Gravitational Waves from the Early Universe

Lecture 1A: Gravitational Waves, Theory

Kai Schmitz (CERN)

Chung-Ang University, Seoul, South Korea | June 2 – 4

1/21 ◦ 1916: Albert Einstein predicts GWs based on his general theory of relativity ◦ 2016: The LIGO/Virgo Collaboration announces the detection of GW150914 ◦ 2017: Nobel Prize in Physics awarded to Rainer Weiss, Barry Barish, and Kip Thorne → Milestone in fundamental physics, triumph of general relativity → Discovery of a new class of astrophysical objects: heavy black holes in binary systems

First direct detection of gravitational waves

[Nicolle Rager Fuller for sciencenews.org]

2/21 ◦ 2016: The LIGO/Virgo Collaboration announces the detection of GW150914 ◦ 2017: Nobel Prize in Physics awarded to Rainer Weiss, Barry Barish, and Kip Thorne → Milestone in fundamental physics, triumph of general relativity → Discovery of a new class of astrophysical objects: heavy black holes in binary systems

First direct detection of gravitational waves

[Nicolle Rager Fuller for sciencenews.org]

◦ 1916: Albert Einstein predicts GWs based on his general theory of relativity

2/21 ◦ 2017: Nobel Prize in Physics awarded to Rainer Weiss, Barry Barish, and Kip Thorne → Milestone in fundamental physics, triumph of general relativity → Discovery of a new class of astrophysical objects: heavy black holes in binary systems

First direct detection of gravitational waves

[Nicolle Rager Fuller for sciencenews.org]

◦ 1916: Albert Einstein predicts GWs based on his general theory of relativity ◦ 2016: The LIGO/Virgo Collaboration announces the detection of GW150914

2/21 → Milestone in fundamental physics, triumph of general relativity → Discovery of a new class of astrophysical objects: heavy black holes in binary systems

First direct detection of gravitational waves

[Nicolle Rager Fuller for sciencenews.org]

2/21 → Discovery of a new class of astrophysical objects: heavy black holes in binary systems

First direct detection of gravitational waves

[Nicolle Rager Fuller for sciencenews.org]

◦ 1916: Albert Einstein predicts GWs based on his general theory of relativity ◦ 2016: The LIGO/Virgo Collaboration announces the detection of GW150914 ◦ 2017: Nobel Prize in Physics awarded to Rainer Weiss, Barry Barish, and Kip Thorne → Milestone in fundamental physics, triumph of general relativity

2/21 First direct detection of gravitational waves

[Nicolle Rager Fuller for sciencenews.org]

2/21 ◦ Rapid and impressive progress: 50 events during observing runs O1, O2, and O3a ◦ Mergers: 47 binary black holes, 2 binary neutron stars, 1 black hole + ? ◦ 1 primary BH in the supernova mass gap, 1 remnant intermediate-mass black hole

Gravitational-Wave Transient Catalog 2 (GWTC-2)

[LIGO/Virgo Collaboration: 2010.14527] [Frank Elavsky, Aaron Geller for the LIGO/Virgo Collaboration]

3/21 ◦ Mergers: 47 binary black holes, 2 binary neutron stars, 1 black hole + ? ◦ 1 primary BH in the supernova mass gap, 1 remnant intermediate-mass black hole

Gravitational-Wave Transient Catalog 2 (GWTC-2)

[LIGO/Virgo Collaboration: 2010.14527] [Frank Elavsky, Aaron Geller for the LIGO/Virgo Collaboration] ◦ Rapid and impressive progress: 50 events during observing runs O1, O2, and O3a

3/21 ◦ 1 primary BH in the supernova mass gap, 1 remnant intermediate-mass black hole

Gravitational-Wave Transient Catalog 2 (GWTC-2)

[LIGO/Virgo Collaboration: 2010.14527] [Frank Elavsky, Aaron Geller for the LIGO/Virgo Collaboration] ◦ Rapid and impressive progress: 50 events during observing runs O1, O2, and O3a ◦ Mergers: 47 binary black holes, 2 binary neutron stars, 1 black hole + ?

