Finding Cosmic Inflation Eiichiro Komatsu (Max-Planck-Institut fr - - PowerPoint PPT Presentation

Finding Cosmic Inflation

Eiichiro Komatsu (Max-Planck-Institut für Astrophysik) “Gravity and Black Holes”, Cambridge July 3, 2017

Cook’s Branch

Heaven in Texas, and probably the most unexpected place to meet Stephen often

A Remarkable Story

• Observations of the cosmic

microwave background and their interpretation taught us that galaxies, stars, planets, and

• urselves originated from tiny

fluctuations in the early Universe

• But, what generated the initial fluctuations?

Leading Idea

• Quantum mechanics at work in the early Universe
• “We all came from quantum fluctuations”
• But, how did quantum fluctuations on the microscopic

scales become macroscopic fluctuations over large distances?

• What is the missing link between small and large

scales?

Mukhanov & Chibisov (1981); Hawking (1982); Starobinsky (1982); Guth & Pi (1982); Bardeen, Turner & Steinhardt (1983)

Cosmic Inflation

• Exponential expansion (inflation) stretches the wavelength
• f quantum fluctuations to cosmological scales

Sato (1981); Guth (1981); Linde (1982); Albrecht & Steinhardt (1982) Quantum fluctuations on microscopic scales

Inflation!

Key Predictions

• Fluctuations we observe today in CMB and the matter

distribution originate from quantum fluctuations during inflation

ζ

scalar mode

hij

tensor mode

• There should also be ultra long-wavelength

gravitational waves generated during inflation

Starobinsky (1979)

We measure distortions in space

• A distance between two points in space

d`2 = a2(t)[1 + 2⇣(x, t)][ij + hij(x, t)]dxidxj

X

i

hii = 0

• ζ : “curvature perturbation” (scalar mode)
• Perturbation to the determinant of the spatial metric
• hij : “gravitational waves” (tensor mode)
• Perturbation that does not alter the determinant

We measure distortions in space

• A distance between two points in space

d`2 = a2(t)[1 + 2⇣(x, t)][ij + hij(x, t)]dxidxj

X

i

hii = 0

• ζ : “curvature perturbation” (scalar mode)
• Perturbation to the determinant of the spatial metric
• hij : “gravitational waves” (tensor mode)
• Perturbation that does not alter the determinant

scale factor

Finding Inflation

• Inflation is the accelerated, quasi-exponential expansion.

Defining the Hubble expansion rate as H(t)=dln(a)/dt, we must find

¨ a a = ˙ H + H2 > 0 ✏ ≡ − ˙ H H2 < 1

• For inflation to explain flatness of spatial geometry of our
• bservable Universe, we need to have a sustained period
• f inflation. This implies ε=O(N–1) or smaller, where N is

the number of e-folds of expansion counted from the end

• f inflation:

N ≡ ln aend a = Z tend

t

dt0 H(t0) ≈ 50

Have we found inflation?

• Have we found ε << 1?
• To achieve this, we need to map out H(t), and show that it

does not change very much with time

• We need the “Hubble diagram” during inflation!

✏ ≡ − ˙ H H2

Fluctuations are proportional to H

• Both scalar (ζ) and tensor (hij) perturbations are

proportional to H

• Consequence of the uncertainty principle
• [energy you can borrow] ~ [time you borrow]–1 ~ H
• KEY: The earlier the fluctuations are generated, the more

its wavelength is stretched, and thus the bigger the angles they subtend in the sky. We can map H(t) by measuring CMB fluctuations over a wide range of angles

Fluctuations are proportional to H

• We can map H(t) by measuring CMB fluctuations over a

wide range of angles

• 1. We want to show that the amplitude of CMB fluctuations

does not depend very much on angles

• 2. Moreover, since inflation must end, H would be a

decreasing function of time. It would be fantastic to show that the amplitude of CMB fluctuations actually DOES depend on angles such that the small scale has slightly smaller power

