1. Searching for Axionic Blue Isocurvature Perturbations
Daniel J. Chung
10/26/2017
w/ Hojin Yoo
w/ Amol Upadhye
1711.XXXXX w/ Amol Upadhye

2. Quantum attractor solutions in inflationary dynamics
 initial conditions for cosmology (initial inhomogeneities)
Initial conditions (at leading order in pert) are either
adiabatic: intuitively, a single dynamical mode
fixes the inhomogeneities
isocurvature: multiple independent dynamical modes
fix the inhomogeneities
(e.g. dark matter)
(e.g. photons)
Operationally, the classification can be written as
(DM-photon isocurvature)

3. (DM-photon isocurvature)
adiabatic density isocurvature density
inhomogeneity inhomogeneity
It is easy to destroy isocurvature perturbations.
i.e. Multiple d.o.f. is insufficient for isocurvature
perturbations to be observationally relevant.

4. The dark matter needs to be hidden to generate isocurvature:
example 1:
If there are many more than , and forward reactions occur
makes enough blue s.t.
adiabatic again
O( )
DM must be sufficiently secluded to prevent this.

5. example 2:
If thermal equilibrium (now forward and backward), then
Prevention of thermalization is
easy to estimate in terms of
particle picture:
If , then
This is easily satisfied by canonical
QCD axion models.

6. Recap: If the field is
secluded from the inflaton
secluded from thermal products of inflaton
light during inflation
sufficiently stable to effect cosmo fluid initial conditions
isocurvature perturbations in addition to
adiabatic perturbations.
Nonthermal dark matter candidates such as the axions
can satisfy this.

7. fermionic isocurvature perturbations: Chung, Yoo, Zhou 13
Isocurvature Sample
Multi-field inflation: Axenides, Brandenberger, Turner 83; Linde 85; Starobinsky 85; Silk, Turner 87; Polarski, Starobinsky 94; Linde, Mukhanov 97; Langlois 99
Axions: Turner, Wilczek, Zee 83; Steinhardt, Turner 83; Axenides, Brandenberger, Turner 83; Linde 84, 85;
Seckel, Turner 85; Efstathiou, Bond 86; Hogan, Rees 88; Lyth 90; Linde, Lyth 90; Turner, Wilczek 91; Linde 91; Lyth 92; Kolb, Tkachev 94; Fox, Pierce, Thomas 04; Beltran, Garcia-Bellido, Lesgourgues 06; Hertzberg, Tegmark, Wilczek 08; Kasuya, Kawasaki 09; Marsh, Grin, Hlozek, Ferreira 13; Choi, Jeong, Seo 14
B-genesis/Affleck-Dine: Bond, Kolb, Silk 82; Enqvist, McDonald 99, 00; Kawasaki, Takahashi 01;
Kasuya, Kawasaki, Takahashi 08; Harigaya, Kamada, Kawasaki, Mukaida, Yamada 14; Hertzberg, Karouby 14
SUSY Moduli: Yamaguchi 01; Moroi and Takahashi 01; Lemoine, Martin, Yokoyama 09; Iliesiu, Marsh, Moodley, Watson 13
WIMPZILLAs: Chung, Kolb, Riotto, Senatore 04; Chung, Yoo 11; Chung, Yoo, Zhou 13
fermionic isocurvature perturbations: Chung, Yoo, Zhou 13
dark energy isocurvature: Malquarti, Liddle 02; Gordon, Hu 04
I would like to begin by displaying a little reference page on isocurvature perturbations to give credit to some of the work that have influenced me.