3/21 Gravitational-Wave Transient Catalog 2 (GWTC-2)

3/21 ◦ GWs propagate freely after their production; new window onto the (early) Universe ◦ Allow us to probe energies far beyond the reach of other experiments / observations The journey has just begun: GWs will turn into indispensable tool for astrophysics and cosmology and advance to a primary probe of fundamental physics in the 21st century.

A new era in astrophysics and cosmology

[NASA]

4/21 ◦ Allow us to probe energies far beyond the reach of other experiments / observations The journey has just begun: GWs will turn into indispensable tool for astrophysics and cosmology and advance to a primary probe of fundamental physics in the 21st century.

A new era in astrophysics and cosmology

[NASA]

◦ GWs propagate freely after their production; new window onto the (early) Universe

4/21 The journey has just begun: GWs will turn into indispensable tool for astrophysics and cosmology and advance to a primary probe of fundamental physics in the 21st century.

A new era in astrophysics and cosmology

[NASA]

◦ GWs propagate freely after their production; new window onto the (early) Universe ◦ Allow us to probe energies far beyond the reach of other experiments / observations

4/21 A new era in astrophysics and cosmology

[NASA]

◦ GWs propagate freely after their production; new window onto the (early) Universe ◦ Allow us to probe energies far beyond the reach of other experiments / observations The journey has just begun: GWs will turn into indispensable tool for astrophysics and cosmology and advance to a primary probe of fundamental physics in the 21st century.

4/21 Lecture Part A Part B Wed: Gravitational waves, theory Gravitational waves, experiments Thu: Big-bang and inﬂationary cosmology Phase transitions Fri: Cosmic defects Current developments and outlook Literature: ◦ M. Maggiore, Gravitational Waves. Vol. 1: Theory and Experiments, Oxford University Press (2007), ISBN-13: 978-0198570745. ◦ M. Maggiore, Gravitational Waves. Vol. 2: Astrophysics and Cosmology, Oxford University Press (2018), ISBN-13: 978-0198570899. ◦ C. Caprini and D. G. Figueroa, Cosmological Backgrounds of Gravitational Waves, Class. Quant. Grav. 35 (2018) 163001, [arXiv:1801.04268]. ◦ J. D. Romano, N. J. Cornish, Detection methods for stochastic gravitational-wave backgrounds: a uniﬁed treatment, Living Rev. Rel. 20 (2017), no. 1 2, [arXiv:1608.06889].

Contact: [email protected] Slides (slightly longer version): https://doi.org/10.5281/zenodo.4678779

Syllabus

Aim of this set of lectures: Highlight some of the exciting new physics scenarios that we might be able to discover in the GW sky, with a focus on GWs from the early Universe

5/21 Literature: ◦ M. Maggiore, Gravitational Waves. Vol. 1: Theory and Experiments, Oxford University Press (2007), ISBN-13: 978-0198570745. ◦ M. Maggiore, Gravitational Waves. Vol. 2: Astrophysics and Cosmology, Oxford University Press (2018), ISBN-13: 978-0198570899. ◦ C. Caprini and D. G. Figueroa, Cosmological Backgrounds of Gravitational Waves, Class. Quant. Grav. 35 (2018) 163001, [arXiv:1801.04268]. ◦ J. D. Romano, N. J. Cornish, Detection methods for stochastic gravitational-wave backgrounds: a uniﬁed treatment, Living Rev. Rel. 20 (2017), no. 1 2, [arXiv:1608.06889].

Contact: [email protected] Slides (slightly longer version): https://doi.org/10.5281/zenodo.4678779

Syllabus

5/21 Contact: [email protected] Slides (slightly longer version): https://doi.org/10.5281/zenodo.4678779

Syllabus

Aim of this set of lectures: Highlight some of the exciting new physics scenarios that we might be able to discover in the GW sky, with a focus on GWs from the early Universe Lecture Part A Part B Wed: Gravitational waves, theory Gravitational waves, experiments Thu: Big-bang and inﬂationary cosmology Phase transitions Fri: Cosmic defects Current developments and outlook Literature: ◦ M. Maggiore, Gravitational Waves. Vol. 1: Theory and Experiments, Oxford University Press (2007), ISBN-13: 978-0198570745. ◦ M. Maggiore, Gravitational Waves. Vol. 2: Astrophysics and Cosmology, Oxford University Press (2018), ISBN-13: 978-0198570899. ◦ C. Caprini and D. G. Figueroa, Cosmological Backgrounds of Gravitational Waves, Class. Quant. Grav. 35 (2018) 163001, [arXiv:1801.04268]. ◦ J. D. Romano, N. J. Cornish, Detection methods for stochastic gravitational-wave backgrounds: a uniﬁed treatment, Living Rev. Rel. 20 (2017), no. 1 2, [arXiv:1608.06889].