Data Analysis

• Decompose the observed

temperature fluctuation into a set

• f waves with various wavelengths
• Show the amplitude of waves as a

function of the (inverse) wavelengths

Long Wavelength Short Wavelength

180 degrees/(angle in the sky) Amplitude of Waves [μK2]

WMAP Collaboration

Power spectrum, explained

Amplitude of Waves [μK2]

180 degrees/(angle in the sky)

Density of Hydrogen & Helium

Amplitude of Waves [μK2]

180 degrees/(angle in the sky)

Density of All Matter

180 degrees/(angle in the sky) Amplitude of Waves [μK2]

Long Wavelength Short Wavelength

Removing Ripples: Power Spectrum of Primordial Fluctuations

180 degrees/(angle in the sky) Amplitude of Waves [μK2]

Long Wavelength Short Wavelength

Removing Ripples: Power Spectrum of Primordial Fluctuations

180 degrees/(angle in the sky) Amplitude of Waves [μK2]

Long Wavelength Short Wavelength

Removing Ripples: Power Spectrum of Primordial Fluctuations

180 degrees/(angle in the sky) Amplitude of Waves [μK2]

Long Wavelength Short Wavelength

Let’s parameterise like

Wave Amp. ∝ `ns−1

180 degrees/(angle in the sky) Amplitude of Waves [μK2]

Long Wavelength Short Wavelength

Wave Amp. ∝ `ns−1

WMAP 9-Year Only: ns=0.972±0.013 (68%CL)

2001–2010

WMAP Collaboration

1000 100

South Pole Telescope [10-m in South Pole] Atacama Cosmology Telescope [6-m in Chile]

Amplitude of Waves [μK2]

ns=0.965±0.010

2001–2010

WMAP Collaboration

1000 100

South Pole Telescope [10-m in South Pole] Atacama Cosmology Telescope [6-m in Chile]

Amplitude of Waves [μK2]

2001–2010

ns=0.961±0.008

~5σ discovery of ns<1 from the CMB data combined with the distribution of galaxies

WMAP Collaboration

Residual

Planck 2013 Result!

180 degrees/(angle in the sky)

Amplitude of Waves [μK2]

2009–2013

ns=0.960±0.007

First >5σ discovery of ns<1 from the CMB data alone [Planck+WMAP]

• Thus, in principle, a rapidly-varying H(t) can be

compensated by varying ε or cs

Have we seen ε<<1?

• Note quite. ζ is basically proportional to H(t), but the pre-

factor can depend on time

• If there was only one dominant energy field during

inflation [single-field inflation]:

Garriga & Mukhanov (1999)

⇣ = (2✏cs)−1/2 × H

propagation speed

• f the fluctuation

We want more supporting evidence

• ζ does not quite probe H(t) directly because its property

depends on the property of matter fields present during inflation

• E.g., Connection between ζ and H(t) can be

complicated if we have more than one field during inflation

• We need another probe measuring H(t) more directly
• “Extraordinary claim requires extraordinary evidence”

Here comes gravitational waves

• Gravitational waves are not coupled to scalar matter at

the linear order. (More later on other forms of matter.) Thus, its vacuum fluctuation is connected directly to H(t)

Starobinsky (1979)

hij = √ 2eij MPl × H

independent of time!

prim

Finding nearly scale-invariant GW

• We wish to find primordial gravitational waves from

inflation by measuring its nearly scale-invariant spectrum:

hhij(k)hij,∗(k)i / knt

with

|nt| ⌧ 1

prim prim

nt = −2✏ < 0

In most models,

Theoretical energy density

Watanabe & EK (2006)

GW entered the horizon during the radiation era GW entered the horizon during the matter era

Spectrum of GW today

Spectrum of GW today

Watanabe & EK (2006) CMB PTA Interferometers

Wavelength of GW ~ Billions of light years!!!

Theoretical energy density

Since we have not found a signature of GW in CMB yet…

• Let’s talk about other tests of inflation before talking

about how to find GW in the future mission

• Gaussianity: Further support for quantum fluctuations
• Isotropy test: Was there a vector field during inflation?

