Download Effective Theories and Modified Gravity

Effective Theories and
Modified Gravity
A Bad Cop wrapped in a Wet Blanket
standing in a Cold Shower
Cliff Burgess
Outline
• Effective Field Theory and Gravity
• EFT and theoretical error
• Decoupling
• EFT in time-dependent settings
• Decoupling and when it fails
• Gravitational exceptionalism
• Learning from UV completions
• The case for modifying gravity
• What microphysics can teach us
Ole Miss 2014
Outline
• Effective Field Theory and Gravity
• EFT and theoretical error
• Decoupling
• EFT in time-dependent settings
• Decoupling and when it fails
• Gravitational exceptionalism
• Learning from UV completions
• The case for modifying gravity
• What microphysics can teach us
Ole Miss 2014
Outline
• Effective Field Theory and Gravity
• EFT and theoretical error
• Decoupling
• EFT in time-dependent settings
• Decoupling and when it fails
• Gravitational exceptionalism
• Learning from UV completions
• The case for modifying gravity
• What microphysics can teach us
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Part I
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EFT and theoretical error
Decoupling
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EFT and theoretical error
Quantum field theory is a precision science:
e.g. QED:
𝑎𝜇 = 1159652188.4 4.3 10−12 (exp)
𝑎𝜇 = 1159652140 27.1 10−12
(th)
QED’s renormalizability is an important part of its
calculability, and so also underpins the theory error
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EFT and theoretical error
• General Relativity is also a precision science:
e.g. solar system tests, binary pulsar, ...
Ole Miss 2014
EFT and theoretical error
• General Relativity is also a precision science:
e.g. solar system tests, binary pulsar, ...
𝑑𝑃/𝑑𝑡 = −2.408(10) 10−12 (exp)
𝑑𝑃/𝑑𝑡 = −2.40243 5 10−12 (th)
This comparison meaningless if size of quantum
effects is unknown
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EFT and theoretical error
Cannot afford a ‘split-brain’ mentality:
ℏ=1
𝐺𝑁 = 0
ℏ=0
𝐺𝑁 = 1
Quantum corrections can be calculable within an
effective field theory framework, even for gravity
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EFT and theoretical error
e.g. for graviton-graviton scattering about a static
background in GR:
1
1 2
2
2
2 +. .
𝐿 = 𝜕ℎ +
ℎ 𝜕ℎ +
ℎ
𝜕ℎ
2
𝑀𝑝
𝑀𝑝
+
gives
𝐴𝑛𝑜 𝑙𝑜𝑜𝑝𝑠 ≅
𝑄2
𝑀𝑝
2
About flat space Q
is the CM energy
(de Witt)
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EFT and theoretical error
• The loop integral diverges, and higher-order loops
diverge more and more (because the coupling has
dimensions of negative powers of mass)
𝐴1𝑙𝑜𝑜𝑝 =
𝐴2𝑙𝑜𝑜𝑝 =
𝑄2
𝑀𝑝
4
𝑑4 𝑝
𝑝6
2𝜋 4 𝑝2 + 𝑄 2
𝑄2
𝑑4 𝑝
𝑀𝑝 6
2𝜋
4
2
4
𝑝10
𝑝2 + 𝑄 2
7
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EFT and theoretical error
All divergences cannot be absorbed into Newton’s
constant
2
M
L
c3 3
p
2


R  c1R  c2 R R  2 R  
2
m
g
Divergences can be absorbed if GR is just first term in
a derivative expansion that includes all possible local
interactions allowed by symmetries
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EFT and theoretical error
Predictive provided we regard calculations as expansions
in 𝑄 2 /𝑚2 since no negative powers of Q arise in the
loop expansion.
e.g. L-loop contribution to graviton scattering at energy Q
involving E external lines (in dimensional regularization) and
𝑉𝑖𝑘 vertices involving 𝑖 fields and 𝑘 derivatives:
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EFT and theoretical error
• Leading order, 𝑄2 /𝑀𝑝 𝐸−2 , corresponding to:
• L = 0 and Vik = 0 unless k = 2 ;
• i.e. classical GR: tree graphs using only interactions with
two derivatives.