5/21 Slides (slightly longer version): https://doi.org/10.5281/zenodo.4678779

Syllabus

Contact: [email protected]

5/21 Syllabus

Contact: [email protected] Slides (slightly longer version): https://doi.org/10.5281/zenodo.4678779

5/21 Notation and conventions

Natural units: c = ~ = 1

Minkowski metric: ηµν = diag (−1, +1, +1, +1) Physical constants: √ ◦ Planck mass M = 1/ G ' 1.22 × 1019 GeV Pl √ 18 ◦ Reduced Planck mass mPl = MPl/ 8π ' 2.44 × 10 GeV

Abbreviations:

ALP Axion-like particle MD Matter domination BBN Big-bang nucleosynthesis PBH Primordial black holes BOS Blanco-Pillado–Olum–Shlaer PISC Peak-integrated sensitivity curve BSM Beyond the Standard Model PLISC Power-law-integrated sensitivity curve CCR Canonical commutation relation QCD Quantum chromodynamics CMB Cosmic microwave background QFT Quantum ﬁeld theory DM Dark matter RD Radiation domination DOF Degree of freedom SFOPT Strong ﬁrst-order phase transition EOM Equation of motion SGWB Stochastic gravitational-wave background EOS Equation of state SM Standard Model EW Electroweak SMBH Supermassive black hole FLRW Friedmann–Lemaˆıtre–Robertson–Walker SNR Signal-to-noise ratio GW Gravitational wave SVT Scalar–vector–tensor KAGRA Kamioka Gravitational Wave Detector TOA Time of arrival LHS Left-hand side TT Transverse-traceless LIGO Laser Interferometer Gravitational-Wave Observatory UTC Unequal-time correlator LRS Lorenz–Ringeval–Sakellariadou VD Vacuum domination LUNA Laboratory for Underground Nuclear Astrophysics VOS Velocity-dependent one-scale

6/21 Outline Lecture 1A

1. GWs in linerarized gravity

2. Propagation of GWs in the expanding Universe

3. Stochastic backgrounds of cosmological GWs

4. Summary

7/21 ◦ Radiative: Dynamical evolution governed by wave equation ◦ Gauge-invariant: Physical DOFs, independent of choice of coordinate system

In linearized gravity over a ﬁxed, ﬂat Minkowski background:

gµν (x) = ηµν + hµν (x) , |hµν | 1 , hµν = hνµ (1)

GR is invariant under general coordinate transformations. Transformation behavior of hµν :

0µ µ µ 0 0 x = x + ξ (x) ⇒ hµν (x ) = hµν (x) − ∂µξν (x) − ∂ν ξµ(x) (2)

Einstein tensor Gµν (LHS of Einstein’s ﬁeld equations) to ﬁrst order in hµν :

1 1 α α β α Gµν = Rµν − gµν R = ∂α∂ν h¯ + ∂ ∂µh¯ − h¯ − ηµν ∂α∂ h¯ (3) 2 2 µ να µν β in terms of the trace-reversed metric perturbation:

1 µ µν h¯ = hµν − ηµν h , h = h = η hνµ (4) µν 2 µ

Deﬁnition of gravitational waves

GWs are: Radiative and gauge-invariant tensor perturbations of the spacetime metric gµν

1. GWs in linerarized gravity 8/21 ◦ Gauge-invariant: Physical DOFs, independent of choice of coordinate system

In linearized gravity over a ﬁxed, ﬂat Minkowski background:

gµν (x) = ηµν + hµν (x) , |hµν | 1 , hµν = hνµ (1)

GR is invariant under general coordinate transformations. Transformation behavior of hµν :