8. a small sample of observational focus:
Gordon, Pritchard 09; Kawasaki, Sekiguchi, Takahashi 11; Takeuchi, Chongchitnan 13; Sekiguchi, Tashiro, Silk, Sugiyama 13; Holder, Nollett, Engelen 09; Grin, Dore, Kamionkowski 11; Hikage, Kawasaki, Sekiguchi, Takahashi 12; Planck 2013 results XXII; Chlub and Grin 13; Valiviita 17; Grin, Hanson, Holder, Dore, Kamionkowski 13; Dent, Easson, Tashiro 12; Kawasaki and Yokoyama 14; Valviita, Muhonen 03; Ferrer, Rasanen, Valviita 04; Kurki-Suonio, Muhonen, Valviita 05; Keskitalo, Kurki-Suonio, Muhonen, Valviita 07; Beltran, Garcia-Belliodo, Lesgourgues, Riazuelo 04; Beltran, Garcia-Bellido, Lesgourgues, Viel 05; Sollom, Challinor, Hobson 09
Fluid dynamics and general field theory parameterizations:
Bardeen 1980; review of Kodama, Sasaki 84; review of Malik, Wands 09; Kolb and Turner, EARLY UNIVERSE
+ Many more papers on curvaton and non-Gaussianities; apologies for omissions

9. Traditionally, axions were thought to be oscillating about the PQ
symmetry breaking vacuum during inflation:
isocurvature
spectral index
Goldstone theorem
scale invariant

10. Hence, most papers before 2009 were regarding scale invariant
axionic isocurvature perturbations
Why blue instead of scale invariant ?

11. Similar to the tensor to scalar ratio r, it can tell us something
qualitative about high energy theory and field history.
[ ]
Suppose there are spectator fields of constant mass
(model “independent”)
minimum e-folds
for cosmology
Look up reasons for different parts of the curve.
Measuring is a signature of a time dependent mass of a
scalar field.

12. Initial conditions are somewhat generic for flat directions.
Inflaton is generically out of equilibrium. Then why not
the Peccei-Quinn symmetry breaking order parameter?
The eta problem of inflaton  eta opportunity for axionic
blue isocurvature spectrum.
e.g. Kaehler induced mass terms during inflation:
3) Theoretically well-motivated
Axion solves the strong CP problem of Standard Model,
can be the dark matter, and can naturally generate blue
isocurvature perturbations.
deform away from NG theorem

13. Axionic sector hidden from inflation:
Example: Kasuya, Kawasaki
Axionic sector hidden from inflation: PQ
charged under R
PQ symmetry and R symmetry in SUSY:
Large is motivated because there is a flat direction.
i.e. quartic terms are Khaeler suppressed

14. Axionic sector hidden from inflation:
Example: Kasuya, Kawasaki
Axionic sector hidden from inflation: PQ
charged under R
Khaeler induced mass terms during inflation:
e.g. eta problem is generic
good for blue spectrum

15. Break in the
spectrum is
generic.

16. 4) Scale invariant isocurvature is a small effect.
2013
Small even if correlated:
adiabatic
cross
isocurvature

17. What is the largest ABI spectrum consistent with data?
Assuming that axion isocurvature spectrum that is strongly
blue (instead of scale invariant )
is sufficiently interesting, let’s focus on phenomenology:
phenomenology is gravitational effects on CMB and large
scale structure.
Given that the blue spectrum is most naturally generated
in axion type of models as deformations from scale invariance, we
call strongly blue isocurvature of this type
ABI = axionic blue isocurvature
What is the largest ABI spectrum consistent with data?

18. Step 1:
Since analytic predictions leave a gap [ ]
e.g.
Compute seminumerical approximation: [ D.C.,
Upadhye]

19. Step 1:
Since analytic predictions leave a gap [ ]
e.g.
Compute seminumerical approximation: [ D.C.,
Upadhye]

20. Somewhat challenging: Some fields are varying 10 orders of magnitude [
DC and Upadhye]
Solve for a lattice of k
values.
Somewhat challenging:
Some fields are varying
10 orders of magnitude
and scale factors
are varying
26 orders of magnitude

21. Compute using the numerical results the following for
a lattice of k values:
Construct a 3-parameter fitting function:

