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Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Problem of definition: Jordan vs Einstein frame, beyond slow-roll
Godfrey Leung
godfrey.leung@apctp.org
Asia-Pacific Centre for Theoretical Physics
collaboration work with Jonathan White[arXiv:1509.xxxxx]
3rd - 5th August 2015APCTP-TUS joint workshop on Dark Energy
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
Einstein’s General Relativity
In Einstein gravity, it is assumed matter is minimally coupled to gravity
S =
∫d4x√−g(M2
pR/2) + Sm[gµν , φ]
R= Ricci Scalar, Sm = matter action
For example, single scalar field φ
Sm =
∫d4x√−g[−
1
2gµν∂µφ∂νφ+ V (φ)
]
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
Einstein’s General Relativity (con’t)
Strong/weak equivalence principle
Strong: The gravitational motion of a small test body depends only on its initialposition in spacetime and velocity, and not on its constitution.
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
Non-minimal Coupled Models
Can we violate the equivalence principle? Yes
Example, introducing non-minimal coupling to gravity
SJ =
∫d4x√−g(f (ψ)M2
pR/2) + Sm[gµν , ψ]
generic in modified gravity and unified theories, such as string theory, f(R),Chameleons, TeVeS...
conformally related to
SE =
∫d4x√−gE (M2
pR/2) + Sm[(gµν)E , ψ]
by the conformal transformation gµν → (gµν)E = f (ψ)gµν
They are mathematically equivalent
Question: But are they physically equivalent?
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
Non-minimal Coupled Models
Can we violate the equivalence principle? Yes
Example, introducing non-minimal coupling to gravity
SJ =
∫d4x√−g(f (ψ)M2
pR/2) + Sm[gµν , ψ]
generic in modified gravity and unified theories, such as string theory, f(R),Chameleons, TeVeS...
conformally related to
SE =
∫d4x√−gE (M2
pR/2) + Sm[(gµν)E , ψ]
by the conformal transformation gµν → (gµν)E = f (ψ)gµν
They are mathematically equivalent
Question: But are they physically equivalent?
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
Physics should be frame independent!
Conformal transformation = field redefinition
More precisely, conformal transformation = change of scale
1 meter is only meaningful with respect to a reference scale
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
But...
In cosmology, density fluctuations are usually quantified in terms of ζ q
ζ ≡ −ϕ+Hδρ
ρ
can be defined in both conformal frames, where ρ is the effective energy densityfrom Gµν = Tµν/M2
p
For instance
dimensionless and gauge invariant, but not frame independent as we will see...
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
But...
In cosmology, density fluctuations are usually quantified in terms of ζ q
ζ ≡ −ϕ+Hδρ
ρ
can be defined in both conformal frames, where ρ is the effective energy densityfrom Gµν = Tµν/M2
p
For instance
dimensionless and gauge invariant, but not frame independent as we will see...
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
Aside: Isocurvature perturbation
Perturbation is purely adiabatic if δP = Pρδρ. Not always true though...
Entropic/isocurvature perturbations= perturbations ⊥ background trajectory, natural in multifield inflation models
some hints in 2013 Planck results
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
Aside: Isocurvature perturbation
Perturbation is purely adiabatic if δP = Pρδρ. Not always true though...
Entropic/isocurvature perturbations= perturbations ⊥ background trajectory, natural in multifield inflation models
some hints in 2013 Planck results
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
Inequivalence of ζ in Einstein and Jordan frames
It was found that ζ is frame-dependent in the presence of isocurvatureperturbation [White et al. 12, arXiv:1205.0656], [Chiba and Yamaguchi 13]
reason: isocurvature perturbation is frame-dependent (artificial)
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Einstein’s General RelativityNon-minimal Coupled ModelsFrame dependence or Independence?Caveat: Curvature Perturbation ζIsocurvature Perturbationζ 6= ζ in general
Inequivalence of ζ in Einstein and Jordan frames
Examples, in multifield models
ζ − ζ ≈ AJKKJK + BJK KJK , KJK ≡ δφJ φK − δφK φJ
where
KJK is a measure of the isocurvature perturbation
ζ − ζ → 0 only if isocurvature vanishes in general
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
ζ ↔ ζRelation between observables
Relation between ζ and ζ
linear and non-linear order
using the separate universe assumption, we can write ζ and ζ in terms of the δNformalism [White, Minamitsuji and Sasaki et al.]
ζ =NI δφI +NIJδφ
IRδφJ
R+
ζ =NI δφI + NIJδφ
IRδφJ
R+ ...
