Constraining Gravitational Waves from Inflation

Ema DimastrogiovanniThe University of New South Wales

BEYOND19 - Warsaw— July 5th 2019

Outline

Brief intro + current bounds

Particle sources during inflation

Tensor fossils

Polarized Sunyaev–Zeldovich tomography

1810.09463 - Deutsch, ED, Fasiello, Johnson, Muenchmeyer1707.08129 - Deutsch, ED, Johnson, Muenchmeyer, Terrana

1806.05474 - ED, Fasiello, Hardwick, Assadullahi, Koyama, Wands

1608.04216 - ED, Fasiello, Fujita 1411.3029 - Biagetti, ED, Fasiello, Peloso

1708.01587 - Biagetti, ED, Fasiello1407.8204 - ED, Fasiello, Jeong, Kamionkowski

1504.05993 - ED, Fasiello, Kamionkowski

1906.07204 - ED, Fasiello, Tasinato

Outline

Particle sources during inflation

Tensor fossils

Polarized Sunyaev–Zeldovich tomography

Brief intro + current bounds

about 13.8 billion years

The universe over time

P⇣(k) =1

8⇡2

1

✏

H2

M2pl

✓k

k⇤

◆ns�1

2.2⇥ 10�9

0.968± 0.006

[k⇤ = 0.05Mpc�1, 68%C.L.]

from Planck measurements of CMB anisotropies

�� ! ⇣ ! �T

Two-point statistics of primordial perturbations: scalars

�ii = @i�ij = 0 two polarization states of the graviton

Gravitational waves

ds2 = �dt2 + a2(t) (�ij + �ij) dxidxj

• homogeneous solution: GWs from vacuum fluctuations

GW background from inflation

• inhomogeneous solution: GWs from sources

⇧TTij /

scalars

{@i�@j�}TT

vectors

{EiEj +BiBj}TT

tensors

{�ij}TT

anisotropic stress-energy

tensor�Tij(from )

�ij + 3H �ij + k2�ij = 16⇡G⇧TT

ij

red tilt: amplitude decreases

as we go towards smaller scales

energy scale of inflation

Pvacuum� (k) =

2

⇡2

H2

M2pl

✓k

k⇤

◆nT

nT = �r/8

• homogeneous solution: GWs from vacuum fluctuations

and non-chiral!!!

GW background from inflation

r ⌘ P�

P⇣

10�1610�18 10�14 10�12 10�10 10�8 10�6 10�4 10�2 1 102wave

frequency (Hz)

space-based interferometers

terrestrialinterfer.

time between wave peaks

(age of the universe) (hours) (secs) (millisecs)

CMB anisotropies

Scales — Experiments

(years)

pulsar timing arrays (PTA)

Observational bounds/sensitivities

Implications for model building:

threshold for large field inflation

��

Mpl&

⇣r⇤8

⌘1/2N⇤ &

⇣ r⇤0.01

⌘1/2

H . 5⇥ 1013 GeVupper bound on the

energy scale of inflation

r[0.05Mpc�1] < 0.06

BICEP2/KECK+Planck

Next generation:BICEP, SPT-3G,

Simons Obs., LiteBIRD, CMB-S4,

PICO

�(r) ! 0.0005

Observational bounds/sensitivities

Outline

Particle sources during inflation

Tensor fossils

Polarized Sunyaev–Zeldovich tomography

Brief intro + current bounds

�ij + 3H �ij + k2�ij = 16⇡G⇧TT

ij

INFLATON SPECTATOR SECTOR+

negligible energy density compared to the inflaton

P tot� = P vacuum

� + P spectator�

• inhomogeneous solution: GWs from sources

GW background from inflation

Why additional fields?

Interesting for phenomenology: qualitatively different signatures w.r.t. basic single-field inflation testable!

Natural from a top-down perspective: plenty of candidates from string theory (e.g. moduli fields, axions, Kaluza-Klein modes, gauge fields…)

Scalar field (I)

Spectator fields with small sound speed

P (X,�)

subdominant at the background levelrelevant for perturbations

[Biagetti, Fasiello, Riotto 2012, Biagetti, ED, Fasiello, Peloso 2014, Fujita, Yokoyama, Yokoyama, 2014]

�ij + 3H �ij + k2�ij = 16⇡G⇧TT

ij

Pspectator� / 1

cns

H4

M4P

Sourced may be comparable to vacuum fluctuations

Breaking standard r—H relation: r = f(✏, cs)

small sound speed: from integrating out heavy fields

c2s =PX

PX + �20PXX

< 1

X ⌘ � (@�)2

/ @i�@j�

Scalar field (II)

[Chung et al., 2000, Senatore et al, 2011, Pearce et al, 2017]

Auxiliary scalars with time-varying mass

graviton (and scalar fluctuations!) production

g2

2(�� �⇤)

2 �2

particle burst when inflaton crosses over value�⇤

features in the power spectra

Axion-Gauge fields models: genesis

✏ ⌘M2

p

2

V

0

V

!2

, ⌘ ⌘ M2pV

00

V

… but flatness may be spoiled by radiative corrections!