22. Some of the parameters defining fitting function through interpolation:

23. 2) Fit to Planck and BOSS (SDSS) DR11 data representing the
strongest constraints on cosmological power spectrum.
[1711.XXXXX DC and Upadhye]
a) Is there a hint for ABI in the data?
b) What is the largest amplitude allowed by these data?
MCMC fitting procedure:
Follow the fitting procedure of Amol Upadhye ]:
Planck likelihood code:
BOSS DR11 redshift-space galaxy power spectrum:

24. Prior:
Planck and BOSS fit results:

25. Our fit is the first with this kind of break: [1711.XXXXX]
Since 1 sigma is consistent with zero, there is no statistically
significant hint.
However, it is encouraging that the data is “starting” to prefer
a nonzero ABI amplitude when marginalizing over the
spectral index

26. Intuition for the fit:
The BOSS & Planck data overlap in large k-space sensitivities
This is where the isocurvature is kicking in and being constrained.
The more constraining of the
two data sets is Planck.
In the best fit, the break
is slightly below BAO scale

27. b) The main effect of BOSS is
Decrease and while preserving

28. c) Although the “best fit” does not care much about BOSS, there
is a hint that goes from 1-sigma to 2-sigma when the model prior
is fixed to be HI-BLUE (large break k and large spectral index).
For example: w/ and
2-sigma hint!
Second encouraging result!

29. Back to best fit: What are the fundamental axion parameters
consistent with the best fit region?
Upper bound
Trh=10^6 GeV
Fa > Tmax
Lower bound
Trh=10^8 GeV
Fa > Tmax

30. Back to best fit: What are the fundamental axion parameters
consistent with the best fit region?
All dark matter can be axions!

31. Possibly a low l and high l anomaly is showing up in this fit:
[ Addison et al.]
2.5 sigma discrepancy

32. [1711.XXXXX]

33. How much room is there for discovery in future data?
With the curent 2-sigma boundary with the break scale set at , the isocurvature power can be 40 times larger than the adiabatic power. [Contrast that
with the adiabatic phenomenology of O(1)%]
primordial measure without transfer function
There is good prospects for seeing the break in future experiments such as the SKA.

34. Summary:
Axionic (strongly) blue isocurvature spectrum with a break (called ABI spectrum) is interesting because
qualitative information about time dependent mass if nI > 2.4 is measured
Initial conditions are generic
theoretically well motivated
scale invariant spectrum is constrained to be a small effect with little prospect for future data improving constraints.
Carried out the first data fit which shows encouraging but statistically insignificant 1-sigma fit preference
With theoretical prior of HIGH-BLUE, there is a 2-sigma hint.

35. (summary continued)
For the best fit, all of the dark matter can be axions.
With the break at , the isocurvature can be around 40X bigger than the adiabatic amplitude at k=10
With future experiments such as SKA and LSST, there is
large room for discovery.

36. End

37. backup slides

38. Scale invariant isocurvature
CMB
adiabatic
cross
isocurvature
2013

39. matter power spectra
[Takeuchi, Chongchitnan 13]
minihalos  21 cm experiments such as SKA and FFTT

40. Bound on constant mass:
Raising is desired
Lowering
Ne is desired
[ ]
Varied to
satisfy nonlinear
nonlocal 13 constraints
Measuring is a signature of a time dependent mass of a
scalar field.

41. CMB spectral distortion
[Dent, Easson, Tashiro 12]
PIXIE
sensitivity

42. What are DM-photon isocurvature perturbations?
DM inhomogeneities different from photons.
Scale invariant ( ) isocurvature is well constrained
by large scale (CMB scale) data.
Future data will most strongly improve short scale info.
Blue ( ) is easy to hide on CMB scales
and may be seen by future data.
Question: What will we learn about high energy theory if we
observe blue isocurvature perturbations?

43. High energy theory dependence:
(correlation functions during inflation) X (transfer function)
quantum: correlators in the presence of
gravitational constraints.
Most of isocurvature literature is regarding a scale invariant spectrum.
Hence, we have formulated some useful theorems to deal with
blue isocurvature perturbations.
[DC, H. Yoo ]

44. Linear spectator isocurvature field:
Assume that coherent oscillations form CDM
We formulate some useful theorems regarding this

45. Conservation is valid through end of inflation and through reheating.
Theorem 1: A classically conserved quantity for spectator isocurvature with the
dominant interactions given by gravity and is
measures the deviation from the Newtonian gauge
Reason: an approximate symmetry in the limit that mass term dominates.
Conservation is valid through end of inflation and through
reheating.