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EFT and theoretical error
• Next-to-leading order, 𝑄4 /𝑀𝑝 𝐸 , corresponding to one of two
cases:
• L = 1 and Vik = 0 unless k = 2 ;
• L = 0 and Vik = 0 unless k = 2 or 4 and Vi4 = 1 ;
• i.e. one-loop GR, plus tree graphs including one
‘counterterm’ taken from the R2 interactions .
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EFT and theoretical error
• Points to notice:
• Must do Q/M expansion: if not even the semi-classical
approximation breaks down!
• The reference scale M is a physical (renormalized) mass
and never a cutoff
• Cutoffs are guaranteed never to appear in a low-energy
theory, since they always cancel in physical results.
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EFT and theoretical error
about cutoffs, :
• PointsMore
to notice:
• Must do Q/M expansion: if not even the semi-classical
EFT canbreaks
depend
on  since it is
approximation
down!
generated by integrating out physics
E >
• Thewith
reference
scale M is a physical (renormalized) mass
and never a cutoff
onceareused
in physical
• Cutoffs
guaranteed
neverprediction
to appear inone
a low-energy
theory,
since they E
always
in physical
results.
also integrates
< ,cancel
at which
point 
drops out of physical quantities.
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EFT and theoretical error
•
Can calculate coeff a
about cutoffs, : purely within LE
PointsMore
to notice:
theory
by tracking
• Must do Q/M expansion: if not even
the semi-classical
logfor
cutoff dependence.
 is often
a useful
approximation
breaks
down!proxy
following how physics depends on a real
• Thephysical
reference mass
scale MMis a physical (renormalized) mass
and never a cutoff
LE:are guaranteed
A = a lnnever
(/m)
• Cutoffs
to appear in a low-energy
theory,
always
HE:since they
A=
a ln cancel
(M/)in physical results.
phys:
A = a ln (M/m)
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EFT and theoretical error
•
Calculation of coeff a
about cutoffs, : purely within LE
PointsMore
to notice:
theory
needn’t track
• Must do Q/M expansion: if not even
the semi-classical
Tracking
powers
of  iscoeff b or c of physical
approximation
breaks
down!
result.
is often less useful
• The reference scale M is a physical (renormalized) mass
LE:a cutoff A = a 2  b m2
and never
2  a 2
HE:are guaranteed
A = c Mnever
• Cutoffs
to appear in a low-energy
theory,
since they
always
phys:
A=
c M2cancel
 b min2 physical results.
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Decoupling
But how to interpret the non-GR terms in the action?
2
M
L
c3 3
p
2


R  c1R  c2 R R  2 R  
2
m
g
As would be obtained if we ‘integrate out’ a collection of
particles with 𝑚2 ≫ 𝑄 2
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Decoupling
Mass in loop
scale,
so:are there?
Whatsets
other
scales
lightest mass dominates for dim > 4 terms
eg: inmass
4D inflation,
given
and<M4p
heaviest
dominates
for Hdim
there is also v2 = H Mp plus mass
𝑐3 coupling to𝑎𝑘
of any other particles
= 𝑘
2
2
inflaton
2
𝑚
16𝜋 𝑚𝑘
In extra dimensional inflation there is
Notice in particular 𝑀𝑝 is the least important scale when it
mKK < Ms < Mp , and so on…
appears also
in a denominator
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Decoupling
Decoupling: at low energies heavy particles always
contribute suppressed by their mass (once the dim < 5
couplings are appropriately renormalized)
mass M
𝛿𝐿~𝑀 𝑅 + 𝑅2 ln 𝑀 + 𝑅3 𝑀2 + ⋯
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EFT summary
• Quantum effects in gravity are calculable
• Must recognize the implicit low-energy expansions:
Q/𝑀𝑝 and Q/𝑚
• In particular this justifies domain of classical
approximation
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EFT summary
• Quantum effects in gravity are calculable
• Must recognize the implicit low-energy expansions:
Q/𝑀𝑝 and Q/𝑚
• Never get anything but a series in local powers of
fields and derivatives in this way: f ( R) ?