0µ µ µ 0 0 x = x + ξ (x) ⇒ hµν (x ) = hµν (x) − ∂µξν (x) − ∂ν ξµ(x) (2)

Einstein tensor Gµν (LHS of Einstein’s ﬁeld equations) to ﬁrst order in hµν :

1 1 α α β α Gµν = Rµν − gµν R = ∂α∂ν h¯ + ∂ ∂µh¯ − h¯ − ηµν ∂α∂ h¯ (3) 2 2 µ να µν β in terms of the trace-reversed metric perturbation:

1 µ µν h¯ = hµν − ηµν h , h = h = η hνµ (4) µν 2 µ

Deﬁnition of gravitational waves

GWs are: Radiative and gauge-invariant tensor perturbations of the spacetime metric gµν ◦ Radiative: Dynamical evolution governed by wave equation

1. GWs in linerarized gravity 8/21 In linearized gravity over a ﬁxed, ﬂat Minkowski background:

gµν (x) = ηµν + hµν (x) , |hµν | 1 , hµν = hνµ (1)

GR is invariant under general coordinate transformations. Transformation behavior of hµν :

0µ µ µ 0 0 x = x + ξ (x) ⇒ hµν (x ) = hµν (x) − ∂µξν (x) − ∂ν ξµ(x) (2)

Einstein tensor Gµν (LHS of Einstein’s ﬁeld equations) to ﬁrst order in hµν :

1 1 α α β α Gµν = Rµν − gµν R = ∂α∂ν h¯ + ∂ ∂µh¯ − h¯ − ηµν ∂α∂ h¯ (3) 2 2 µ να µν β in terms of the trace-reversed metric perturbation:

1 µ µν h¯ = hµν − ηµν h , h = h = η hνµ (4) µν 2 µ

Deﬁnition of gravitational waves

GWs are: Radiative and gauge-invariant tensor perturbations of the spacetime metric gµν ◦ Radiative: Dynamical evolution governed by wave equation ◦ Gauge-invariant: Physical DOFs, independent of choice of coordinate system

1. GWs in linerarized gravity 8/21 GR is invariant under general coordinate transformations. Transformation behavior of hµν :

0µ µ µ 0 0 x = x + ξ (x) ⇒ hµν (x ) = hµν (x) − ∂µξν (x) − ∂ν ξµ(x) (2)

Einstein tensor Gµν (LHS of Einstein’s ﬁeld equations) to ﬁrst order in hµν :

1 1 α α β α Gµν = Rµν − gµν R = ∂α∂ν h¯ + ∂ ∂µh¯ − h¯ − ηµν ∂α∂ h¯ (3) 2 2 µ να µν β in terms of the trace-reversed metric perturbation:

1 µ µν h¯ = hµν − ηµν h , h = h = η hνµ (4) µν 2 µ

Deﬁnition of gravitational waves

In linearized gravity over a ﬁxed, ﬂat Minkowski background:

gµν (x) = ηµν + hµν (x) , |hµν | 1 , hµν = hνµ (1)

1. GWs in linerarized gravity 8/21 Einstein tensor Gµν (LHS of Einstein’s ﬁeld equations) to ﬁrst order in hµν :

1 1 α α β α Gµν = Rµν − gµν R = ∂α∂ν h¯ + ∂ ∂µh¯ − h¯ − ηµν ∂α∂ h¯ (3) 2 2 µ να µν β in terms of the trace-reversed metric perturbation:

1 µ µν h¯ = hµν − ηµν h , h = h = η hνµ (4) µν 2 µ

Deﬁnition of gravitational waves

In linearized gravity over a ﬁxed, ﬂat Minkowski background:

gµν (x) = ηµν + hµν (x) , |hµν | 1 , hµν = hνµ (1)

GR is invariant under general coordinate transformations. Transformation behavior of hµν :

0µ µ µ 0 0 x = x + ξ (x) ⇒ hµν (x ) = hµν (x) − ∂µξν (x) − ∂ν ξµ(x) (2)

1. GWs in linerarized gravity 8/21 in terms of the trace-reversed metric perturbation:

1 µ µν h¯ = hµν − ηµν h , h = h = η hνµ (4) µν 2 µ

Deﬁnition of gravitational waves

In linearized gravity over a ﬁxed, ﬂat Minkowski background:

gµν (x) = ηµν + hµν (x) , |hµν | 1 , hµν = hνµ (1)