δφIR
= flat-gauge field perturbations in Einstein frame
observables can be expressed in terms of δN coefficients, e.g. in Jordan frame
ns − 1 = −2(εH)∗ −2
NIN I+
2
3H2∗
NKN L
NINJS IJ∗
[∇K ∇LV − RKLPQ
dφP
dt
dφQ
dt
]∗
Pζ = N INI
(H
2π
)2
∗
and r =8
N INI
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
ζ ↔ ζRelation between observables
Relation between ζ and ζ (con’t)
general relation between N and N
N =
∫ ω
RHdη =
∫ ω
R
(H −
f ′
2f
)dη = N(ω,R)−
1
2ln
(fω
fR
)consider a simplified case where f ′ ≈ 0 at the time of interest (late time)
using the δN formalism, the first and second order δN coefficients are related by
NI ≈NI −(
1
2+ c
)(fJ
f
)
(∂φJ∂φI∗
)ω
NIJ ≈NIJ −(
1
2+ c
)[(fKL
f−
fK fL
f 2
)
(∂φK∂φI∗
)ω
(∂φL∂φJ∗
)ω
+
(fK
f
)
(∂2φK∂φI∗∂φ
J∗
)ω
]we have assumed εf ≡ |f ′/Hf | 1
c ≡ Hρ′ρ , equals to −1/3 (matter era) and −1/4 (radiation era)
ζ − ζ can be arbitrarily large depending on f , but how about observables?
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
ζ ↔ ζRelation between observables
Difference between primordial observables beyond slowroll
Defining the fractional difference between the power spectra amplitudes and thespectral indices
∆Pζ ≡Pζ − PζPζ
and slightly different definition for ns and fNL
ns − 1 + 2(εH)∗
ns − 1 + 2(εH)∗= (1 + ∆Pζ)−1(1 + ∆ns)
fNL
fNL
= (1 + ∆Pζ)−2 (1 + ∆fNL)
using the asymptotic relation between the δN coefficients, we therefore have
∆Pζ =−1
NP NP
[(1 + 2c)
(fK
f
)
(∂φK∂φI∗
)ω
N I
−(
1
2+ c
)2 ( fK fL
f 2
)
(∂φK∂φI∗
)ω
(∂φL∂φJ∗
)ω
S IJ∗
]and similarly for ∆ns and ∆fNL
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Model considered
we consider the multifield model with the following action in the Jordan frame
SJ =
∫d4x√−gf (Φ)R −
1
2GIJ(Φ)gµν∂φI∂φJ − V (Φ)
or in the Einstein frame
SE =
∫d4x√−g
M2pR
2−
1
2SIJ(Φ)gµν∂φI∂φJ − V (Φ)
note that the field space metric and scalar potential are related by
SIJ =M2
p
2f
(GIJ + 3
fI fJ
f
)and V =
VM4p
4f 2
after inflation ends, reheating is modeled by adding a friction to the field EOM inEinstein frame
(φI )′′ + ΓIJK (φJ)′(φK )′ + 2(H+ a(I )Γ)(φI )′ + a2S IJ VJ = 0
ρ′γ + 4Hργ =(I )Γ
aSIJ(φI )′(φJ)′
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Simple example: Non-minimal coupling f ′ = 0 always
to illustrate, we consider the class of two-field models
simple explicit example: f = f (χ) and χ is a frozen spectator field
Sχχ ≡M2
p
2f
(Gχχ + 3
f 2χ
f
)and V = V (φ)M2
p
we further assume Gφχ = 0 and Gφφ = f such that there is no mixing in thekinetic term
Einstein frame results = simple chaotic inflation
ns − 1 = −6ε∗ + 2η∗ , r = 16ε∗
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Simple example: Non-minimal coupling f ′ = 0 always (con’t)
results during slow-roll inflationary regime, consistent with previous analytic work
0.964
0.966
0.968
0.97
0.972
0.974
0.976
0.978
0.98
0.982
0 10 20 30 40 50 60
Spectralindex
N
Jordan, β = 1, χ∗ = 10−3Mpl
Jordan, β = 10, χ∗ = 10−3Mpl
Jordan, β = 20, χ∗ = 10−3Mpl
Jordan, β = 10, χ∗ = 10Mpl
Einstein
0
0.05
0.1
0.15
0.2
0 10 20 30 40 50 60
Ten
sor-to-scalarratio
N
Jordan, β = 1, χ∗ = 10−3Mpl
Jordan, β = 10, χ∗ = 10−3Mpl
Jordan, β = 20, χ∗ = 10−3Mpl
Jordan, β = 10, χ∗ = 10Mpl
Einstein
model choice: V (φ) = 12m2φ2, 2f /M2
p = e−βχ/Mp with φ∗ = 15.0Mp
observables seem coincide at the end of inflation
however, ζ still evolve in Jordan frame...
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Simple example: Non-minimal coupling f ′ = 0 always (con’t)
how about beyond slow-roll, particularly after reheating?
for this particular model, since the non-minimal coupled field χ is frozen, thingssimplify
the fractional difference
∆Pζ =1
16
(1
N2φ
)(fχ
f
)2
∗Sχχ∗
ns − ns
ns= ∆Pζ
[ns − 1 + 2(εH)∗
ns
]difference can only be large if ∆Pζ O(1)
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Simple example: Non-minimal coupling f ′ = 0 always (con’t)
how about beyond slow-roll, particularly after reheating?