Generic requirement for inflation: nearly flat potential: ✏, |⌘| ⌧ 1

� ! �+ c

V (') = ⇤4 [1� cos ('/f)]

Flatness protected by axionic shift symmetryNatural Inflation [Freese, Frieman, Olinto 1990]

f & MPAgreement with observations requires:

undesirable constraint on the theory[Kallosh, Linde, Susskind, 1995, Banks et al, 2003]

V (�)

�

Axion-Gauge fields models: motivation

naturally light inflaton

��

4fF F

sub-Planckian axion decay constant

support reheating

Anber - Sorbo 2009, Cook - Sorbo 2011, Barnaby - Peloso 2011, Barnaby - Namba - Peloso 2011Adshead - Wyman 2011, Maleknejad - Sheikh-Jabbari, 2011, ED - Fasiello - Tolley 2012ED - Peloso 2012, Namba - ED - Peloso 2013, Adshead - Martinec -Wyman 2013, ED - Fasiello - Fujita 2016Agrawal - Fujita - Komatsu 2017, Thorne - Fujita - Hazumi - Katayama - Komatsu - Shiraishi ’17Caldwell - Devulder 2017, …

LspectatorP�,vacuum P�,sourced

L = Linflaton � 1

2(@�)2 � U(�)� 1

4FF +

��

4fF F

Inflaton field dominates energy density of the universe

Spectator sector contribution to curvature fluctuations negligible

[ED-Fasiello-Fujita 2016]

Axion-Gauge fields models: SU(2)

Aa0 = 0

Aai = aQ�ai

slow-roll background attractor solution

�Aai = tai + ... TT-component A �

One helicity of the gauge field fluctuations is amplified from coupling with axion the same helicity of the tensor mode is amplified

�R

tRgauge field(L)

�

A

[ED-Fasiello-Fujita 2016]

Axion-Gauge fields models: SU(2)

Axion-Gauge fields models: signatures

Non-Gaussianity

Chirality

Scale dependence

Scale-dependence

basic single-field inflation axion-gauge fields models

nT ' �r/8

(nearly flat spectrum)

[ED-Fasiello-Fujita 2016, Thorne et al, 2017]

detectably large and running nT

bump may occur at small scales

Chirality

basic single-field inflation axion-gauge fields models

�L = �R �L 6= �Rnon-chiral chiral

hTBi, hEBi 6= 0hTBi, hEBi = 0(parity conservation)

Detectable at 2 by LiteBIRD for r > 0.03 [Thorne et al, 2017]

�

basic single-field inflation axion-gauge fields models

�L = �R �L 6= �Rnon-chiral chiral

Interferometers: need advanced design with multiple (non co-planar) detectors [Thorne et al. 2017, Smith-Caldwell 2016]

Chirality

Non-Gaussianity: beyond the power spectrum

k1

k2

k3

h��1k1��2k2��3k3i = (2⇡)3�(3)(k1 + k2 + k3)B

�1�2�3� (k1, k2, k3)

tensor bispectrum

shape:

amplitude: fNL =B

P 2⇣

Tensor non-Gaussianity

k1

k2

k3

from interactions of the tensors with other fields or from self-interactions

�

�

�

[Agrawal - Fujita - Komatsu 2017]detectable by upcoming CMB space missions

axion-gauge fields models

�

�

� �

�

�

AA

A

A A

basic single-field inflation

fNL = O(r2)

too small for detection

�

�

�

Tensor non-Gaussianity

fNL = r2 · 50✏B

Mixed (scalar-tensor) non-Gaussianity

testing interactions of tensors and matter fields

A

�

� ⇣

[ED - Fasiello - Hardwick - Koyama - Wands 2018, Fujita - Namba - Obata 2018]

potentially observable!