GR is invariant under general coordinate transformations. Transformation behavior of hµν :

0µ µ µ 0 0 x = x + ξ (x) ⇒ hµν (x ) = hµν (x) − ∂µξν (x) − ∂ν ξµ(x) (2)

Einstein tensor Gµν (LHS of Einstein’s ﬁeld equations) to ﬁrst order in hµν :

1 1 α α β α Gµν = Rµν − gµν R = ∂α∂ν h¯ + ∂ ∂µh¯ − h¯ − ηµν ∂α∂ h¯ (3) 2 2 µ να µν β

1. GWs in linerarized gravity 8/21 Deﬁnition of gravitational waves

In linearized gravity over a ﬁxed, ﬂat Minkowski background:

gµν (x) = ηµν + hµν (x) , |hµν | 1 , hµν = hνµ (1)

GR is invariant under general coordinate transformations. Transformation behavior of hµν :

0µ µ µ 0 0 x = x + ξ (x) ⇒ hµν (x ) = hµν (x) − ∂µξν (x) − ∂ν ξµ(x) (2)

Einstein tensor Gµν (LHS of Einstein’s ﬁeld equations) to ﬁrst order in hµν :

1 1 α α β α Gµν = Rµν − gµν R = ∂α∂ν h¯ + ∂ ∂µh¯ − h¯ − ηµν ∂α∂ h¯ (3) 2 2 µ να µν β in terms of the trace-reversed metric perturbation:

1 µ µν h¯ = hµν − ηµν h , h = h = η hνµ (4) µν 2 µ

1. GWs in linerarized gravity 8/21 The divergence of the metric perturbation thus transforms as:

0µ¯0 0 µ¯ ∂ hµν (x ) = ∂ hµν (x) − ξν (x) (6)

Use gauge freedom to impose the Lorentz gauge condition. That is, choose ξµ such that µ¯ 0µ¯0 0 ξµ(x) = ∂ hµν (x) ⇒ ∂ hµν (x ) = 0 (7)

Analogous to Lorenz gauge in electrodynamics. Solution for ξµ can always be constructed α αβ using the Green’s function of the d’Alembertian operator = ∂α∂ = η ∂α∂β . Einstein tensor of the linearized theory in Lorentz gauge: 1 Gµν = − h¯ (8) 2 µν

Residual gauge freedom in Lorentz gauge: four functions ξµ satisfying ξµ = 0.

Lorentz gauge

Under inﬁnitesimal coordinate transformations:

¯0 0 ¯ α hµν (x ) = hµν (x) + ξµν (x) , ξµν (x) = ηµν ∂αξ (x) − ∂µξν (x) − ∂ν ξµ(x) (5)

1. GWs in linerarized gravity 9/21 Use gauge freedom to impose the Lorentz gauge condition. That is, choose ξµ such that µ¯ 0µ¯0 0 ξµ(x) = ∂ hµν (x) ⇒ ∂ hµν (x ) = 0 (7)

Residual gauge freedom in Lorentz gauge: four functions ξµ satisfying ξµ = 0.

Lorentz gauge

Under inﬁnitesimal coordinate transformations:

¯0 0 ¯ α hµν (x ) = hµν (x) + ξµν (x) , ξµν (x) = ηµν ∂αξ (x) − ∂µξν (x) − ∂ν ξµ(x) (5)

The divergence of the metric perturbation thus transforms as:

0µ¯0 0 µ¯ ∂ hµν (x ) = ∂ hµν (x) − ξν (x) (6)

1. GWs in linerarized gravity 9/21 Analogous to Lorenz gauge in electrodynamics. Solution for ξµ can always be constructed α αβ using the Green’s function of the d’Alembertian operator = ∂α∂ = η ∂α∂β . Einstein tensor of the linearized theory in Lorentz gauge: 1 Gµν = − h¯ (8) 2 µν

Residual gauge freedom in Lorentz gauge: four functions ξµ satisfying ξµ = 0.

