Large-scale Structure: the numerical version

Dragan Huterer

ICTP Trieste/SAIFR Cosmology School

January 18-29, 2021

Lecture 1: LSS basics. Growth function and power spectrum.

Ann Arbor, Michigan

Michigan Stadium (115,000)

University of Michigan

LCTP focuses on: 1. Particle theory 2. Particle pheno 3. Cosmology

Also: working with three Brazilian students at the moment (U. Andrade, R. van Mertens; O. Alves ←@Michigan)

Fairly old research group photo

Emphasis of my lectures

1. Concepts made simple/intuitive 2. Developing experience with numerical work

I want to enable to you to code up calculations FROM SCRATCH

and produce useful, research-ready quantities.

Probably the least theoretical lectures in this school Probably the most numerical lectures in this school⇒

How do we describe the large-scale structure and constrain cosmological model?

Harvard-Cfa survey; de Lapparent, Geller & Huchra (1986)

SDSS/BOSS surveySDSS/BOSS collaboration

•Ground photometric: ‣Kilo-Degree Survey (KiDS) ‣Dark Energy Survey (DES) ‣Hyper Supreme Cam (HSC) ‣Large Synoptic Survey Telescope (LSST)

•Ground spectroscopic: ‣Hobby Eberly Telescope DE Experiment (HETDEX) ‣Prime Focus Spectrograph (PFS) ‣Dark Energy Spectroscopic Instrument (DESI)

•Space: ‣Euclid ‣Wide Field InfraRed Space Telescope (WFIRST)

Ongoing or upcoming LSS experiments:

Dark Energy Survey

• New camera on 4m telescope in Chile • Observations 2013-2019 • ~700 scientists worldwide • Analyses in progress (first major results Aug 2017)

Dark Energy Survey (DES)

Cerro Tololo, ChileBlanco

Telescope

Overdensity deltaThe basic* “observable” quantity is the overdensity

�(x, t) ⌘ ⇢(x, t)� ⇢̄⇢̄

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Note that it satisfies −1 ≤ δ ≤ ∞. [A corollary of that is that, for large delta ≳ 1,

the distribution of delta is always non-Gaussian.]

We will specialize in small fluctuations (| δ|≪ 1) from here on. In that limit, the statistical distribution of delta may or may not be Gaussian:

(* this doesn’t capture the overall number of galaxies, which is

described separately by the mass function)

•Standard inflationary theory (single scalar field, always slow-rolls, in Einstein gravity, on a beautiful day in June) predicts tiny (nonzero but basically unobservable) Non-G - so basically the “sky is Gaussian”

•Searches for so-called primordial NG are at the forefront of research in cosmology - finding one would be finding Holy Grail

Fourier-space overdensity

�k(t) =1pV

Z�(r, t)e�ik·rd3r

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Note:

•𝛿(r) is dimensionless, but then •𝛿k is not, and has units of L3/2, or [k]−3/2

⇒ so k3|𝛿k|2 is dimensionless ⇒ more on that a bit later

Two-point correlation functionConsider a point process (with point particles in 3D space) with

underlying constant density n. Probability of finding a particle in volume dV is

dP = ndV

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Probability of finding two particles in respective volumes dV1, dV2 is

dP = n2(1 + ⇠(r12)) dV1dV2

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where ksi (r12) is the excess probability for clustering. It is the two-point correlation function.

Related quantities can be written as a function of ksi. For example,

dP (2|1) = n(1 + ⇠(r12)) dV2

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hNi = nV + nZ

⇠(r)dV.

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Two-point correlation function 𝛏(r)

Intuitive understanding of 2pt correlation function: It is the excess probability for clustering.

“If I am sitting on one galaxy, I am more (or less) likely to find one nearby, then if I were sitting in space (between galaxies)”

What functional form does 𝛏(r) take?