2
M
L
c3 3
p
2


R  c1R  c2 R R  2 R  
2
m
g
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EFT summary
• Quantum effects
in gravity
are calculable
These
are usually
useful for cosmology
and particle
physics,
and when
used
• Must recognize
the implicit
low-energy
expansions:
make these unusually sensitive to UV
Q/𝑀𝑝 and Q/𝑚
physics.but a series in local powers of
• Never get anything
fields and derivatives in this way: f ( R) ?
• Heavy particles generically decouple. Yet theories with lowdimension interactions – like scalars with 𝑚2 𝜑 2 terms – can
be sensitive to very heavy particles.
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EFT summary
• Quantum effects
inare
gravity
area calculable
These
usually
problem for theories
that break
lorentzlow-energy
invarianceexpansions:
in the
• Must recognize
the implicit
UV, since they spread to all particles
Q/𝑀𝑝 and Q/𝑚
throughbut
loops
• Never get anything
a series in local powers of
fields and derivatives in this way: f ( R) ?
• Heavy particles generically decouple. Yet theories
with low-dimension interactions – like scalars with
𝑚2 𝜑 2 terms – can be sensitive to very heavy
particles.
• Ditto for lorentz-violating diffs btwn 𝜑 2 and 𝛻𝜑 2
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Part II
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Time dependent situations
Gravitational exceptionalism
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Decoupling and Horizons
• Effective field theory as applied to geometries having
horizons (eg: black holes, inflation) introduces some
new issues
• Adiabatic time slicing is not the static one
• These are similar to those which arise for timedependent background fields, F(t)
• Slow evolution
• No level crossing
1 dF

F dt
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Time dependent situations
• Can effective theories be used for time-dependent
problems?
• Q1: when do effective theories capture the time dependence
of evolution in the full theory?
• Since EFTs have higher-derivative interactions, why
aren’t there new runaway solutions?
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Time dependent situations
• Can effective theories be used for time-dependent
problems?
• Q1: when do effective theories capture the time dependence
of evolution in the full theory?
• Since EFTs have higher-derivative interactions, why
aren’t there new runaway solutions?
• Q2: Can an effective theory be set up to describe the
fluctuations about a time-dependent background?
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Time dependent situations
• Time dependent solutions to EFT only need agree with
adiabatic evolution in the full theory
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Time dependent situations
• Time dependent solutions to EFT only need agree with
adiabatic evolution in the full theory
For example, for the theory:
𝐿 = 𝜑2 + 𝜓 2 − 𝑀2 𝜓 2 − 𝑔 𝜑2 𝜓
Feynman tree graphs give
𝐿𝑒𝑓𝑓
2
𝑔
4+⋯
= 𝜑2 −
𝜑
2𝑀2
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Time dependent situations
• Time dependent solutions to EFT only need agree with
adiabatic evolution in the full theory
𝑀2 𝜓
𝑔 2
= − 𝜑
2
But the equation
𝜓+
has solutions
𝑔
2
𝜓= −
𝜑
+⋯
2
2𝑀
and so equivalent way to get 𝐿𝑒𝑓𝑓 (at tree level) is
𝐿𝑒𝑓𝑓 (𝜑) = 𝐿 𝜑, 𝜓(𝜑)
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Time dependent situations
• Time dependent solutions to EFT only need agree with
adiabatic evolution in the full theory
After all
𝑔 2
2
𝜓+ 𝑀 𝜓= − 𝜑
2
also has non-adiabatic solutions
𝜓=
𝑐𝑘
𝑘=∓
𝑒 𝑖𝑘𝑀𝑡
𝑔
2+⋯
−
𝜑
2𝑀2
and so equations of motion of the EFT only agree with
the adiabatic solutions
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Time dependent situations
• This is also why runaway solutions do not arise
• Higher-order equations of motion normally acquire new,
often runaway, solutions:
• e.g. if
𝐿 = 𝑞 2 + 𝑞 2 /𝑀2 then 𝑞 + 𝑞 /𝑀2 = 0
.
and
so 𝑞 𝑡 = 𝑎 + 𝑏𝑡 + 𝑐 𝑒 𝑀𝑡 + 𝑑𝑒 −𝑀𝑡
• Key point: must perturb in the effective interactions, since
they are only required to reproduce the full physics to fixed
order in 1/𝑀
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Time dependent situations
• This resembles models that inflate based on highercurvature interactions:
L  M R  z R 
2
p
2
• inflates with 𝑅 = 𝐻 2 = 𝑀𝑝 2 /𝜁
• gives the right primordial density fluctuations provided z ~
108.
