1. The Pursuit of primordial non-Gaussianity in the large scale structure of the universe
Donghui Jeong
Texas Cosmology Center and Astronomy Department
University of Texas at Austin
Astronomy Colloquium, Seoul National University, 27 May 2010

2. Contents Introduction Part I. Gaussian halo bias
Part II. Non-Gaussian halo bias
II.1. Effect on the galaxy power spectrum
II.2. Effect on the galaxy bispectrum
II.3. Effect on the galaxy-galaxy, galaxy-CMB lensing
This is the contents of my talk. After brief introduction, I will first talk about how HETDEX works with ideal, linear power spectrum. Then, I will introduce the real world issues such as the non-linearities, and the HETDEX window function. Finally, I will talk little bit about the extra cosmology we can do with HETDEX.

3. The concordance cosmological model
Our universe is flat, and dominated by cosmological constant
From “WMAP+BAO+H0” in Komatsu et al. (2010):
Let’s get started. Now we have a concordance cosmological model. In this model, our universe is almost flat, and dominated by cosmological constant now. From the observation of cosmic microwave background radiation, BAO and the Hubble key project, we can measure the following cosmological parameters very accurately. H0 means that the currently universe is expanding with 70.4 km/s at 1 Mpc separation. And these numbers are the energy budget of the universe. This pie chart summarize the energy budget. Currently three quarters of the total energy is dark energy, and rest of them are matter. Among them, about 20 percent is the dark matter, and the ordinary matters, which we loosely called as ‘baryons’ are only 5 percent Finally, .

4. From inflation to now

5. Four biggest challenges in cosmology
We do not know the nature of the building blocks!
Inflation How does our Universe begin, or what drives inflation?
Dark Matter What is/are the dark matter(s) made of?
Dark Energy What drives the current acceleration? Is this really the cosmological constant?
Dark Matter, Dark Energy Is General Relativity a valid theory in the cosmological scales?

6. The answer will come from galaxy surveys
On-going, near future surveys
Baryon Oscillation Spectroscopic Survey (0.2 < z < 0.7, 10,000 sq. deg.)
WiggleZ (0.2 < z < 1.0, 1,000 sq. deg.)
Hobby-Eberly Telescope Dark Energy eXperiment (1.9 < z < 3.5, 420 sq. deg.) : Start observing from Fall 2011!!
Surveys which may happen in the future
BigBOSS (0.2 < z < 0.7)
Euclid (0 < z < 2.0)
Cosmic Inflation Probe (1 < z < 6)
SUMIRE/LAS (?? < z < ??)
and more to be proposed

7. Comparing galaxy surveys
dV/dz for 1 str. (~3300 sq. deg.) survey area
~3 [Gpc/h]3
~6 [Gpc/h]3
~1.5 [Gpc/h]3
~1.3 [Gpc/h]3
420 sq. deg.
1000 sq. deg.
10,000 sq. deg.
7600 sq. deg.

9. I. Power spectrum
Probability of finding two galaxies at separation r is given by the two-point correlation function
P(k) is the Fourier transform of ξ(r)
Or, in terms of density contrast, δ(k),
dV1 r
dV2

10. II. Bispectrum
Probability of finding three galaxies at separation (r, s, t) is given by the two, and three-point correlation function
B(k,k’) is the Fourier transform of ζ(r,s).
Or, in terms of density contrast,
dV1 s r t
dV3
dV2

11. Inflation sets the initial condition
Seed fluctuations predicted by most inflation models are nearly scale invariant and obey nearly Gaussian statistics, which are often parametrized as
Initial power spectrum
Initial bispectrum
Here, primordial curvature perturbation is defined as (local type)
Primordial curvature erturbation
Gaussian random field

12. Constraining inflation models
From WMAP7+BAO+H0 (Komatsu et al. 2010)

13. Primordial non-Gaussianity
Well-studied parameterization is “local” non-Gaussianity :
Current best measurement of fNL
From CMB (Komatsu et al, 2010)
From SDSS power spectra (Slosar et al, 2009)
Therefore, initial condition is Gaussian to ~0.04% level! Thus, I am talking about a very tiny non-Gaussianity!
Primordial curvature perturbation
Gaussian random field

14. Single-field Theorem (Consistency relation)
For ANY single-field inflation models, where there is only one degree of freedom during inflation, Maldacena(2003);Seery&Lidsey(2005);Creminelli&Zaldarriaga(2004)
With the current limit of ns=0.96, fNL has to be about for single field inflation.

