *p _{x}*,

*p*, and

_{y}*p*are the spatial momenta in the direction of the Cartesian

_{z}*x*,

*y*, and

*z*axes.

Now, in order to convert this Hamiltonian to its quantum mechanical form, the momentum variable in this equation is replaced by the result for quantum momentum (which we derived earlier on the Quantum Casino page):

As this is a momentum **operator**, we need something for it to
operate **on**. So we have to again introduce this strange concept
of a **wavefunction**, ,
extending through space. Our equation for the Hamiltonian
**operator** (there is now a circumflex over the *H*) now
becomes:

The Hamiltonian, *H*, is really the total energy, *E*, so the
Hamiltonian operator is the energy operator. But we already derived an
expression for the energy operator on the Quantum
Casino page:

So substituting for this value, our equation now becomes:

which is the famous Schrödinger equation! This Hamiltonian approach is how
Erwin Schrödinger first derived the equation in his 1926 paper *"Quantization
as an Eigenvalue Problem"* (original German paper here).

In 1965, John Wheeler and Bryce DeWitt wondered what would happen if you
applied this "standard" method of quantization (called *canonical
quantization*) to the force of gravity. Let's see what happens:

The first principle to note is that of *general covariance* (sometimes
called *diffeomorphism invariance*). According to the principle of
general covariance, a theory must give the same results not matter how you move
its coordinate system around in the universe (general covariance arose from the
requirements of general relativity). To put it another way, general covariance
says that no set of coordinates is special: there is no absolute coordinate
system, everything is relative. It can be seen that general covariance
inevitably arises from another very simple principle: **there is nothing
outside the universe**, or, to put it another way, there are no absolute
axes of reference for space or time outside the universe by which we can make
our measurements.

Hence, it is impossible to define a position for our "universe object" (and hence no potential energy) and it is impossible to define a speed (and hence no kinetic energy). The Hamiltonian allows a plot of the variation in total energy of a system as the components of a system take different configurations (e.g., galaxies moving around in our "universe object"). So in this case, the Hamiltonian (kinetic + potential energy) is zero:

If we are going to quantize gravity using the
canonical quantization method, we need to convert this Hamiltonian to an
operator (as described in the derivation of the Schrödinger equation above). But
if we're converting the Hamiltonian to an operator, we once again need something
for it to operate **on**: we need a wavefunction. So another new
concept must now be introduced: the *wavefunction of the universe*. The
principle of the "wavefunction of the universe" imagines the entire universe as
a single object, a quantum object. Michio Kaku explains it well: *"When the
universe was born, it was smaller than an electron, which is a quantum object
that can exist simultaneously in many states. So the universe must also be a
quantum object and exist in many states."* (see here).
Whether it is valid to consider the entire universe as a single object subject
to the laws of quantum mechanics we shall consider in a later section on this
page, but for now we can apply our Hamiltonian operator to our "wavefunction of
the universe":

This is the *Wheeler-DeWitt equation* - a sort of Schrödinger equation
for the gravitational field. It is the most famous equation in quantum
gravity.

(A variation on this canonical quantization of gravity eventually leads to
the recent, highly-speculative field of *loop quantum gravity* - see this
*Physics World
article* by Carlo Rovelli).

#### Time and the Wheeler-DeWitt Equation

There's something remarkable about the Wheeler-DeWitt equation, and it can be seen if we expand the Hamiltonian operator:

Or, expressed in words, *the rate of change of the state of the universe
with respect to time is zero*. The universe isn't changing with time! But we
look around us and we see things changing all the time: people are walking,
birds are flying. So is the equation wrong? Well, no. What the equation is once
again telling us is that there is no external time reference by which we can
measure the progress of time within the universe: **there is no clock
outside the universe!** As Andrei Linde explains: *"The notion of
evolution is not applicable to the universe as a whole since there is no
external observer with respect to the universe, and there is no external clock
that does not belong to the universe"* (see page 25 of Andre Linde's arXiv
paper hep-th/0211048).

The Wheeler-DeWitt equation suggests a model in which all of time is laid-out
(just as the space dimension is laid-out), and all times are equally real: there
is no special "now", no distinction between past and future. In fact, "past" and
"present" do not exist - the movement of time is considered to be just an
illusion of human perception (the Wheeler-DeWitt equation reveals how the
universe does not change with time as there is no external time reference by
which we can measure the progress of time within the universe: **there is
no clock outside the universe**).

Most physicists would favour this model as the most accurate representation of time. It is also called block time because all of spacetime can be viewed as being laid-out as an unchanging four-dimensional block:

For more information about this, see this *Scientific
American article* by Paul Davies.

Eternal Life

It might come as a surprise that this orthodox "block universe" view of time in fact leads us to conclude that we possess a form of eternal life! This is a consequence of the principle that in the block time model all periods of time are equally real. If a loved one dies, you might take some comfort from the knowledge that this period of time in which your loved one is dead has, in fact, no greater reality than the time when your loved one was alive. According to physics, it is just as valid to consider your loved one as alive as it is to consider them dead!

Einstein took comfort from this knowledge when his lifelong friend Michele Besso died. He wrote a letter consoling Besso's family:

"Now he has departed from this strange world a little ahead of me. That means nothing. People like us, who believe in physics, know that the distinction between past, present, and future is only a stubbornly persistent illusion."Of course, the flip-side is that you're already dead!