[Values of Temperatures in the Sky Minus 2.725 K] / [Root Mean Square]

Fraction of the Number of Pixels Having Those Temperatures Quantum Fluctuations give a Gaussian distribution of temperatures. Do we see this in the WMAP data?

[Values of Temperatures in the Sky Minus 2.725 K] / [Root Mean Square]

Fraction of the Number of Pixels Having Those Temperatures

YES!!

Histogram: WMAP Data Red Line: Gaussian

WMAP Collaboration

Testing Gaussianity

• Since a Gauss distribution

is symmetric, it must yield a vanishing 3-point function

[Values of Temperatures in the Sky Minus 2.725 K]/ [Root Mean Square] Fraction of the Number of Pixels Having Those Temperatures

Histogram: WMAP Data Red Line: Gaussian

hδT 3i ⌘ Z ∞

−∞

dδT P(δT)δT 3

• More specifically, we measure

this by averaging the product

• f temperatures at three

different locations in the sky

hδT(ˆ n1)δT(ˆ n2)δT(ˆ n3)i

Lack of non-Gaussianity

• The WMAP data show that the distribution of temperature

fluctuations of CMB is very precisely Gaussian

• with an upper bound on a deviation of 0.2% (95%CL)

ζ(x) = ζgaus(x) + 3 5fNLζ2

gaus(x) with fNL = 37 ± 20 (68% CL)

• The Planck data improved the upper bound by an order of

magnitude: deviation is <0.03% (95%CL)

fNL = 0.8 ± 5.0 (68% CL)

WMAP 9-year Result Planck 2015 Result

• Consider that there existed a homogeneous vector field at

the beginning of inflation

• Energy density of the vector field is tiny compared to

the “inflaton” field φ driving inflation

• With an appropriate setting, this vector field makes the

inflationary expansion anisotropic if

Vector field during inflation?

Aµ = (0, u(t), 0, 0)

A1: Preferred direction in space at the initial time with f=exp(cφ2/2) Watanabe, Kanno & Soda (2009, 2010)

Fµν ≡ ∂µAν − ∂νAµ

• How large can be during inflation?
• In single scalar field theories, Einstein’s equation gives
• But, a vector field yields anisotropic stress in the stress-

energy tensor, sourcing a sustained period of anisotropic inflation

Anisotropic Inflation

ds2 = −dt2 + e2Ht h e−2β(t)dx2 + e2β(t)(dy2 + dz2) i ˙ β/H ˙ β ∝ e−3Ht T i

j = Pδi j + πi j

π1

1 = −2

3V, π2

2 = π3 3 = 1

3V with ¨ β + 3H ˙ β = 1 3V

sourced by anisotropic stress Watanabe, Kanno & Soda (2009, 2010)

Observational Consequence

• Anisotropic inflation breaks rotational invariance, making

the scalar power spectrum depend on a direction of the wavenumber

Ackerman, Carroll & Wise (2007); Watanabe, Kanno & Soda (2010)

P(k) → P(k) = P0(k) h 1 + g∗(k)(ˆ k · ˆ E)2i

is a preferred direction in space

ˆ E

• The model predicts g∗(k) = −O(1) × 24IkN 2

k

• I is the energy density fraction of a vector field divided by ε

I ≡ 4 ✓∂φU U ◆−2 ρA U

Slowly-varying function of time ζ ζ

Signature in the CMB

• The effect of this “quadrupolar modulation” of the power

spectrum on the CMB can be understood intuitively. It turns a circular hot/cold spot of the CMB into an elliptical

• ne:

P(k) → P(k) = P0(k) h 1 + g∗(k)(ˆ k · ˆ E)2i preferred direction, E g*<0

• This is a local effect, rather than a global one. The power

spectrum measured at any location in sky is modulated by (ˆ

k · ˆ E)2

ζ ζ

A Beautiful Story

• In 2007, Ackerman, Carroll, and Wise proposed g* as a

powerful probe of anisotropic inflation

• In 2009, Groeneboom and Eriksen reported a significant

detection, g*=0.15±0.04, in the WMAP data at 94 GHz

• Wow! A new observable proposed by theorists was

looked for in the data, and was found. Beautiful.