15. Implication of non-Gaussianity
Therefore, any detection of fNL would rule out all the single field models regardless of
The form of potential
The form of kinetic term (or sound speed)
The initial vacuum state
We can detect primordial non-Gaussianity from
CMB bispectrum
High-mass cluster abundance
Scale dependent bias
Galaxy power spectrum
Galaxy bispectrum (Jeong & Komatsu, 2009b)
Weak gravitational lensing (Jeong, Komatsu, Jain, 2009)

16. Galaxies are biased tracers
Cosmological theory (or N-body simulation) tells us about the dark matter distribution, not about the galaxy distribution.
What we observe from survey are galaxies, not dark matter.
Bias : How is galaxy distribution related to the matter distribution?

17. From dark matter to galaxies
In order to calculate the bias from the first principle, we need to understand the complicated galaxy formation theory such as
Dark matter halo formation
Merger history
Chemistry and cooling
Background radiation (UV)
Feedback (SN, AGN, …)
(and even more)
Each item is the separate research topic.

18. Bias : a simple approach (works only on large scales)
Galaxies are formed inside of dark matter halos.
Halos form at the peak of density field!
But, not every peak forms a halo.
The peak has to have sufficient over-density.
Let’s say there is a threshold over-density, δc, above which dark matter clumps to form a halo.
A halo of mass M is “a region in the space around the peak of smoothed density field (depends on M) whose over-density is greater than the critical over-density.”
Simplest assumption :
Every dark matter halo hosts a galaxy, or galaxies.

19. Recipes : Finding galaxies (2D example)
Smoothing
(or picking up a certain mass M)
Galaxies x y
Find peaks
above threshold
Critical over-density surface

20. Contents Introduction Part I. Gaussian halo bias
Part II. Non-Gaussian halo bias
II.1. Effect on the galaxy power spectrum
II.2. Effect on the galaxy bispectrum
II.3. Effect on the galaxy-galaxy, galaxy-CMB lensing
This is the contents of my talk. After brief introduction, I will first talk about how HETDEX works with ideal, linear power spectrum. Then, I will introduce the real world issues such as the non-linearities, and the HETDEX window function. Finally, I will talk little bit about the extra cosmology we can do with HETDEX.

21. Gaussian density field
One point statistics : The PDF of filtered density field(y) at a given point is completely determined by r.m.s. of fluctuation, :
Two point statistics : The covariance of filtered density field is given by the two point correlation function.
Note that

22. Cumulative mass function
Fraction of collapsed objects with scale larger than R

23. Question asked by Kaiser (1984)
What is the probability of finding two galaxies separated by a distance r?
Joint Probability that two points x and x+r have galaxies at the same time.
where and y1 y2 r

24. Galaxy bias is linear on large scales!
Galaxy correlation function can be calculated as
on large scales, when, δc≫σR (high peak approximation).
On large enough scale, galaxy correlation function is simply proportional to the matter correlation function.
This relation is called a linear bias.

25. Galaxy bias : 2nd method
Linear bias means the linear relation between matter density contrast and the galaxy density contrast.
One can change the question to the following: For a given large scale over-dense region, what is the over-density (excess number) of galaxies in that region?

26. Peak-background split method
Decomposing a density field as peak (on galaxy scale) and background (on matter fluctuation scale).
1D schematic example (peaks in the over-dense region) x
Peak
Background
Large scale over-density +
(Gaussian Random field)
(Offset from cosmic mean)

27. Galaxies with/without BG
Positive (negative) background effectively reduces (increases) the threshold over-density.
Threshold over-density
3 Peaks without background
11 Peaks with background
= cosmic mean density of galaxies
(e.g. mass function)
= mass function with a positive offset, or reduced threshold!!
(see, dashed line in the left figure)

28. All we need is a Mass function
Therefore, mass function determines the bias!
Example: for Press-Schetchter mass function, bias is given by

29. How accurate is it?
Only qualitatively useful. (e.g. Jeong & Komatsu (2009))
Prediction from theory (Sheth-Tormen mass function)
measured from Millennium simulation
Higher peaks
Lower peaks
Lower peaks
Higher peaks

30. Contents Introduction Part I. Gaussian halo bias
Part II. Non-Gaussian halo bias
II.1. Effect on the galaxy power spectrum
II.2. Effect on the galaxy bispectrum
II.3. Effect on the galaxy-galaxy, galaxy-CMB lensing
This is the contents of my talk. After brief introduction, I will first talk about how HETDEX works with ideal, linear power spectrum. Then, I will introduce the real world issues such as the non-linearities, and the HETDEX window function. Finally, I will talk little bit about the extra cosmology we can do with HETDEX.