#### The Wavefunction of the Universe

In our discussion on quantum gravity we have introduced the concept of a
"wavefunction of the universe". The idea is that the universe has a
corresponding wavefunction which describes the probabilities of it having
different states (forms). This is meant to be analogous to the well-established
formulation in quantum mechanics whereby a quantum particle has a corresponding
wavefunction which describes the probabilities of it appearing in a certain
location, say. Michio Kaku explains it well: *"When the universe was born, it
was smaller than an electron, which is a quantum object that can exist
simultaneously in many states. So the universe must also be a quantum object and
exist in many states."* (see here). The
idea of a wavefunction for the universe was first proposed in the Hartle and
Hawking "No Boundary" proposal, though it has since been used in many models of
quantum cosmology.

However, it is possible to imagine several objections to this belief that the
entire universe can be treated as if it was a quantum particle. In his online
physics textbook *Motion Mountain*, Christoph Schiller draws attention to
the many differences between the universe and a particle. He comes to the
conclusion that there can be no such thing as a wavefunction of the universe:
*"Beware of anybody who claims to know something about the wavefunction of
the universe. Just ask him: If you know the wavefunction of the universe, why
aren't you rich?"* (see page 835 of Motion
Mountain).

Schiller's first complaint is that the universe cannot have a "state" because
a "state" is defined as being the form of a system **at a given
time**. However, that would appeat to require some aspect of the universe
to be able to change with time when the universe is viewed "from outside", i.e.,
the universe could appear to change when viewed by some form of external
observer. However, as we discovered earlier in our discussion on quantum
gravity, **there is nothing outside the universe**, certainly no
external clock by which changes in the universe might be measured (this
unchanging universe was revealed by the Wheeler-DeWitt equation in the previous
section).

This idea of a state as being the form of a system at a particular moment in
time is even used (perhaps unwittingly) by Stephen Hawking in a description of
the wavefunction of the universe on page 43 of his arXiv paper hep-th/9409195: *"It is useful
to introduce a concept that can describe the state of the universe at
one time"* (emphasis added). But we know that the wavefunction is
not dependent on time! There's no external time axis. So when we talk about a
"wavefunction of the universe" we are really considering a "block universe"
state in which all of time is laid-out and all times are equally real (see the
previous section and the section on "Time" on page 11 of Carlo Rovelli's arXiv
paper gr-qc/9903045 which
considers this time-independent notion of "state").

We can also see that Michio Kaku's previous convincing quote falls into the same trap by inferring some notion of time external to the universe (by his use of the word "simultaneously" implying some sort of external time reference):

The basic idea behind Michio Kaku's logic is that the universe, being
composed of particles which obey the laws of quantum mechanics, could itself be
considered a quantum object. This is fair enough. But an anology which treats
the universe as though it is an actual particle is unfortunate: the entire
universe can never behave in the same way as a particle. As was revealed
earlier, **there is nothing outside the universe**, no external
axes, so a "universe particle" can never have a location, or a momentum - both
of which are mainstays in the quantum behaviour of a particle. Likewise, we
cannot think of the universe as having a "time" (as shown by the Wheeler-DeWitt
equation) or an energy (*"the energy of a closed universe is equal to
zero"* - quoted from Alexander Vilenkin's arXiv paper gr-qc/9302016). So we should never
think of the universe as a particle, in a quantum superposition (as Michio Kaku
suggests). If you need proof, try using that "universe particle" in the
double-slit experiment! Where would you get the slits from?!

So if we shouldn't think of the universe as a particle, how should we imagine
it? Surely when we are considering the wavefunction of the entire universe
**as a whole** then we should not be just considering it as a
single point-particle moving around in a background of spacetime. Surely we
should consider it as *"the collection of all matter and radiation,
plus all of spacetime"* (see page 835 of Motion
Mountain). But is it valid to apply quantization techniques to spacetime
itself? I think an answer can be found if we remember that before we observe a
particle's position, we have to consider it to be in superposition of many
different possible positions. We should then remember John Wheeler's description
of general relativity:

*"Matter tells spacetime how to curve, spacetime tells matter how to move"*. If that is the case, then when a particle is in a superposition state we should also consider spacetime (the shape of which is determined by the position of the particle) to be in a superposition state! This implies that it

**is**valid to apply the standard quantum mechanical formulation to spacetime (and, therefore, gravity), and it hence is also valid to consider a "wavefunction of the universe". When we quantize gravity we are quantizing the gravitational field and spacetime which forms the very essence of the universe itself (for more on this, see Carlo Rovelli's arXiv paper gr-qc/9903045).

#### Decoherence and the Wavefunction of the Universe

If we're now considering the entire universe to be a single quantum system, a
superposition state described by a wavefunction, then we might wonder why we do
not see superpositions in reality. For example, why do we see not see galaxies
in a superposition state like Schrödinger's Cat, dead and alive at the same
time, so to speak. The process by which classical states emerge from quantum
superpositions is called *decoherence*, and this is considered in depth
in the page on Quantum
Decoherence.

Stephen Hawking considers this a likely solution on page 57 of his
aforementioned arXiv paper hep-th/9409195: *"People
normally try to account for decoherence by interactions with an external system,
such as a heat bath, that is not measured. In the case of the universe, there is
no external system, but I would suggest that the reason we observe classical
behaviour is that we can only see part of the universe."* I believe this
emphasis on an **external** environment in order to achieve
decoherence is rather a misconception. As discussed at the top of the Quantum
Decoherence page (the discussion on rainbows), the crucial factor in
decoherence is that the object being measured, together with the measurement
apparatus, should be considered as **one** entangled system. The
emphasis is on the fact that we should consider the resultant entangled system
as being a **single** system. And this is how we should think of
the universe as a whole: one entangled system. There is no need to postulate any
external environment. A galaxy, for example, would decohere by its interactions
with the other galaxies which surround it.

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