Cosmological theory actually most straightforwardly predicts the Fourier transform of the 2pt function - the power spectrum P(k). Nevertheless, this phenomenological fit worked well since 1980s:

⇠(r) =

✓r

r0

◆�

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with γ ≈ −1.8 and r0 = 5 Mpc (galaxies) or 20 Mpc (clusters).

Definition of 𝛏(r)

⇠(r) =h [ ⇢(x+ r)� h⇢i ] [ ⇢(x)� h⇢i ] ix

h⇢i2

= h �(x+ r) �(x) ix

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So measure points separated by a distance r, and average over the locations of all such points (vector x).

To do this precisely and accurately, you need an estimator. More about that in a bit.

How to measure 𝛏(r)

Two subtle points

⇠(r) =h [ ⇢(x+ r)� h⇢i ] [ ⇢(x)� h⇢i ] ix

h⇢i2

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• The average above is nominally over all fixed spatial positions x in many realizations of the universe (!). We obviously don’t have access to multiple universes, so we reinterpret the average as that over all locations x in our (one) universe. In doing that we have assumed the ergodic theorem. See Thorne-Blandford “Modern Classical Physics” for more on ergodic theorem and its proof.

•The two point function above is formally 𝝃(r) (note, vector r). We usually convert it to 𝝃(r) (scalar r) by assuming the principle of homogeneity - that the universe is the same at every location in space. As far as we know (and as most inflationary models predict), the universe is homogeneous on largest scales, though testing that is cutting-edge area of research).

Three-point correlation function 𝜻(r) You know the drill! Definition:

dP = n3[1 + ⇠(r12) + ⇠(r13) + ⇠(r23) + ⇣(r123))] dV1dV2dV3

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⇣(r, s, |r� s|) = h [ ⇢(x+ r)� h⇢i ] [ ⇢(x+ s)� h⇢i ] [ ⇢(x)� h⇢i ] ixh⇢i3

= h �(x+ r) �(x+ s) �(x) ix

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Explicitly:

Note, it is a function of a triangle configuration. (source of a lot of complexity/complication in working with 3pt fun)

Fourier trans. of the 3-pt function is called the bispectrum. We won't cover it here but it is another hot research topic.

Bispectrum, which is ≈0 in the CMB, can be large (and a huge pain to measure and model) in the LSS.

Fabian’s and Valerie’s lectures covered perturbation theory, how perturbations are created evolve, adiabatic and issocurvature modes.

Here we start with one important result:

Growth of linear perturbations

Specializing in adiabatic modes, late times (cs is tiny), sub-horizon modes, and assuming GR:

This is the key equation that describes the growth of linear, sub-horizon fluctuations in General relativity.

�̈ + 2H �̇ � 4⇡G⇢M (t)� = 0

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Growth of linear perturbations

1. Radiation domination

�̈ + 2H �̇ � 4⇡G⇢M (t)� = 0

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a(t) ∝t2/3, H(t)=dot(a)/a = 2/(3t), 4𝜋G𝝆M=(3/2)H2, sol of the 𝛿 ∝ tn get n(n-1)+(4/3)n−2/3=0, or

�(t) = B1t2/3 +B2t

�1

' a(t) (MD)

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2. Matter domination

a(t) ∝t1/2, H(t)=dot(a)/a = 1/(2t), ignore 4𝜋G𝝆M term get ddot(𝛿)+dot(𝛿)/t=0, or

3. Lambda domination

�(t) = A1 +A2 ln(t)

(RD)