31. From initial curvature to density
Dalal et al. (2008)
Taking Laplacian
grad(φ)=0 at the potential peak
Poisson equation Laplacian(φ)∝δρ=δ<ρ>
Dalal et al.(2008); Matarrese&Verde(2008); Carmelita et al.(2008); Afshordi&Tolly(2008); Slosar et al.(2008);

32. N-body result : effect of nG
375 Mpc/h
Result from Dalal et al. (2008)
“Large positive fNL accelerates the evolution of over-dense regions and retards the evolution of under-dense regions, while large negative fNL has precisely the opposite effect”
80 Mpc/h
fNL =-5000
fNL =-500
fNL =0
fNL =500
Then, what about galaxy bias?
fNL =5000

33. Again, peak-background split
Non-Gaussian density field at the peak has additional contribution from the primordial curvature perturbation!
BG2:curvature
Peak
BG1:density + +
Increase once by large scale density
Increase once more by large scale curvature

34. Galaxies with/without nG
Positive (negative) fNL effectively reduces (increases) the threshold over-density further more.
Remember, δΦ is always positive!
Threshold over-density
11 Peaks with Gaussianity
17 Peaks with non-Gaussianity
= mass function with additional positive offset, or reduced threshold!!

35. Galaxy bias with nG
The primordial non-Gaussianity changes the galaxy power spectrum by
where change of linear bias is given by
~1/k2
Linear bias depends on the scale!!

36. N-body test of P(k) I (Dalal et al. 2008)
Halo-matter correlation

37. N-body test of P(k) II (Desjacques et al. 2009)
Phh(k), Halo-Halo correlation
Pmh(k)

38. Can we detect this? (e.g. HETDEX)
Galaxy power spectrum in real space
ΔfNL=21 (68% C.L.) Comparable to WMAP7!!

39. The galaxy P(k) in redshift space
Peculiar velocity, which further shift the redshift of the galaxy induces yet another change in power spectrum (Kaiser effect)
As a result, power spectrum becomes anisotropic: increase in clustering along line of sight direction μ

40. Prediction for galaxy surveys
Predicted 1-sigma marginalized error of non-linearity parameter (fNL): sq. deg. galaxy survey (CIP) z V
[Gpc/h]3 ng
10-3[h/Mpc]3 B1
ΔfNL (Kmax=0.1 [h/Mpc])
ΔfNL (Kmax=0.2 [h/Mpc])
1.25
17.43
93.120
1.37
6.11
4.99
1.75
20.91
52.738
1.61
5.04
4.02
2.25
22.20
28.501
1.96
4.46
3.49
2.75
22.26
13.217
2.44
4.12
3.16
3.25
21.69
5.067
3.10
3.90
2.94
3.75
20.83
3.744
3.57
3.77
2.79
4.25
19.84
2.586
4.09
3.68
2.69
4.75
18.82
1.537
4.71
3.64
2.63
5.25
17.82
0.7726
5.46
3.65
2.62
5.75
16.87
0.3898
6.23
3.71
2.66
6.25
15.97
0.2058
7.00
3.85
2.76

41. Contents Introduction Part I. Gaussian halo bias
Part II. Non-Gaussian halo bias
II.1. Effect on the galaxy power spectrum
II.2. Effect on the galaxy bispectrum
II.3. Effect on the galaxy-galaxy, galaxy-CMB lensing
This is the contents of my talk. After brief introduction, I will first talk about how HETDEX works with ideal, linear power spectrum. Then, I will introduce the real world issues such as the non-linearities, and the HETDEX window function. Finally, I will talk little bit about the extra cosmology we can do with HETDEX.

42. Bispectrum and non-Gaussianity
Bispectrum is the Fourier space counter part of three point correlation function:
CMB (z~1100) bispectrum is a traditional tool to test the non-Gaussianity, because it should vanish when density field is Gaussian.
The latest limit on fNL is (Smith et al. 2009) fNL= 38±21 (68% C.L.)
Predicted 68% C.L. range of Planck satellite is ΔfNL= 5.

43. The galaxy bispectrum: theory
Jeong & Komatsu (2009b)
The galaxy bispectrum: theory
The galaxy bispectrum consists of four pieces
I. Matter bispectrum due to primordial non-Gaussianity
II. Matter bispectrum due to non-linear gravitational evolution
III. Non-linear galaxy bias
IV. Non-Gaussianity term from peak correlation I II
III IV

44. Triangular configurations

45. Bispectrum of Gaussian Universe
Jeong & Komatsu (2009b)
Bispectrum of Gaussian Universe
We can measure bias from Equilateral and Folded triangles: Δb1/b1= 0.01, Δb2/b2= 0.05 for HETDEX
Bispectrum from non-linear gravitational evolution
Bispectrum from non-linear galaxy bias

46. Linearly evolved primordial bispectrum
Notice the factor of k2 in the denominator.
Sharply peaks at the squeezed configuration!

