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Sean Carroll, Caltech
What We (Don’t) Know About
the Beginning of the Universe
1. What we know about the Big Bang
2. The spacetime viewpoint
3. The quantum viewpoint
What we know about the Big Bang:
1. Something Bang-like happened.
standard GR
(ΛCDM)
today
allowed
histories
[Carroll & Kaplinghat][Planck]
cosmic background radiation primordial nucleosynthesis
The universe 13.8 billion years ago was hot, dense,
expanding very rapidly, and decelerating.
What we know about the Big Bang:
2. Classical GR suggests singularities are generic.
Highly symmetric universes tend
to have an initial singularity (Lemaître).
More strongly, Hawking’s theorem:
compact expanding universes obeying
the Strong Energy Condition (gravity
attracts) always have singularities.
[Donald Menzel, Popular Science, 1932]
But the Strong Energy
Condition can be violated.
And theorists are happy to
consider modifying GR.
What we know about the Big Bang:
3. The early universe had extremely low entropy.
time
early universe
S ~ Sradiation ~ 1088
today
S ~ SBH ~ 10103
future
S ~ SdS
~ 10123
Of all the states that look macroscopically like our present
universe, only a tiny fraction evolved from smooth states.
Most were chaotic, Planckian, singular.
space of
states
“macrostates” = sets
of macroscopically
indistinguishable microstates
Boltzmann, 1870s: entropy counts the number
of states that look the same macroscopically.
Low initial entropy is
an enormous fine-tuning.
Calls out for a robust
explanation.
Inflation doesn’t explain why entropy was initially low.
Inflation: if a patch of
space starts in a false
vacuum, it naturally
accelerates, creates
energy, smooths out,
and reheats into
matter and radiation.
But that initial proto-inflationary state is even lower-entropy
than the conventional hot big Bang (1 < Sinflation < 1015
).
You don’t explain low entropy by positing even lower entropy.
1. What it’s like to have a beginning.
The spacetime viewpoint on the beginning of the universe
size
time
size
time
2. Ways of avoiding a beginning
– eternal universes.
Bouncing
ReproducingHibernating
Cyclic
size
time
size
time
size
time
What it’s like to have a beginning
Don’t ever say the universe “came into existence.”
Sounds like a process within time, rather than the
beginning of time itself.
Rather, there was an initial moment – a time before
which there was no other time.
What “caused” the universe?
Wrong question. Rather: is it plausible that the laws
of physics allow for a universe with a beginning?
(Yes.)
Bouncing cosmologies
Smooth out the singularity, either through new degrees
of freedom (fields, branes, dimensions), or through
intrinsically quantum effects.
Stringy Bounce
[Veneziano]
Quantum
Cosmology
[Bojowald, Ashtekar,
Page, Hartle,
Hawking, Hertog]
de Sitter Bounce
[Aguirre, Gratton]
Ekyprotic Bounce
[Khoury, Ovrut,
Steinhardt, Turok]
Bouncing cosmologies have an entropy puzzle:
•If entropy grows monotonically, requires
infinite fine-tuning.
•If entropy has a minimum at the bounce, why?
size
time
entropy
?
?
Cyclic cosmologies
Repeat the bounce over and over.
[Turok, Steinhardt; Penrose]
Hibernating cosmologies
Universe is quiescent and quasi-stationary into
the eternal past; at some point undergoes a phase
transition and begins to expand.
[Brandenberger, Vafa] [Greene, Hinterbichler, Judes, Parikh]
String gas cosmology Primordial degravitation
Both cyclic and hibernating cosmologies have an
entropy catastrophe:
•Entropy grows monotonically for all time. Requires
infinite fine-tuning in the infinite past.
size
time
entropy
[Farhi, Guth, Guven]
Reproducing cosmologies
Imagine a “parent” universe that
is itself quiescent and high-entropy.
But through some mechanism it can give birth to new
offspring universes, with initially low entropy.