46. Theorem 3: Quantum correlator.+ +
Technical challenge of the theorem proof: quantization in the presence of
gravitational constraint.

47. Quantization error in the presence of mixing
corollary: Suppose a sudden transition of mass is made during inflation.
Quantization error in the presence of mixing
This leads to

48. Application 1: Is there a lower bound on blue isocurvature perturbation from
DM fraction?
Puzzle: According to theorem 2
O(1) for blue
spectrum
Hence, we naively expect
True ?
If this were true, then one can place a lower bound on the isocurvature perturbations:

49. No, because of theorem 3:
as ,
thus is not order unity
How does this work in detail in an example?
There is a secular growth from a source term effect.

50. gravitational interactions.
illustration: Consider a 2 field model in spatially flat gauge
gravitational
interactions.
Expand in and truncate to O(0).
shall we drop?
truncated

51. Intuitive lesson: Even if the spectator field quantum fluctuates
shall we drop?
truncated
No. Secularly grows to become an important source term for long time evolution.
secular source piece precisely cancels
Intuitive lesson: Even if the spectator field quantum fluctuates
independently on short length sclaes, gravitational interactions
makes the spectator field grow an adiabatic field profile.

52. Application 2: Is a blue spectrum a signature for
Hubble scale massive scalar fields?
Why can’t we trivially generate a measurable large blue
spectrum with a constant mass field?
Blue spectral energy density dilutes away!

53. What is the maximum measurable
spectral index that can be attributed to
a constant mass spectator?
Answer: 2.4

54. 13 constraints
perturbativity
adiabatic slow-roll
dominates
Sufficient number of efolds
Dim 5 or 6 Planck
suppressed
operators exist
Reheating energy conservation

55. 13 constraints continued
Coherent oscillations
occur before reheating
(not an important bd)
Overlaps with constr 5
Tensor bound
Dark matter bound
Isocurvature
experimental
bound at max k

56. 13 constraints continued 2
Small k exp bound
Linear spectator
No transplanckian
Validity of 9 if not using 9’
(classicalized modes)

57. Bound on constant mass:
Raising is desired
Lowering
Ne is desired
[ ]
Varied to
satisfy nonlinear
nonlocal 13 constraints
Measuring is a signature of a time dependent mass of a
scalar field. (SKA seems to be able to clearly measure 3)
[Takeuchi, Chongchitnan 13]

58. Pressure for Hi=He
Lowers Hi before
saturating r

59. Up Trh  up Hi due to E conservation
up epsilon  split Hi and He
Increases Ne

60. Measuring is a signature of a time dependent mass of a scalar field.
What kind of theories can lead to meaurable ?
1) Need : e.g. supergravity eta problem helps you
2) Need the mass to shut off to not dilute away the energy density.
Displaced PQ symmetry breaking is a natural model satisfying this.
Goldstone theorem
is evaded because not
in vacuum.

61. Emphasize Generic Nature
Goldstone theorem
is evaded because not
in vacuum.
Two broad choices with axions:
PQ symmetry breaking VEV is in equilibrium during infl
PQ symmetry breaking VEV is out of equilibrium during infl
Leads to blue spectrum with a plateau

62. Axionic sector hidden from inflation:
Application 3: A clarification/correction of [Kasuya, Kawasaki ]
Axionic sector hidden from inflation: PQ
charged under R
PQ symmetry and R symmetry in SUSY:
Large is motivated because there is a flat direction.
i.e. quartic terms are Khaeler suppressed