47. N-body test of matter bispectrum
Nishimichi et al. (in preparation)
20 runs of (8 Gpc/h)3 simulations
Matter bispectrum
large scale
squeezed
fNL

48. Non-Gaussian peak correlation terms
Jeong & Komatsu (2009)
Non-Gaussian peak correlation terms
The galaxy bispectrum also depends on trispectrum (four point function) of underlying mass distribution!!

49. Matter trispectrum I. TΦ
For local type non-Gaussianity,
Primordial trispectrum is given by
For more general multi-field inflation, trispectrum is

50. Shape of TΦ terms Both of TΦ terms peak at squeezed configurations.
fNL2 term peaks more sharply than gNL term!!

51. Matter trispectrum II. T1112
Trispectrum generated by non-linearly evolved primordial non-Gaussianity.

52. Shape of T1112 terms T1112 terms also peak at squeezed configurations.
T1112 terms peak almost as sharp as gNL term.

53. Are new terms important? (z=0)
Jeong & Komatsu (2009)
Are new terms important? (z=0)

54. N-body galaxy bispectrum
Nishimichi et al. (in preparation)
20 runs of (8 Gpc/h)3 simulations
Galaxy bispectrum
large scale
squeezed
fNL

55. Even more important at high-z!! (z=3)
Jeong & Komatsu (2009)
Even more important at high-z!! (z=3)

56. Prediction for galaxy surveys
Jeong (2010)
Prediction for galaxy surveys
Predicted 1-sigma marginalized error of non-linearity parameter (fNL) from the galaxy bispectrum alone z V
[Gpc/h]3 ng
10-5[h/Mpc]3 b1
ΔfNL (SK07)
ΔfNL (JK09)
SDSS-LRG
0.315
1.48
136
2.17
60.38
5.43
BOSS
0.35
5.66
26.6
1.97
31.96
3.13
HETDEX
2.7
2.96 27
4.10
20.39
2.35
CIP
2.25
6.54
500
2.44
8.96
0.99
ADEPT
1.5
107.3
93.7
2.48
5.65
0.92
EUCLID
1.0
102.9
156
1.93
5.56
0.77

57. Contents Introduction Part I. Gaussian halo bias
Part II. Non-Gaussian halo bias
II.1. Effect on the galaxy power spectrum
II.2. Effect on the galaxy bispectrum
II.3. Effect on the galaxy-galaxy, galaxy-CMB lensing
This is the contents of my talk. After brief introduction, I will first talk about how HETDEX works with ideal, linear power spectrum. Then, I will introduce the real world issues such as the non-linearities, and the HETDEX window function. Finally, I will talk little bit about the extra cosmology we can do with HETDEX.

58. fNL from Weak gravitational lensing
Jeong, Komatsu, Jain (2009)
fNL from Weak gravitational lensing
Picture from M. Takada (IPMU)

59. Mean tangential shear Mean tangential shear is given by
It is often written as
where, Σc is the “critical surface density” R G

60. Mean tangential shear, status
Mean tangential shear from SDSS Sheldon et al. (2009)
What about larger scales?

61. BAO in mean tangential shear
Jeong, Komatsu, Jain (2009)
BAO in mean tangential shear
Errors are dominated by cosmic variance.
(SDSS LRG)
RBAO=106.9 Mpc/h
Δχ2 = 0.85

62. BAO in mean tangential shear
Jeong, Komatsu, Jain (2009)
BAO in mean tangential shear
Result from the thin (delta function) lens plane
(LSST cluster)
RBAO=106.9 Mpc/h
Δχ2 = 1.34
Even larger scale?

63. fNL in mean tangential shear (LRG)
Jeong, Komatsu, Jain (2009)
fNL in mean tangential shear (LRG)

64. fNL in mean tangential shear (LSST)
Jeong, Komatsu, Jain (2009)
fNL in mean tangential shear (LSST)

65. CMB anisotropy as a backlight
Unlensed
Lensed
Picture from Hu & Okamoto (2001)

66. Jeong, Komatsu, Jain (2009)
Galaxy-CMB lensing, z=0.3

67. Jeong, Komatsu, Jain (2009)
Galaxy-CMB lensing, z=0.8

68. Cluster-CMB lensing, z=5
Jeong, Komatsu, Jain (2009)
Cluster-CMB lensing, z=5
High-z population provide a better chance of finding fNL.

69. Conclusion
Detecting primordial non-Gaussianity will rule out all single field inflation models. We can estimate the primordial non-Gaussianigy from large scale structure of the universe.
For Gaussian case, large scale bias is a constant, but for non-Gaussian case, it depends on the wave-number sharply.
The galaxy power spectrum and the galaxy bispectrum are very promising tools to prove primordial non-Gaussianities!!
Weak gravitational lensing on large scales can provides independent cross-checks of non-Gaussianity.