Subsequent Events

• In 2010, Groeneboom et al. reported the opposite sign,

g*=–0.18±0.04, in the WMAP data at 41 GHz (not 94)

• The best-fitting preferred direction in sky was the ecliptic
• pole. Did not seem cosmological…
• Elliptical beam (point spread function) of the WMAP was a

culptrit!

• instrument electronics
• attitude control/propulsion
• command/data handling
• battery and power control

60K 90K

• WMAP visits ecliptic poles from many different

directions, circularising beams

• WMAP visits ecliptic planes with 30% of possible angles

Ecliptic Poles # of observations in Galactic coordinates

41GHz 94GHz

Planck 2013 Data

• We also found a significant detection from the Planck

temperature data: g*=–0.111±0.013

• This is also consistent with the beam ellipticity of Planck
• g* is consistent with zero after subtracting the beam effect

Kim & EK (2013)

g*=0.002±0.016 (68%CL) g*(raw)=–0.111±0.013 (68%CL)

Kim & EK (2013)

What does this mean for anisotropic inflation?

• g* is consistent with zero, with 95%CL upper bound of |g*|

<0.03

• Comparing this with the model prediction, we find

Naruko, EK & Yamaguchi (2016)

˙

• H ≈ V

U ≈ ✏I < 5 × 10−9

Breaking of rotational symmetry is tiny, if any!

• The “natural” value is either 10–2 or exp(-3N)=exp(-150)!

ds2 = −dt2 + e2Ht h e−2β(t)dx2 + e2β(t)(dy2 + dz2) i

Recap so far

• With WMAP we found super-horizon, adiabatic, and

Gaussian primordial fluctuations with ns<1

• The Planck data confirmed all of our findings, and

significantly tightened the limits and strengthened ns<1

• We found no evidence for breaking of rotational

invariance during inflation after correcting for instrumental effects, and put a stringent bound

• All the data are wonderfully consistent with the

predictions of single-field slow-roll inflation models

But we want to find more about inflation!

Back to Gravitational Waves

• Next frontier in the CMB research
• 1. Find evidence for nearly scale-invariant gravitational

waves

• 2. Once found, test Gaussianity to make sure (or not!)

that the signal comes from vacuum fluctuation

• 3. Constrain inflation models

New Research Area!

Measuring GW

d`2 = dx2 = X

ij

ijdxidxj d`2 = X

ij

(ij + hij)dxidxj

• GW changes distances between two points

Laser Interferometer

Mirror Mirror detector

No signal

Laser Interferometer

Mirror Mirror

Signal!

detector

Laser Interferometer

Mirror Mirror

Signal!

detector

LIGO detected GW from a binary blackholes, with the wavelength

• f thousands of kilometres

But, the primordial GW affecting the CMB has a wavelength of billions of light-years!! How do we find it?

Detecting GW by CMB

Isotropic electro-magnetic fields

Detecting GW by CMB

GW propagating in isotropic electro-magnetic fields

hot hot cold cold c

• l

d c

• l

d h

• t

h

• t

Detecting GW by CMB

Space is stretched => Wavelength of light is also stretched

hot hot cold cold c

• l

d c

• l

d h

• t

h

• t

Detecting GW by CMB Polarisation

electron electron Space is stretched => Wavelength of light is also stretched

hot hot cold cold c

• l

d c

• l

d h

• t

h

• t

Detecting GW by CMB Polarisation

Space is stretched => Wavelength of light is also stretched

63

• No detection of polarisation from primordial

GW yet

• Many ground-based and balloon-borne

experiments are taking data now

The search continues!!