AAACGnicdVDLSgMxFM34rPVVdekmWJSKUCZFtF0I9bFwqWJV6JSSyaQ1mMmMyR2xDP0ON/6KGxeKuBM3/o0ZraCiB+7lcM69JPf4sRQGXPfNGRoeGR0bz03kJ6emZ2YLc/MnJko04w0WyUif+dRwKRRvgADJz2LNaehLfupf7Gb+6RXXRkTqGHoxb4W0q0RHMApWaheIF3AJtASreGULb7cJXrO94kllJc/Lr3iXCQ2wF/rRdVo62lvttwtFt+y6LiEEZ4RsbriW1GrVCqliklkWRTTAQbvw4gURS0KugElqTJO4MbRSqkEwyft5LzE8puyCdnnTUkVDblrpx2l9vGyVAHcibUsB/lC/b6Q0NKYX+nYypHBufnuZ+JfXTKBTbaVCxQlwxT4f6iQSQ4SznHAgNGcge5ZQpoX9K2bnVFMGNs28DeHrUvw/OamUyXq5drherO8M4sihRbSESoigTVRH++gANRBDN+gOPaBH59a5d56c58/RIWews4B+wHl9B8eMna4=

a(t) ∝exp(H𝛬t), H(t)=H𝛬=const, ignore 4𝜋G𝝆M term get ddot(𝛿)+2H𝛬dot(𝛿)=0, or �(t) = C1 + C2 exp(�2H⇤t)

' const (⇤D)

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

Growth of linear perturbations�̈ + 2H �̇ � 4⇡G⇢M (t)� = 0

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LoLr4OaxdCqY9HwmLxDc3QNzy6Rq</latexit>

�(t) /

8>><

>>:

RD : ln(t)

MD : a(t)

⇤D : const

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•RD and MD growth trends have been confirmed for a long time by basic cosmological observations

•ΛD growth suppression, even though it started “yesterday”, has clearly been observed in the cosmological data!

Growth of linear perturbationsCalculating the general solutions

D(a) =�(a)

�(a = 1)

AAACDXicdZDLSgMxFIYz9VbrbdSlm2AV2k2ZlGLbRaGoC5cVbCt0SsmkmTY0cyHJCGWYF3Djq7hxoYhb9+58GzNtFRU9EPj4/3M4Ob8TciaVZb0bmaXlldW17HpuY3Nre8fc3evIIBKEtknAA3HtYEk582lbMcXpdSgo9hxOu87kLPW7N1RIFvhXahrSvodHPnMZwUpLA/PovICLsAFtV2AS20PKFdZK8oUNVEwGZt4qWZaFEI IpoOqJpaFer5VRDaLU0pUHi2oNzDd7GJDIo74iHEvZQ1ao+jEWihFOk5wdSRpiMsEj2tPoY4/Kfjy7JoHHWhlCNxD6+QrO1O8TMfaknHqO7vSwGsvfXir+5fUi5db6MfPDSFGfzBe5EYcqgGk0cMgEJYpPNWAimP4rJGOsc1E6wJwO4fNS+D90yiVUKdUvK/nm6SKOLDgAh6AAEKiCJrgALdAGBNyCe/AInow748F4Nl7mrRljMbMPfpTx+gH8FJpO

Linear growth rate D

Growth suppression factor g

(implicitly via D)

D(a) ⌘ ag(a)g(a = 1)

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

D org

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.

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Growth of linear perturbationsCalculating the general solutions

D(a) =�(a)

�(a = 1)

AAACDXicdZDLSgMxFIYz9VbrbdSlm2AV2k2ZlGLbRaGoC5cVbCt0SsmkmTY0cyHJCGWYF3Djq7hxoYhb9+58GzNtFRU9EPj4/3M4Ob8TciaVZb0bmaXlldW17HpuY3Nre8fc3evIIBKEtknAA3HtYEk582lbMcXpdSgo9hxOu87kLPW7N1RIFvhXahrSvodHPnMZwUpLA/PovICLsAFtV2AS20PKFdZK8oUNVEwGZt4qWZaFEI IpoOqJpaFer5VRDaLU0pUHi2oNzDd7GJDIo74iHEvZQ1ao+jEWihFOk5wdSRpiMsEj2tPoY4/Kfjy7JoHHWhlCNxD6+QrO1O8TMfaknHqO7vSwGsvfXir+5fUi5db6MfPDSFGfzBe5EYcqgGk0cMgEJYpPNWAimP4rJGOsc1E6wJwO4fNS+D90yiVUKdUvK/nm6SKOLDgAh6AAEKiCJrgALdAGBNyCe/AInow748F4Nl7mrRljMbMPfpTx+gH8FJpO