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Time dependent situations
• What seems odd is that successive terms in the
curvature expansion are equal size (and not because
the first term was particularly small)
• Should the dimensionless part of the coefficient of the 𝑅3
term be order 1 or order 𝜁 2 ?
• Time dependence is 𝑎(𝑡)~𝑒 𝐻𝑡 with 𝐻~𝑀𝑝 /√𝜁 seems hard
to understand as agreeing with full theory order by order in
1/𝑀 ~ 𝜁/𝑀𝑝
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Time dependent situations
• Can effective theories be used for expansions about
time-dependent backgrounds?
• At face value there is a problem: we distinguish light from
heavy states from one another using energy, but lose
energy conservation when expanding about a timedependent background.
• Can nonetheless work if background is adiabatically
evolving since in this case can define an adiabatic notion of
energy at a given time, 𝐸(𝑡)
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Time dependent situations
• For EFT built around adiabatic evolution:
• Must check whether the low-energy criterion, 𝐸(𝑡) ≪
𝑀(𝑡) remains true as a function of time:
eg: must
avoid level crossing:
M
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Gravitational exceptionalism
Gravitational exceptionalism:
Q: Isn’t gravity a unique situation for which insights from
EFTs in other areas do not apply?
In particular: doesn’t inflation (or Hawking radiation)
violate decoupling by stretching initially short-wavelength
modes out to longer distances?
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Gravitational exceptionalism
• Descent of modes is not special to gravity:
• similar effects can also happen in other time-dependent
settings (like the collapse of Landau levels 𝐸 = 𝑛 𝜔 with
𝜔 ∝ 𝐵 when magnetic field 𝐵 is turned off).
B
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Gravitational exceptionalism
• Descent of modes is not special to gravity:
• Normally the appearance of such states has no low-energy
effects, so long as the time dependence is adiabatic so states
descend in their adiabatic vacuum.
• All known examples of trans-Planckian phenomena rely on
breakdown of adiabaticity. (Some also violate lorentz
invariance in the far UV, but this is not required, as may be
seen from rolling scalar examples.)
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Gravitational exceptionalism
Trans-planckianism
• Less precise version: Since we don’t understand
quantum gravity (at short distances), how do we
know that trans-Planckian physics decouples from
long-wavelength physics?
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Gravitational exceptionalism
Trans-planckianism
• Less concrete version: Since we don’t understand quantum
gravity (at short distances), how do we know that transPlanckian physics decouples from long-wavelength physics?
Cannot answer for sure without knowing the physics
above the Planck scale. but:
• Trans-planckian physics appears to decouple in string theory
• If decoupling fails must explain why any understanding of
nature is possible at all (ie there is an uncontrolled
theoretical error)
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Exceptionalism summary
• Time dependence can be captured within an EFT
• Must require the time dependence to be adiabatic
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Exceptionalism summary
• Time dependence can be captured within an EFT
• Must require the time dependence to be adiabatic
• Must check that large mass hierarchies do not
become small over time
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Exceptionalism summary
• Time dependence can be captured within an EFT
• Must require the time dependence to be adiabatic
• Must check that large mass hierarchies do not
become small over time
• Cosmology appears not to introduce qualitatively
new considerations beyond those associated with
time-dependence.
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Part III
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The case for modifying gravity
Microphysics as a crucial clue
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Dark Matter/Energy or Modified GR?
• Evidence for Dark Matter
comes from many sources:
• Mass in galaxies
• Mass in clusters of galaxies
• Temperature fluctuations in
the CMB
• Start of galaxy formation.
• Evidence for Dark Energy
is less robust, but would
involve modifications to
gravity over large distances.
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Dark Matter/Energy or Modified GR?