Current Situation

1989–1993 2001–2010 2009–2013 202X–

ESA

2025– [proposed]

JAXA

+ possibly NASA

LiteBIRD

2025– [proposed]

ESA

2025– [proposed]

JAXA

+ possibly NASA

LiteBIRD

2025– [proposed]

Polarisation satellite dedicated to measure CMB polarisation from primordial GW, with a few thousand super-conducting detectors in space

ESA

2025– [proposed]

JAXA

+ possibly NASA

LiteBIRD

2025– [proposed]

Down-selected by JAXA as

• ne of the two missions

competing for a launch in 2025

Tensor-to-scalar Ratio

• We really want to find this! The current upper bound is

r<0.07 (95%CL)

r ⌘ hhijhiji hζ2i

BICEP2/Keck Array Collaboration (2016)

2007

WMAP 3-Year Data

2009

WMAP 5-Year Data

2011

WMAP 7-Year Data

2013

WMAP 9-Year Data

2013

WMAP 9-Year Data + ACT + SPT

2013

WMAP 9-Year Data + ACT + SPT + BAO

WMAP(temp+pol)+ACT+SPT+BAO+H0 WMAP(pol) + Planck + BAO

ruled

• ut!

WMAP Collaboration

WMAP(temp+pol)+ACT+SPT+BAO+H0 WMAP(pol) + Planck + BAO

ruled

• ut!

WMAP Collaboration with non-minimal coupling: EK & Futamase (1999)

Inflaton Potential，V(φ) Inflaton Field Value，φ/MPlanck

Z d4x√−g R 2 − 1 2(∂φ)2 − λ 4 φ4

• Z

d4x√−g 1 2(1 + ξφ2)R − 1 2(∂φ)2 − λ 4 φ4

• ns=0.94, r=0.32

ns=0.96, r=0.005

EK & Futamase (1999)

WMAP(temp+pol)+ACT+SPT+BAO+H0 WMAP(pol) + Planck + BAO

ruled

• ut!

ruled out! ruled out! ruled out! ruled out!

Polarsiation limit added: r<0.07 (95%CL)

Planck Collaboration (2015); BICEP2/Keck Array Collaboration (2016)

Are GWs from vacuum fluctuation in spacetime, or from sources?

• Homogeneous solution: “GWs from vacuum fluctuation”
• Inhomogeneous solution: “GWs from sources”
• Scalar and vector fields cannot source tensor fluctuations

at linear order

• SU(2) gauge field can!

⇤hij = −16πGπij

Maleknejad & Sheikh-Jabbari (2013); Dimastrogiovanni & Peloso (2013); Adshead, Martinec & Wyman (2013)

GW from Axion-SU(2) Dynamics

• φ: inflaton field
• χ: pseudo-scalar “axion” field. Spectator field (i.e.,

negligible energy density compared to the inflaton)

• Field strength of an SU(2) field :

Dimastrogiovanni, Fasielo & Fujita (2017)

Scenario

• The SU(2) field contains tensor, vector, and scalar

components

• The tensor components are amplified strongly by a

coupling to the axion field

• But, only one helicity is amplified => GW is chiral

(well-known result)

• Brand-new result: GWs sourced by this mechanism are

strongly non-Gaussian!

Agrawal, Fujita & EK (to appear on arXiv in the next couple of weeks)

Large bispectrum in GW from SU(2) fields

• ΩA << 1 is the energy density fraction of the gauge field
• Bh/Ph2 is of order unity for the vacuum contribution
• Gaussianity offers a powerful test of whether the

detected GW comes from the vacuum fluctuation or from sources

BRRR

h

(k, k, k) P 2

h(k)

≈ 25 ΩA

Agrawal, Fujita & EK (to appear on arXiv in the next couple of weeks) Aniket Agrawal (MPA) Tomo Fujita (Stanford->Kyoto)

Summary

• Single-field inflation looks good: all the CMB data support it
• Next frontier: Using CMB polarisation to find GWs from

inflation

• With LiteBIRD we plan to reach r~10–3, which is 100 times

smaller than the current bound

• GW from vacuum or sources? An exciting window to new

physics