Inflationary GWs from vacuum fluctuations

One or more of these predictions may be easily violated beyond

the minimal set-up!

• Energy scale of inflation: V 1/4infl ⇡ 1016GeV (r/0.01)1/4

• Scalar field excursion (Lyth bound): ��/MP & (r/0.01)1/2

• Non-chiral:

H ⇡ 2⇥ 1013p

r/0.01GeV

nT ' �2✏ = �r/8• Red tilt:

• Nearly Gaussian: fNL ⌧ 1

PL = PR

Outline

Particle sources during inflation

Tensor fossils

Polarized Sunyaev–Zeldovich tomography

Brief intro + current bounds

amplitude of long-wavelength modes coupled with amplitude of short-wavelength modes

long wavelength

short wavelength

short wavelengthk3 k2

k1

Squeezed non-Gaussianity

31

Soft limits and fossils

squeezed 3pf affects the 2pf

• No squeezed non-Gaussianity h�~k1�~k2

i = �(3)(~k1 + ~k2)P (k1)diagonal

2p correlation

• Squeezed non-Gaussianity

~K ~k1

~k2

there is also a

off-diagonal

contribution!h�~k1

�~k2i ~K = �(3)(~k1 + ~k2 + ~K)

⇥f(~k1,~k2)A(K)

short-wavelength modes

long-wavelength mode

[ED, Fasiello, Jeong, Kamionkowski - 2014, ED, Fasiello, Kamionkowski - 2015, Biagetti, ED, Fasiello - 2017]

Soft limits and fossils

�

⇣

⇣~K ~k1

~k2comes fromIf

constrain tensor modes amplitude/interactions with induced quadrupole anisotropy

super-Hubble K:

~k0

1~k

0

2

~k2

~k1~k

00

1

~k00

2

~k000

2

~k000

1

~K

estimate tensor modes amplitudefrom off-diagonal correlations

sub-Hubble K:

P⇣(k,xc)|�L = P⇣(k)⇣1 +Q`m(xc,k)k`km

⌘

Soft limits and fossils

�~K ~k1

~k2from

constrain tensor modes amplitude/interactions with induced quadrupole anisotropy

super-Hubble K:

�

�

P�(k,xc)|�L = P�(k)⇣1 +Q`m(xc,k)k`km

⌘

Soft limits and fossils

�~K ~k1

~k2from

constrain tensor modes amplitude/interactions with induced quadrupole anisotropy

super-Hubble K:

�

�

P�(k,xc)|�L = P�(k)⇣1 +Q`m(xc,k)k`km

⌘

Important remark: primordial bispectrum highly suppressed on small scales(superposition of signals from a large number of Hubble patches

+ Shapiro time-delay)[Bartolo, De Luca, Franciolini, Lewis, Peloso, Riotto 2018]

[ED, Fasiello, Tasinato 2019]

Soft limits and fossils

�~K ~k1

~k2from

constrain tensor modes amplitude/interactions with induced quadrupole anisotropy

super-Hubble K:

�

�

Crucial observable for tensor non-Gaussianityat interferometer scales!

P�(k,xc)|�L = P�(k)⇣1 +Q`m(xc,k)k`km

⌘

Why is squeezed non-Gaussianity

so important?

37

kN+1

k1k2

k3

kN

limkN+1!0

⇠ 0[ ]SINGLE-FIELD (single-clock) inflation: soft-limits not observable

Intuitive understanding :

[Maldacena 2003, Creminelli, Zaldarriaga 2004]

Soft limits in inflation

Soft mode rescales background for hard modes Effect can be gauged away!

Super-horizon modes freeze-out Standard initial conditions

Extra fields Soft limits reveal(extra) fields mediating

inflaton or graviton interactions

squeezed bispectrum delivers info on mass spectrum!!!

energy

Hm�

�⇣, �

⇣, �

⇣, �

39

Soft limits in inflation

probe for (extra) fields, pre-inflationary dynamics, (non-standard) symmetry patterns

Non-Bunch Davies initial states[Holman - Tolley 2007, Ganc - Komatsu 2012, Brahma - Nelson - Shandera 2013, …]

Broken space diffs (e.g. space-dependent background)[Endlich et al. 2013, ED - Fasiello - Jeong - Kamionkowski 2014, …]

[Chen - Wang 2009, ED - Fasiello - Kamionkowski 2015, ED - Emami 2016, Biagetti - ED - Fasiello 2017, …]

Extra fields

40

Soft limits in inflation

Learning about primordial gravitational waves through non-Gaussian effects

Local observables affected by long modes (anisotropic effects / off-diagonal correlations)

Effects from “squeezed” tensor-scalar-scalar bispectrum particularly effective at constraining inflation!