• Modifications of gravity
does a poor job describing
observations
• Mass in galaxies
• Mass in clusters of galaxies
• Temperature fluctuations in
the CMB
• Start of galaxy formation.
• Both require modification
at large distances: very hard
to do sensibly
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Modifications in the UV
• Short-distance modification to
gravity is very likely, since
predictability breaks down at high
energies.
• Sensible UV modifications seem
difficult to get (but not impossible).
• String theory provides an example
UV modification.
• Difficult to test since their details
decouple from low-energy physics
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Modifications in the IR
• Long-distance modifications to gravity are very
difficult to construct without violating fundamental
principles like unitarity, cluster decomposition, etc.
(Weinberg)
• Eg: gauge invariance emerges as a requirement for
massless spin-two particle coupled to stress energy
massless spin 2  C     h  
Lint  h T


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Modifications in the IR
• Long-distance modifications to gravity are very
difficult to construct without violating fundamental
principles like unitarity, cluster decomposition, etc.
• Eg: gauge invariance emerges as a requirement for
massless spin-two particle coupled to stress energy
• After decades of investigation, only known
consistent IR modifications are scalar/vector/tensor
theories (possibly in higher dimensions).
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Microphysics as a crucial clue
• The puzzle of ‘non-decoupling’ in cosmology
• Decoupling: usually small-distance physics is not required
to understand long-distance physics
Why should UV physics have anything to say about cosmology?
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Microphysics as a crucial clue
• The puzzle of ‘non-decoupling’ in cosmology
• Most cosmological models rely on long-distance features
that do depend on the details of short-distance physics
• the existence of scalars with small masses, m ≤ H < M
• the existence of small vacuum energies, G ≤ H2
• the existence of exotic low-E states or dynamics
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Microphysics as a crucial clue
• Non-decoupling as a double-edged sword
• If short-distance physics does not decouple, why are
cosmological predictions possible without first solving
quantum gravity?
i.e. are inflationary interpretations of the CMB robust?
• The dangers of ‘Planck slop’
• Even if heavy physics decouples, why can’t small Plancksuppressed interactions ruin cosmology?
𝐿𝑒𝑓𝑓 = 𝑉0 −
𝑉0
𝑀𝑝 2
𝜑2 + ⋯
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Microphysics as a crucial clue
• Both questions require a UV completion to answer.
Seek:
• A mechanism for explaining why quantum corrections do
not ruin required light-scalar properties or vacuum energy
• Understanding of presence/absence of Planck slop and its
implications
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Summary
• Quantum corrections are calculable in GR much like in
any other non-renormalizable theory.
• Quantum effects are typically negligibly small (with
inflationary cosmology a notable exception!)
• Allows efficient tracking of finite-size effects in, eg,
radiation during binary in-spiralling.
• UV modifications to gravity decouple from lowenergy observables
• IR modifications to gravity are difficult to make
consistently
• Scalar, vector, tensor theories (possibly in higher D)
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Summary
• EFTs provide our only current method for comparing
gravitational predictions with data
• Must check the validity of low-energy, classical and
adiabatic approximations.
Ole Miss 2014
Summary
• EFTs provide our only current method for comparing
gravitational predictions with data
• Must check the validity of low-energy, classical and
adiabatic approximations.
• Leads to prejudice for local theories that include all
possible interactions consistent with symmetries in a
systematic derivative expansion
Ole Miss 2014
Summary
• EFTs provide our only current method for comparing
gravitational predictions with data
• Must check the validity of low-energy, classical and
adiabatic approximations.
• Leads to prejudice for local theories that include all
possible interactions consistent with symmetries in a
systematic derivative expansion
• Most heavy physics decouples: robust cosmology
Ole Miss 2014
Summary
• EFTs provide our only current method for comparing
gravitational predictions with data
• Must check the validity of low-energy, classical and
adiabatic approximations.
• Leads to prejudice for local theories that include all
possible interactions consistent with symmetries in a
systematic derivative expansion
• Most heavy physics decouples: robust cosmology
• Suggests some features (technical naturalness, etc)
are important criterion for most cosmologies.
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