Lorentz gauge

Under inﬁnitesimal coordinate transformations:

¯0 0 ¯ α hµν (x ) = hµν (x) + ξµν (x) , ξµν (x) = ηµν ∂αξ (x) − ∂µξν (x) − ∂ν ξµ(x) (5)

The divergence of the metric perturbation thus transforms as:

0µ¯0 0 µ¯ ∂ hµν (x ) = ∂ hµν (x) − ξν (x) (6)

Use gauge freedom to impose the Lorentz gauge condition. That is, choose ξµ such that µ¯ 0µ¯0 0 ξµ(x) = ∂ hµν (x) ⇒ ∂ hµν (x ) = 0 (7)

1. GWs in linerarized gravity 9/21 Einstein tensor of the linearized theory in Lorentz gauge: 1 Gµν = − h¯ (8) 2 µν

Residual gauge freedom in Lorentz gauge: four functions ξµ satisfying ξµ = 0.

Lorentz gauge

Under inﬁnitesimal coordinate transformations:

¯0 0 ¯ α hµν (x ) = hµν (x) + ξµν (x) , ξµν (x) = ηµν ∂αξ (x) − ∂µξν (x) − ∂ν ξµ(x) (5)

The divergence of the metric perturbation thus transforms as:

0µ¯0 0 µ¯ ∂ hµν (x ) = ∂ hµν (x) − ξν (x) (6)

Use gauge freedom to impose the Lorentz gauge condition. That is, choose ξµ such that µ¯ 0µ¯0 0 ξµ(x) = ∂ hµν (x) ⇒ ∂ hµν (x ) = 0 (7)

Analogous to Lorenz gauge in electrodynamics. Solution for ξµ can always be constructed α αβ using the Green’s function of the d’Alembertian operator = ∂α∂ = η ∂α∂β .

1. GWs in linerarized gravity 9/21 Residual gauge freedom in Lorentz gauge: four functions ξµ satisfying ξµ = 0.

Lorentz gauge

Under inﬁnitesimal coordinate transformations:

¯0 0 ¯ α hµν (x ) = hµν (x) + ξµν (x) , ξµν (x) = ηµν ∂αξ (x) − ∂µξν (x) − ∂ν ξµ(x) (5)

The divergence of the metric perturbation thus transforms as:

0µ¯0 0 µ¯ ∂ hµν (x ) = ∂ hµν (x) − ξν (x) (6)

Use gauge freedom to impose the Lorentz gauge condition. That is, choose ξµ such that µ¯ 0µ¯0 0 ξµ(x) = ∂ hµν (x) ⇒ ∂ hµν (x ) = 0 (7)

1. GWs in linerarized gravity 9/21 Lorentz gauge

Under inﬁnitesimal coordinate transformations:

¯0 0 ¯ α hµν (x ) = hµν (x) + ξµν (x) , ξµν (x) = ηµν ∂αξ (x) − ∂µξν (x) − ∂ν ξµ(x) (5)

The divergence of the metric perturbation thus transforms as:

0µ¯0 0 µ¯ ∂ hµν (x ) = ∂ hµν (x) − ξν (x) (6)

Use gauge freedom to impose the Lorentz gauge condition. That is, choose ξµ such that µ¯ 0µ¯0 0 ξµ(x) = ∂ hµν (x) ⇒ ∂ hµν (x ) = 0 (7)

Residual gauge freedom in Lorentz gauge: four functions ξµ satisfying ξµ = 0.

1. GWs in linerarized gravity 9/21 which are invariant under Lorentz-gauge-preserving coordinate transformations in vacuum: ¯0 0 ¯ Tµν = 0 ⇒ hµν (x ) = hµν (x) = 0 (10)

Use this residual gauge freedom to impose four conditions and thus completely ﬁx the gauge:

h¯ = 0 ⇒ h¯µν = hµν (11) µ 0 ∂ hµ0 = 0 ,h i0 = 0 ⇒ ∂ h00 = 0 (12)

We can therefore choose h00 = V (x) = 0, which leads to the transverse-traceless gauge:

hµ0 = 0 , h = 0 , ∂ihij = 0 (13)

Physical DOFs: 10 components of the symmetric metric perturbation hµν − 4 gauge DOFs to ﬁx Lorentz gauge − 4 residual gauge DOFs to ﬁx transverse-traceless gauge = 2 physical transverse polarization states h+ and h×