for this particular model, since the non-minimal coupled field χ is frozen, thingssimplify
the fractional difference
∆Pζ =1
16
(1
N2φ
)(fχ
f
)2
∗Sχχ∗
ns − ns
ns= ∆Pζ
[ns − 1 + 2(εH)∗
ns
]difference can only be large if ∆Pζ O(1)
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Simple example: Non-minimal coupling f ′ = 0 always (con’t)
Beyond slow-roll regime
0.964
0.966
0.968
0.97
0.972
0.974
0.976
0.978
0.98
55.5 56 56.5 57 57.5 58 58.5 59 59.5 60
Spectralindex
N
Jordan, β = 1, χ∗ = 10−3Mpl
Jordan, β = 10, χ∗ = 10−3Mpl
Jordan, β = 20, χ∗ = 10−3Mpl
Jordan, β = 10, χ∗ = 10Mpl
Einstein
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
55.5 56 56.5 57 57.5 58 58.5 59 59.5 60
Ten
sor-to-scalarratio
N
Jordan, β = 1, χ∗ = 10−3Mpl
Jordan, β = 10, χ∗ = 10−3Mpl
Jordan, β = 20, χ∗ = 10−3Mpl
Jordan, β = 10, χ∗ = 10Mpl
Einstein
with reheating parameter Γφ = 0.1(m/√
2)
evolution terminates when Ωγ > 0.9999
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Difference is negligible after reheating
we see the fractional difference between observables are negligible even ζ − ζ islarge
why?
recall
∆Pζ =−1
NP NP
[(1 + 2c)
(fK
f
)
(∂φK∂φI∗
)ω
N I
−(
1
2+ c
)2 ( fK fL
f 2
)
(∂φK∂φI∗
)ω
(∂φL∂φJ∗
)ω
S IJ∗
]
reason: Einstein frame field space metric S IJ∗ also depends on f
Sχχ ≡M2
p
2f
(Gχχ + 3
f 2χ
f
)
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Difference is negligible after reheating
we see the fractional difference between observables are negligible even ζ − ζ islarge
why? recall
∆Pζ =−1
NP NP
[(1 + 2c)
(fK
f
)
(∂φK∂φI∗
)ω
N I
−(
1
2+ c
)2 ( fK fL
f 2
)
(∂φK∂φI∗
)ω
(∂φL∂φJ∗
)ω
S IJ∗
]
reason: Einstein frame field space metric S IJ∗ also depends on f
Sχχ ≡M2
p
2f
(Gχχ + 3
f 2χ
f
)
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Caveat ICaveat II
Caveat I: negative Jordan frame field space metric
We may tune (Gχχ)∗ → (f 2χ/f )∗, with Sχχ remains positive
Example: Gχχ = −b1(f 2χ/f ), V (φ) = 1
2m2φ2 and 2f /M2
p = e−βχ/Mp , with b1 ≤ 3
0.964
0.965
0.966
0.967
0.968
0.969
0.97
0.971
55.5 56 56.5 57 57.5 58 58.5 59 59.5 60
Spectralindex
N
Jordan, b1 = 1
Jordan, b1 = 2.5
Jordan, b1 = 2.9
Einstein
0.04
0.06
0.08
0.1
0.12
0.14
0.16
55.5 56 56.5 57 57.5 58 58.5 59 59.5 60Ten
sor-to-scalarratio
N
Jordan, b1 = 1
Jordan, b1 = 2.5
Jordan, b1 = 2.9
Einstein
only in the very fine-tuned limit the difference becomes significant
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Caveat ICaveat II
Caveat II: non-frozen f
more generic case: f evolves
the model choice: V (φ) = 12m2φ2 exp(−λχ2/M2
p ), Sχχ = Sφφ = 1 and
2f /M2p = exp(−0.5λχ2/M2
p ).
λ = 0.05, 0.06, initial conditions χ∗ = 10−3Mp and φ∗ = 15.0Mp .
0.7
0.75
0.8
0.85
0.9
0.95
1
55 56 57 58 59 60 61 62
Spectralindex
N
Einstein, λ = 0.05
Jordan, λ = 0.05
Einstein, λ = 0.06
Jordan, λ = 0.06
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
55 56 57 58 59 60 61 62
Ten
sor-to-scalarratio
N
Einstein, λ = 0.05
Jordan, λ = 0.05
Einstein, λ = 0.06
Jordan, λ = 0.06
special case: potential ∼ ridge like, initial conditions close to top of the ridge
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled ModelsAsymptotic relation beyond slow roll
Simple explicit exampleCounter-example
Discussion and Conclusion
Discussion and Conclusion
Take home message
conventional definition of curvature perturbation is a not frame-dependentquantity
in theory, using the wrong definition can lead to very different results
e.g. ζ − ζ can be arbitrarily large
however asymptotically the difference between observables are negligible ingeneral after reheating
possible to realise counter examples, but need fine-tuned initial conditions
Ongoing and Future Directions
study the correlation between large (local) non-Gaussianity and the fractionaldifference
decay rates are generically modulated in non-minimal coupled models even insimple perturbative reheating
Γ→ Γ(χ)
At quantum level? see Steinwachs
Godfrey Leung Frame (In)dependence, Non-minimal coupled models
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