63. Axionic sector hidden from inflation:
Application 2: A clarification/correction of [Kasuya, Kawasaki ]
Axionic sector hidden from inflation:
PQ symmetry and R symmetry in SUSY:
Large is natural because there is a flat direction.
i.e. quartic terms are Khaeler suppressed
Khaeler induced mass terms during inflation:
e.g. eta problem is generic
good for blue spectrum

64. Puzzle: They assume that the following freezes automatically
Application 2: A clarification/correction of [Kasuya, Kawasaki ]
wrong- will explain
Puzzle: They assume that
the following freezes automatically
upon horizon exit

65. different same fields fields 1) Why is this frozen? (correct)
Contrast this with our theorem 1
[Kasuya, Kawasaki ]
theorem 1:
different
fields
same
fields
1) Why is this frozen?
(correct)
2) Why massless?
Large limit: action simplifies
took
massless
form
(wrong)

66. Why is frozen?
Non-vacuum axion mass and PQ
natural initial PQ charge density:
charge dilutes

67. difference: quantization induced mode versus homogeneous solution
Comparison of our result with Kasuya, Kawasaki
contrast
Also invalid at n=4
difference: quantization induced mode
versus homogeneous solution

68. Green:
Blue:
Orange:

69. n=4 is impossible for simple formula
wrong
They assumed growing mode
modes become imag
classical stochastic amplitude that grows only if n<4

70. An Example of a Correction
remarkable
Reason for staying away from a more dramatic correction region:
loss of analytic control

71. Gap in Analytic Control
Gap in analytics: mass transition occurs before classicalization.
The features in this part depends
on the details of the transition.

72. Measuring is a signature of a time dependent mass of a
scalar field.
Contrast with an axion:
instead of

73. The right matter power spectrum to look for is with a bump
Looking for correlated isocurvature and CMB signatures.
[In progress with Amol Uphadye.]

74. Summary
Blue CDM-photon isocurvature pertrubations are attractive in
terms of observability.
Three theorems useful for blue isocurvature perturbations
from spectator scalar fields were presented. [ ]
theorems 1: conserved quantity
theorem 3: constrained quantization and secular effect
error bounds given
Applications
gravity induced secular growth effects imprint adiabatic
information onto the isocurvature fields.
A measurement of a blue spectral index larger than 2.4 gives
evidence for a field with time dependent mass during inflation
[ ]
improved the computation of (time-dep mass)

75. more backup

76. tension
nontrivial blue spec range
using flat directions
large from
constr (B)
(A)
quantum angle fluctations are
small compared to classical one
small from
constr (C)
(B)
linear isocurvature
tadpoles
i) large n
ii) large from (B)
minimum k range of blue
spec (large scale to pivot)
and quantization as a
decoupled spectator
(C) X

77. Largest H Allowed by Decoupled Linear Spectator?
Quantization decoupling constraint:

78. Application 3: Is a blue spectrum a signature for
Hubble scale massive scalar fields?
used a time dependent mass during inflation
 k range of blueness limited
What about isocurvature with constant mass during inflation to extend the k range?
challenge: The energy density typically dilutes away.
e.g.

79. Although rolls down to the minimum, decays to a final point that depends on
the initial conditions.
Fortunately, for most of its evolution, the misalignment angle relaxes approximately at the
same rate because of the dominance of quadratic potential.

80. CMB
Scale invariant isocurvature
adiabatic
cross
isocurvature
2013

81. This is very similar to why we need interactions for the
a measurable fermionic isocurvature perturbations
[DC, H. Yoo, P. Zhou ]

82. This is merely a restatement of the results in
Theorem 2: Isocurvature correlator is equal to the S correlator.
merit of the theorem: defines the sense in which arises from
averaging of fluid quantities
This is merely a restatement of the results in
[Gordon, Wands, Bassett, Maartens astro-ph/ ;
Polarski , Starobinsky astro-ph/ ]

83. General boundary condition is a superpostion:
isocurvature
adiabatic

84. The cosmological effect of CDM isocurvature
on cosmology is primarily gravitational.
Newtonian intuition:
Potential force and pressure push fluid components such
as around.
e.g.
This makes CMB sensitive primarily on large scales.