Finding Cosmic Inflation

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

ourselves 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 of 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

h ij

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

= a

²

(t)[1 + 2⇣ (x, t)][

_ij

+ h

_ij

(x, t)]dx

ⁱ

dx

^j

X

i

h

_ii

= 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

= a

²

(t)[1 + 2⇣ (x, t)][

_ij

+ h

_ij

(x, t)]dx

ⁱ

dx

^j

X

i

h

_ii

= 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 + H

²

> 0 ✏ ⌘ H ˙

H

²

< 1

•

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

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

N ⌘ ln a _end

a =

Z _t

_end

t

dt ⁰ H (t ⁰ ) ⇡ 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 ˙

H

²

< 1

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

of 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 W aves [ μ K

2

]

WMAP Collaboration

Power spectrum, explained

Amplitude of W aves [ μ K 2 ]

180 degrees/(angle in the sky)

Density of Hydrogen & Helium

Amplitude of W aves [ μ K 2 ]

180 degrees/(angle in the sky)

Density of All Matter

180 degrees/(angle in the sky) Amplitude of W aves [ μ K

2

]

Long Wavelength Short Wavelength

Removing Ripples:

Power Spectrum of

Primordial Fluctuations

180 degrees/(angle in the sky) Amplitude of W aves [ μ K

2

]

Long Wavelength Short Wavelength

Removing Ripples:

Power Spectrum of

Primordial Fluctuations

180 degrees/(angle in the sky) Amplitude of W aves [ μ K

2

]

Long Wavelength Short Wavelength

Removing Ripples:

Power Spectrum of

Primordial Fluctuations

180 degrees/(angle in the sky) Amplitude of W aves [ μ K

2

]

Long Wavelength Short Wavelength

Let’s parameterise like

Wave Amp. / ` ^{n s 1}

180 degrees/(angle in the sky) Amplitude of W aves [ μ K

2

]

Long Wavelength Short Wavelength

Wave Amp. / ` ^{n s 1}

WMAP 9-Year Only:

n s =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 W aves [ μ K

2

]

n s =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 W aves [ μ K

2

]

2001–2010

n s =0.961±0.008

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

distribution of galaxies

WMAP Collaboration

Res id ua l

Planck 2013 Result!

180 degrees/(angle in the sky)

Amplitude of W aves [ μ K

2

]

2009–2013

n s =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✏c _s ) ^1/2 ⇥ H

propagation speed of 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)

h _ij =

p 2e _ij

M _Pl ⇥ 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:

h h _ij (k)h ^{ij, ⇤} (k) i / k ^{n t}

with

| n _t | ⌧ 1

prim

prim

n

_t

= 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 T emperatur es

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 T emperatur es

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

h T

³

i ⌘

Z

₁

1

d T P ( T ) T

³

•

More specifically, we measure this by averaging the product of temperatures at three

different locations in the sky

h T (ˆ n

₁

) T (ˆ n

₂

) T (ˆ n

₃

) 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)

⇣ (x) = ⇣

_gaus

(x) + 3

5 f

_NL

⇣

_gaus²

(x)

^with

f

_NL

= 37 ± 20 (68% CL)

•

The Planck data improved the upper bound by an order of magnitude: deviation is <0.03% (95%CL)

f

_NL

= 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?

A

_µ

= (0, u(t), 0, 0)

^A1: Preferred direction in space at the initial time

with f=exp(cφ²/2)

Watanabe, Kanno & Soda (2009, 2010)

F_µ⌫ ⌘ @_µA_⌫ @_⌫A_µ

•

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

ds

²

= dt

²

+ e

^2Ht

h

e

^{2 (t)}

dx

²

+ e

^{2 (t)}

(dy

²

+ dz

²

) i

˙ /H

˙ / e

^3Ht

T

_jⁱ

= P

_jⁱ

+ ⇡

_jⁱ

⇡

₁¹

= 2

3 V , ⇡

₂²

= ⇡

₃³

= 1 3 V with

¨ + 3H ˙ = 1

3 V

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) = P

₀

(k ) h

1 + g

_⇤

(k )(ˆ k · E ˆ )

²

i

is a preferred direction in space

E ˆ

•

The model predicts

g

_⇤

(k ) = O (1) ⇥ 24I

_k

N

_k²

•

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

I ⌘ 4

✓ @ U U

◆

2

⇢

_A

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 one:

P (k ) ! P (k) = P

₀

(k ) h

1 + g

_⇤

(k )(ˆ k · E ˆ )

²

i

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 · E ˆ )

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!

WMAPWMAP Spacecraft Spacecraft

MAP990422

thermally isolated instrument cylinder

secondary reflectors focal plane assembly

feed horns back to back Gregorian optics, 1.4 x 1.6 m primaries

upper omni antenna line of sight

deployed solar array w/ web shielding medium gain antennae

passive thermal radiator

warm spacecraft with:

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

60K

90K

300K

• 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)

−0.15 −0.1 −0.05 0 0.05 g*

with beam correction without beam correction

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)!

ds

²

= dt

²

+ e

^2Ht

h

e

^{2 (t)}

dx

²

+ e

^{2 (t)}

(dy

²

+ dz

²

) 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

= dx

²

= X

ij

ij

dx

ⁱ

dx

^j

d`

²

= X

ij

(

_ij

+ h

_ij

)dx

ⁱ

dx

^j

•

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

of 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

cold cold

hot hot

Detecting GW by CMB

Space is stretched => Wavelength of light is also stretched

hot

hot

cold

cold

cold cold

hot hot

Detecting GW by CMB Polarisation

electron electron

Space is stretched => Wavelength of light is also stretched

hot

hot

cold

cold

cold cold

hot hot

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 one 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 ⌘ h h _ij h ^ij i h ⇣ ² i

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+H

0

WMAP(pol) + Planck + BAO ^ruled

out!

WMAP Collaboration

WMAP(temp+pol)+ACT+SPT+BAO+H

0

WMAP(pol) + Planck + BAO ^ruled

out!

WMAP Collaboration

with non-minimal coupling:

EK & Futamase (1999)

Inflaton Potential ， V( φ )

Inflaton Field Value ， φ/M

Planck

Z

d ⁴ x p

g

 R 2

1

2 (@ ) ²

4

4

Z

d

⁴

x p

g

 1

2 (1 + ⇠

²

)R 1

2 (@ )

²

4

4

n

s

=0.94, r=0.32

n

s

=0.96, r=0.005

EK & Futamase (1999)

WMAP(temp+pol)+ACT+SPT+BAO+H

0

WMAP(pol) + Planck + BAO ^ruled

out!

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!

⇤ h _ij = 16⇡ G⇡ _ij

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

B _h ^RRR (k, k, k )

P _h ² (k ) ⇡ 25

⌦ _A

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