Transverse-traceless gauge In Lorentz gauge, the ﬁeld equations obtain the form of wave equations:

1 ¯ 2 Gµν = 2 Tµν ⇒ hµν = − 2 Tµν (9) mPl mPl

1. GWs in linerarized gravity 10/21 Use this residual gauge freedom to impose four conditions and thus completely ﬁx the gauge:

h¯ = 0 ⇒ h¯µν = hµν (11) µ 0 ∂ hµ0 = 0 ,h i0 = 0 ⇒ ∂ h00 = 0 (12)

We can therefore choose h00 = V (x) = 0, which leads to the transverse-traceless gauge:

hµ0 = 0 , h = 0 , ∂ihij = 0 (13)

Transverse-traceless gauge In Lorentz gauge, the ﬁeld equations obtain the form of wave equations:

1 ¯ 2 Gµν = 2 Tµν ⇒ hµν = − 2 Tµν (9) mPl mPl

which are invariant under Lorentz-gauge-preserving coordinate transformations in vacuum: ¯0 0 ¯ Tµν = 0 ⇒ hµν (x ) = hµν (x) = 0 (10)

1. GWs in linerarized gravity 10/21 We can therefore choose h00 = V (x) = 0, which leads to the transverse-traceless gauge:

hµ0 = 0 , h = 0 , ∂ihij = 0 (13)

Transverse-traceless gauge In Lorentz gauge, the ﬁeld equations obtain the form of wave equations:

1 ¯ 2 Gµν = 2 Tµν ⇒ hµν = − 2 Tµν (9) mPl mPl

which are invariant under Lorentz-gauge-preserving coordinate transformations in vacuum: ¯0 0 ¯ Tµν = 0 ⇒ hµν (x ) = hµν (x) = 0 (10)

Use this residual gauge freedom to impose four conditions and thus completely ﬁx the gauge:

h¯ = 0 ⇒ h¯µν = hµν (11) µ 0 ∂ hµ0 = 0 ,h i0 = 0 ⇒ ∂ h00 = 0 (12)

1. GWs in linerarized gravity 10/21 Physical DOFs: 10 components of the symmetric metric perturbation hµν − 4 gauge DOFs to ﬁx Lorentz gauge − 4 residual gauge DOFs to ﬁx transverse-traceless gauge = 2 physical transverse polarization states h+ and h×

Transverse-traceless gauge In Lorentz gauge, the ﬁeld equations obtain the form of wave equations:

1 ¯ 2 Gµν = 2 Tµν ⇒ hµν = − 2 Tµν (9) mPl mPl

which are invariant under Lorentz-gauge-preserving coordinate transformations in vacuum: ¯0 0 ¯ Tµν = 0 ⇒ hµν (x ) = hµν (x) = 0 (10)

Use this residual gauge freedom to impose four conditions and thus completely ﬁx the gauge:

h¯ = 0 ⇒ h¯µν = hµν (11) µ 0 ∂ hµ0 = 0 ,h i0 = 0 ⇒ ∂ h00 = 0 (12)

We can therefore choose h00 = V (x) = 0, which leads to the transverse-traceless gauge:

hµ0 = 0 , h = 0 , ∂ihij = 0 (13)

1. GWs in linerarized gravity 10/21 Transverse-traceless gauge In Lorentz gauge, the ﬁeld equations obtain the form of wave equations:

1 ¯ 2 Gµν = 2 Tµν ⇒ hµν = − 2 Tµν (9) mPl mPl

which are invariant under Lorentz-gauge-preserving coordinate transformations in vacuum: ¯0 0 ¯ Tµν = 0 ⇒ hµν (x ) = hµν (x) = 0 (10)

Use this residual gauge freedom to impose four conditions and thus completely ﬁx the gauge:

h¯ = 0 ⇒ h¯µν = hµν (11) µ 0 ∂ hµ0 = 0 ,h i0 = 0 ⇒ ∂ h00 = 0 (12)

We can therefore choose h00 = V (x) = 0, which leads to the transverse-traceless gauge:

hµ0 = 0 , h = 0 , ∂ihij = 0 (13)