E.g. spacetime quantum tunneling into disconnected
“baby universes.”
size
time
2 large
dimensions
Alternatively: spontaneous compactification
1 large dimension,
1 compact
2 large
dimensions
1 large dimension,
1 compact
Six-dimensional de Sitter space w/electromagnetic fields
will spontaneously nucleate four-dim de Sitter universes.
[Carroll,
Johnson,
& Randall]
Result: a time-symmetric multiverse
• New universes branch off from the parent universe in
both directions of time. Overall time-symmetric.
• Easier to create new low-entropy universes than high-
entropy ones.
• This might explain why our Big Bang had low entropy.
Reproducing cosmologies don’t have an entropy problem!
•Entropy grows without bound toward past and future.
•There is a middle point of lowest entropy, but it
needn’t be “low” in any objective sense. (Indeed,
it can be locally maximal.)
size
time
entropy
[Carroll & Chen; see also Barbour, Koslowski & Mercati;
Hartle & Hertog; Goldstein, Tomulka & Zanghi; Carroll & Guth]
The quantum viewpoint on the beginning of the universe
Quantum theory describes the evolution of quantum
states living in a Hilbert space H, obeying
Schrödinger’s equation .
We often start with a classical system
and “quantize” it, yielding a quantum
theory “of” that system. But that’s
human convention, not Nature.
Honest quantum questions are
about what happens to vectors
in Hilbert space, evolving under the Schrödinger equation.
time(?)
Derived/
emergent
space
fieldsparticles
causality
light cones
metric collapse/
branching
wave functions
Hilbert space
tensor products
entanglement
Hamiltonian
information
entropy
pointer states
Fundamental
Emergence in QM
locality
Time evolution: the Quantum Eternity Theorem
• Consider a universe described by a quantum state
obeying Schrödinger’s equation
with nonzero energy, governed by laws of
physics that are independent of time.
• Then: the universe is eternal.
(Time t runs from –∞ to +∞.)
[Carroll, 2008, arxiv:0811.3722]
In quantum mechanics, if time is fundamental, it never ends.
Expand the state in energy eigenstates:
Each phase just rotates in a circle;
the set of all of them move in a straight
line through a torus. No singularities, barriers, etc.
A generic quantum universe lasts forever, without
a beginning or an end.
Recurrence theorem: if Hilbert space
H is finite-dimensional, states return
to their starting points infinitely often.
Problems with an eternal quantum universe:
recurrences, fluctuations, Boltzmann brains.
Entropy is usually maximal
(equilibrium). Downward
fluctuations are suppressed:
Almost all observers are minimal
fluctuations: “Boltzmann brains.”
Possible solution: Hilbert space is infinite-dimensional.
There is no recurrence theorem in an infinite-dimensional
Hilbert space. Quantum equivalent of an unbounded
phase space.
The quantum state has infinite room to grow and change.
This is the kind of quantum
theory that might ultimately
have as an emergent classical
spacetime a bouncing or
reproductive cosmologies.
Entropy growing without bound in both directions of time.
Alternative: time is emergent, not fundamental
Loophole for Quantum Eternity Theorem: we live
in a single energy eigenstate. E.g. .
Seems non-generic, but is exactly what we get by
quantizing general relativity: the Wheeler-DeWitt
equation for a wave function of spatial three-metrics.
Where does time come from?
[e.g. Hartle, Hawking]
Time can emerge in quantum mechanics
because we can superpose different states
[Page & Wootters 1983]
Emergent time: a stationary
state is a superposition;
one subsystem serves as
an effective “clock.”
t = 1
t = 2
t = 3
Ordinarily: quantum state
evolves as time passes.
If Hilbert space is infinite-dimensional, emergent “time”
can run forever. No need for a beginning – but there could
be one.
But if Hilbert space is finite-dimensional, there are only a
finite number of possible clock states.
Therefore, time will have a beginning.
