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!