Linear growth rate Growth suppression factor (implicitly via D)

D(a) ⌘ ag(a)g(a = 1)

AAACCHicdVDLSgMxFM3UV62vUZcuDBah3ZRJKbZdCEVduKxgW6EzlEyaaUMzD5NMoQxduvFX3LhQxK2f4M6/MdNWUNED93I4516Se9yIM6ks68PILC2vrK5l13Mbm1vbO+buXluGsSC0RUIeihsXS8pZQFuKKU5vIkGx73LacUfnqd8ZUyFZGFyrSUQdHw8C5jGClZZ65uFFARdtehuzMbQ9gUmCB1qZJrqfouK0Z+atkmVZCCGYElQ9sTSp12tlVIMotTTyYIFmz3y3+yGJfRoowrGUXWRFykmwUIxwOs3ZsaQRJiM8oF1NA+xT6SSzQ6bwWCt96IVCV6DgTP2+kWBfyonv6kkfq6H87aXiX143Vl7NSVgQxYoGZP6QF3OoQpimAvtMUKL4RBNMBNN/hWSIdRxKZ5fTIXxdCv8n7XIJVUr1q0q+cbaIIwsOwBEoAASqoAEuQRO0AAF34AE8gWfj3ng0XozX+WjGWOzsgx8w3j4B4s6YqA==

2d2g

d ln a2+ [5� 3w(a)⌦DE(a)]

dg

d ln a+ 3 [1� w(a)]⌦DE(a)g = 0

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

2nd order ODE, can easily solve on the computer. Works for all w(a)CDM cosmological models!

(w(a) = time-dependent equation of state of DE)

Then �̈ + 2H �̇ � 4⇡G⇢M (t)� = 0

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

becomes

The (matter) power spectrumRemember the overdensity in real and Fourier space

�(~r) =

pV

(2⇡)3

Z�~k e

�i~k~r d3k

�~k =1pV

Z�(~r) ei

~k~r d3r

AAAConicdVHbThsxEPVuaaHphbR95KFWQ6sglWgdUEkekFARUtWHFtQmIMVJ5PVOwIr3UtuLFFn7Yf0N3vgbvNmlKghGsnR8xufMeCbMpNAmCK49/8nK02era88bL16+er3efPN2qNNccRjwVKbqLGQapEhgYISRcJYpYHEo4TScH5b500tQWqTJb7PIYByz80TMBGfGUdPmXxqBNKxNL4FbVWzhT/vYUv1HGTssaOqkuN2lmdia7BQNKhKDK8HULhXzoqCfMUzstqjvtVFJR5OdOaWN+4JlCVJ531a64/2vmcr6YWc1bbaCThAEhBBcArL3JXCg3+91SQ+TMuWiheo4njavaJTyPIbEcMm0HpEgM2PLlBFcgusg15AxPmfnMHIwYTHosV2OuMAfHRPhWarccY0u2f8VlsVaL+LQvYyZudD3cyX5UG6Um1lvbEWS5QYSXhWa5RKbFJf7wpFQwI1cOMC4Eq5XzC+YYty4rTbcEG5/ih8Hw26H7Hb6J7utg6/1ONbQBvqA2oigPXSAvqFjNEDce+8deT+8n/6m/90/8X9VT32v1rxDd8KnN5Cnz2Q=

Consider shifting position r by some 𝛥r. Then

�~k ! �~k ei~k�~r

h�~k�⇤~k0i ! ei(~k�~k

0)�~r h�~k�⇤~k0i

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h�~k �⇤~k0i = (2⇡)3 �(3)(~k � ~k0)P (k)

<latexit sha

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texit>

For a homogeneous universe, this must be independent of 𝛥r! Then we have our definition of power spectrum P(k):

Note, this k is scalar (homogeneity)