Ψ [hij, ψa
] internal “clock” λ(hij, ψa
)
semiclassical
trajectory
superspace = {3-geometries, matter fields}
universe had a
beginning
universe may or
may not have had
a beginning
universe is eternal,
with a finite
recurrence time
(& Boltzmann brains)
universe is eternal,
and need never
experience
recurrence
Was the Big Bang the beginning of the universe?timeis
emergent
timeis
fundamental

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What We (Don't) Know About the Beginning of the Universe

  • 1. Sean Carroll, Caltech What We (Don’t) Know About the Beginning of the Universe 1. What we know about the Big Bang 2. The spacetime viewpoint 3. The quantum viewpoint
  • 2. What we know about the Big Bang: 1. Something Bang-like happened. standard GR (ΛCDM) today allowed histories [Carroll & Kaplinghat][Planck] cosmic background radiation primordial nucleosynthesis The universe 13.8 billion years ago was hot, dense, expanding very rapidly, and decelerating.
  • 3. What we know about the Big Bang: 2. Classical GR suggests singularities are generic. Highly symmetric universes tend to have an initial singularity (Lemaître). More strongly, Hawking’s theorem: compact expanding universes obeying the Strong Energy Condition (gravity attracts) always have singularities. [Donald Menzel, Popular Science, 1932] But the Strong Energy Condition can be violated. And theorists are happy to consider modifying GR.
  • 4. What we know about the Big Bang: 3. The early universe had extremely low entropy. time early universe S ~ Sradiation ~ 1088 today S ~ SBH ~ 10103 future S ~ SdS ~ 10123 Of all the states that look macroscopically like our present universe, only a tiny fraction evolved from smooth states. Most were chaotic, Planckian, singular.
  • 5. space of states “macrostates” = sets of macroscopically indistinguishable microstates Boltzmann, 1870s: entropy counts the number of states that look the same macroscopically. Low initial entropy is an enormous fine-tuning. Calls out for a robust explanation.
  • 6. Inflation doesn’t explain why entropy was initially low. Inflation: if a patch of space starts in a false vacuum, it naturally accelerates, creates energy, smooths out, and reheats into matter and radiation. But that initial proto-inflationary state is even lower-entropy than the conventional hot big Bang (1 < Sinflation < 1015 ). You don’t explain low entropy by positing even lower entropy.
  • 7. 1. What it’s like to have a beginning. The spacetime viewpoint on the beginning of the universe size time size time 2. Ways of avoiding a beginning – eternal universes. Bouncing ReproducingHibernating Cyclic size time size time size time
  • 8. What it’s like to have a beginning Don’t ever say the universe “came into existence.” Sounds like a process within time, rather than the beginning of time itself. Rather, there was an initial moment – a time before which there was no other time. What “caused” the universe? Wrong question. Rather: is it plausible that the laws of physics allow for a universe with a beginning? (Yes.)
  • 9. Bouncing cosmologies Smooth out the singularity, either through new degrees of freedom (fields, branes, dimensions), or through intrinsically quantum effects. Stringy Bounce [Veneziano] Quantum Cosmology [Bojowald, Ashtekar, Page, Hartle, Hawking, Hertog] de Sitter Bounce [Aguirre, Gratton] Ekyprotic Bounce [Khoury, Ovrut, Steinhardt, Turok]
  • 10. Bouncing cosmologies have an entropy puzzle: •If entropy grows monotonically, requires infinite fine-tuning. •If entropy has a minimum at the bounce, why? size time entropy ? ?
  • 11. Cyclic cosmologies Repeat the bounce over and over. [Turok, Steinhardt; Penrose]
  • 12. Hibernating cosmologies Universe is quiescent and quasi-stationary into the eternal past; at some point undergoes a phase transition and begins to expand. [Brandenberger, Vafa] [Greene, Hinterbichler, Judes, Parikh] String gas cosmology Primordial degravitation
  • 13. Both cyclic and hibernating cosmologies have an entropy catastrophe: •Entropy grows monotonically for all time. Requires infinite fine-tuning in the infinite past. size time entropy
  • 14. [Farhi, Guth, Guven] Reproducing cosmologies Imagine a “parent” universe that is itself quiescent and high-entropy. But through some mechanism it can give birth to new offspring universes, with initially low entropy. E.g. spacetime quantum tunneling into disconnected “baby universes.” size time
  • 15. 2 large dimensions Alternatively: spontaneous compactification 1 large dimension, 1 compact 2 large dimensions 1 large dimension, 1 compact Six-dimensional de Sitter space w/electromagnetic fields will spontaneously nucleate four-dim de Sitter universes. [Carroll, Johnson, & Randall]
  • 16. Result: a time-symmetric multiverse • New universes branch off from the parent universe in both directions of time. Overall time-symmetric. • Easier to create new low-entropy universes than high- entropy ones. • This might explain why our Big Bang had low entropy.