Tensor fossils

Quadrupole anisotropy crucial observable for tensor non-Gaussianity at interferometer scales

Outline

Particle sources during inflation

Tensor fossils

Polarized Sunyaev–Zeldovich tomography

Brief intro + current bounds

Large scale structure surveys

Primordial Gravitational Waves

CMB polarization

Interferometers

Polarized Sunyaev-Zel’dovich effect

• Polarization from Thomson scattering of (quadrupolar) radiation by free electrons

• Used to obtain a map of the remote (= locally observed) CMB quadrupole

• Additional information w.r.t. primary CMB (scattered photons from off our past light cone)

Notice:

(Q± iU)(ne)��pSZ

= �p6

10�T

Zd�e a(�e)ne(ne,�e) q

±e↵(ne,�e)

qme↵(ne,�e ! 0) = aT2m

qme↵(ne,�e) =

Zd2n

⇥⇥(ne,�e, n) +⇥T (ne,�e, n)

⇤Y ⇤2m(n)

q±e↵(ne,�) =2X

m=�2

qme↵(ne,�e)±2Y2m(ne)

“Remote” (observed at the location of the scatterer) CMB quadrupole

Polarized Sunyaev-Zel’dovich effect

pSZ tomography

Reconstructing the remote quadrupole field from CMB-LSS cross-correlation:

tracer of electron number density

long-wavelength modulation of

small-scale power

ensemble average over small-scales

(q treated as a fixed deterministic field)

[Kamionkowski, Loeb 1997, Alizadeh, Hirata 2012, Deutsch, ED, Johnson, Muenchmeyer, Terrana - 2017]

D(Q± iU)

��pSZ

�(�e)E⇠ h� q± �i ⇠ q±(ne, �e)h� �i(�e)

pSZ tomography

[A.-S. Deutsch, ED, M.C. Johnson, M. Muenchmeyer, A. Terrana - 2017]

Bin-averaged quadrupole field moments decomposition:

q±↵(ne) =X

`m

aq±↵`m ±2Y`m(ne)

aq,E ↵`m = �1

2

�aq+↵`m + aq�↵

`m

�

aq,B ↵`m = � 1

2i

�aq+↵`m � aq�↵

`m

�{ scalars/tensors

tensorsonly

Optimal unbiased estimator : X = E, B

aq,X ↵`m =

X

`1m1`2m2

⇣WX,E

`m`1m1`2m2aE`1m1

+WX,B`m`1m1`2m2

aB`1m1

⌘�⌧↵`2m2

binned density

field

Primordial gravitational wave phenomenology with pSZ tomography

[A.-S. Deutsch, ED, M. Fasiello, M.C. Johnson, M. Muenchmeyer - 2018]

full set of correlations between primary CMB and reconstructed remote quadrupole field

aT`m primary CMB temperature

E-mode remote quadrupole field

B-mode remote quadrupole field

aqE`m(�)

aqB`m(�)

primary CMB polarizationaB`m

aE`m

{

chirality primordial tensorpower spectrum

[A.-S. Deutsch, ED, M. Fasiello, M.C. Johnson, M. Muenchmeyer - 2018]

Fisher matrix forecast to derive exclusion bounds

• Our parameters:

amplitude

scale-dependence

chirality �c

r

nT

[A.-S. Deutsch, ED, M. Fasiello, M.C. Johnson, M. Muenchmeyer - 2018]

• green: zero-noise cosmic variance limit using primary CMB T, E, B

• red: T, E, B, qE, qB with instrumental noise 1µK � arcmin

0.1µK � arcmin• blue: T, E, B, qE, qB with instrumental noise

• grey: T, E, B, qE, qB with no instrumental noise

Forecasted parameter constraints

[A.-S. Deutsch, ED, M. Fasiello, M.C. Johnson, M. Muenchmeyer - 2018]

Observers: optimize future missions to go after these signals

improvements on constraints on phenomenological models of the tensor sector w.r.t. using the primary CMB (only)

pSZ tomography

can lead to discovery of new physics

testable on a vast range of scales (and from cross-correlations of different probes!)

Primordial gravitational waves

different observables (amplitude, chirality, scale dependence, non-Gaussianity) to characterize them and identify their sources

a very important probe of inflation

Thank you!