Power spectrum P(k)At large scales (k ≪0.01 h Mpc−1)

P (k) / kn with n ' 1 (Harrison, Zel0dovich, Peebles spectrum)

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

•See my lecture notes about the HZP 1969 argument that n≈1 •Inflation (1980⇒) predicts n = 1 − 6𝜀+2𝜂, where 𝜀, 𝜂 are “small” •WMAP, Planck (2000s⇒) measure n =0.965 ±0.004 (P18)

kn

Power spectrum P(k)At small scales (k ≫ 0.01 h Mpc−1)

•So basically P(k) = P(k)large-scales ×T2(k), with T(k)∝k−2

•T(k) is called the transfer function, and it “penalizes” modes that entered the horizon before matter-radiation equality (modes with k>0.01)

•The higher-k the mode is, the earlier (before MR eq) it entered the horizon and the less time it had to grow

•See my lecture notes for a more precise argument

P (k) / kn�4

AAAB/HicdVDLSgMxFM34rPU12qWbYBHqwjIpxba7ohuXFewD2rFk0kwbmskMSUYYhvorblwo4tYPceffmGkrqOiBC4dz7uXee7yIM6Ud58NaWV1b39jMbeW3d3b39u2Dw44KY0lom4Q8lD0PK8qZoG3NNKe9SFIceJx2vell5nfvqFQsFDc6iagb4LFgPiNYG2loF1ql6ekgkmGkQzi9TcVZdTa0i07ZcRyEEMwIqp07hjQa9QqqQ5RZBkWwRGtovw9GIYkDKjThWKk+ciLtplhqRjid5QexohEmUzymfUMFDqhy0/nxM3hilBH0Q2lKaDhXv0+kOFAqCTzTGWA9Ub+9TPzL68far7spE1GsqSCLRX7MoXk0SwKOmKRE88QQTCQzt0IywRITbfLKmxC+PoX/k06ljKrlxnW12LxYxpEDR+AYlAACNdAEV6AF2oCABDyAJ/Bs3VuP1ov1umhdsZYzBfAD1tsnA3iUZw==

kn-4kn

What is the relation between P(k) and 𝛏(r)?They are Fourier transforms of each other because

𝛿k and 𝛿(r) are (Wiener-Khinchin theorem)

Proof:

r12=r1−r2; integrate ∫d3r2=V

Of course that P(k) and 𝛏(r) contain the same information However they are measured differently in practice; methods are subject to

different statistical and systematic errors. Roughly speaking:

•𝛏(r) is easier to measure, esp with “holes” in the survey footprint •P(k) is closer to theory

⇠(r) =1

(2⇡)3

ZP (k)ei

~k~rd3~k =1

2⇡2r

Z 1

0P (k) sin(kr)kdk

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

Consider

Dimensionless power spectrum 𝛥2(k)

Now evaluate the zero-lag correlation function, or variance, 𝛏(r=0)

⇠(0) =1

2⇡2

Z 1

0P (k) lim

r!0

sin(kr)

krk2dk ⌘

Z 1

0�2(k)d ln k

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

where �2(k) ⌘ k3P (k)

2⇡2

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is also called the power spectrum and has the following nice features:

•It is dimensionless (computer coders, rejoice!) •Physically, it is the contribution to variance per log wavenumber

�2(k, a) = A4

25

1

⌦2M

✓k

kpiv

◆n�1 ✓k

H0

◆4[ag(a)]2 T 2(k)Tnl(k)

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Code-friendly power spectrum 𝛥2(k)

A is the primordial amplitude (dimensionless, ~10−10) 𝛺M is the density of matter relative to critical kpiv is some chosen pivot; modern convention is kpiv=0.05 Mpc−1

H0 is the Hubble constant g(a) is the growth suppression factor T(k) is the transfer function, accounts mainly for “turnover” in power spectrum due to radiation-matter transition. Tnl(k) is the prescription for nonlinear clustering; super important on scales k ≿ 0.1 h Mpc−1