  • 17. Reproducing cosmologies don’t have an entropy problem! •Entropy grows without bound toward past and future. •There is a middle point of lowest entropy, but it needn’t be “low” in any objective sense. (Indeed, it can be locally maximal.) size time entropy [Carroll & Chen; see also Barbour, Koslowski & Mercati; Hartle & Hertog; Goldstein, Tomulka & Zanghi; Carroll & Guth]
  • 18. The quantum viewpoint on the beginning of the universe Quantum theory describes the evolution of quantum states living in a Hilbert space H, obeying Schrödinger’s equation . We often start with a classical system and “quantize” it, yielding a quantum theory “of” that system. But that’s human convention, not Nature. Honest quantum questions are about what happens to vectors in Hilbert space, evolving under the Schrödinger equation.
  • 19. time(?) Derived/ emergent space fieldsparticles causality light cones metric collapse/ branching wave functions Hilbert space tensor products entanglement Hamiltonian information entropy pointer states Fundamental Emergence in QM locality
  • 20. Time evolution: the Quantum Eternity Theorem • Consider a universe described by a quantum state obeying Schrödinger’s equation with nonzero energy, governed by laws of physics that are independent of time. • Then: the universe is eternal. (Time t runs from –∞ to +∞.) [Carroll, 2008, arxiv:0811.3722]
  • 21. In quantum mechanics, if time is fundamental, it never ends. Expand the state in energy eigenstates: Each phase just rotates in a circle; the set of all of them move in a straight line through a torus. No singularities, barriers, etc. A generic quantum universe lasts forever, without a beginning or an end.
  • 22. Recurrence theorem: if Hilbert space H is finite-dimensional, states return to their starting points infinitely often. Problems with an eternal quantum universe: recurrences, fluctuations, Boltzmann brains. Entropy is usually maximal (equilibrium). Downward fluctuations are suppressed: Almost all observers are minimal fluctuations: “Boltzmann brains.”
  • 23. Possible solution: Hilbert space is infinite-dimensional. There is no recurrence theorem in an infinite-dimensional Hilbert space. Quantum equivalent of an unbounded phase space. The quantum state has infinite room to grow and change. This is the kind of quantum theory that might ultimately have as an emergent classical spacetime a bouncing or reproductive cosmologies. Entropy growing without bound in both directions of time.
  • 24. Alternative: time is emergent, not fundamental Loophole for Quantum Eternity Theorem: we live in a single energy eigenstate. E.g. . Seems non-generic, but is exactly what we get by quantizing general relativity: the Wheeler-DeWitt equation for a wave function of spatial three-metrics. Where does time come from? [e.g. Hartle, Hawking]
  • 25. Time can emerge in quantum mechanics because we can superpose different states [Page & Wootters 1983] Emergent time: a stationary state is a superposition; one subsystem serves as an effective “clock.” t = 1 t = 2 t = 3 Ordinarily: quantum state evolves as time passes.
  • 26. If Hilbert space is infinite-dimensional, emergent “time” can run forever. No need for a beginning – but there could be one. But if Hilbert space is finite-dimensional, there are only a finite number of possible clock states. Therefore, time will have a beginning. Ψ [hij, ψa ] internal “clock” λ(hij, ψa ) semiclassical trajectory superspace = {3-geometries, matter fields}
  • 27. universe had a beginning universe may or may not have had a beginning universe is eternal, with a finite recurrence time (& Boltzmann brains) universe is eternal, and need never experience recurrence Was the Big Bang the beginning of the universe?timeis emergent timeis fundamental