1. GWs in linerarized gravity 10/21 Decompose δg and δT into functions that transform as irreps (SVT) of spatial rotations:1

δg00 = −2φ (14)

δg0i = ∂iB + Si (15)

δgij = −2 ψ δij + (∂i∂j − 1/3 δij ∆)E + ∂iFj + ∂j Fi + hij (16) δT00 = ρ (17)

δT0i = ∂iu + ui (18)

δTij = p δij + (∂i∂j − 1/3 δij ∆)σ + ∂ivj + ∂j vi + Πij (19)

plus a set of constraint equations. E.g., for the tensor perturbations: ∂ihij = hii = 0. Gauge-invariant linear combinations:

Φ = φ + B˙ − 1/2 E,¨ Θ = −2φ − 1/3 ∆E, Σi = Si − F˙i , hij (20)

Field equations:

ρ + 3p − 3u ˙ ρ 2Si 2Πij ∆Φ = 2 , ∆Θ = − 2 , ∆Σi = − 2 , hij = − 2 (21) 2mPl mPl mPl mPl

The tensor perturbations hij are the only gauge-invariant DOFs satisfying a wave equation.

Scalar–vector–tensor (SVT) decomposition

Next: Identify physical DOFs independent of a particular gauge choice, allow for δTµν 6= 0.

1 Caution: New notation, hij → δgij . hij are now the tensor components of the general spatial perturbations δgij .

1. GWs in linerarized gravity 11/21 δg00 = −2φ (14)

δg0i = ∂iB + Si (15)

δgij = −2 ψ δij + (∂i∂j − 1/3 δij ∆)E + ∂iFj + ∂j Fi + hij (16) δT00 = ρ (17)

δT0i = ∂iu + ui (18)

δTij = p δij + (∂i∂j − 1/3 δij ∆)σ + ∂ivj + ∂j vi + Πij (19)

plus a set of constraint equations. E.g., for the tensor perturbations: ∂ihij = hii = 0. Gauge-invariant linear combinations:

Φ = φ + B˙ − 1/2 E,¨ Θ = −2φ − 1/3 ∆E, Σi = Si − F˙i , hij (20)

Field equations:

ρ + 3p − 3u ˙ ρ 2Si 2Πij ∆Φ = 2 , ∆Θ = − 2 , ∆Σi = − 2 , hij = − 2 (21) 2mPl mPl mPl mPl

The tensor perturbations hij are the only gauge-invariant DOFs satisfying a wave equation.

Scalar–vector–tensor (SVT) decomposition

Next: Identify physical DOFs independent of a particular gauge choice, allow for δTµν 6= 0. Decompose δg and δT into functions that transform as irreps (SVT) of spatial rotations:1

1 Caution: New notation, hij → δgij . hij are now the tensor components of the general spatial perturbations δgij .

1. GWs in linerarized gravity 11/21 Gauge-invariant linear combinations:

Φ = φ + B˙ − 1/2 E,¨ Θ = −2φ − 1/3 ∆E, Σi = Si − F˙i , hij (20)

Field equations:

ρ + 3p − 3u ˙ ρ 2Si 2Πij ∆Φ = 2 , ∆Θ = − 2 , ∆Σi = − 2 , hij = − 2 (21) 2mPl mPl mPl mPl

The tensor perturbations hij are the only gauge-invariant DOFs satisfying a wave equation.

Scalar–vector–tensor (SVT) decomposition

Next: Identify physical DOFs independent of a particular gauge choice, allow for δTµν 6= 0. Decompose δg and δT into functions that transform as irreps (SVT) of spatial rotations:1

δg00 = −2φ (14)

δg0i = ∂iB + Si (15)

δgij = −2 ψ δij + (∂i∂j − 1/3 δij ∆)E + ∂iFj + ∂j Fi + hij (16) δT00 = ρ (17)

δT0i = ∂iu + ui (18)

δTij = p δij + (∂i∂j − 1/3 δij ∆)σ + ∂ivj + ∂j vi + Πij (19)

plus a set of constraint equations. E.g., for the tensor perturbations: ∂ihij = hii = 0.

1 Caution: New notation, hij → δgij . hij are now the tensor components of the general spatial perturbations δgij .

1. GWs in linerarized gravity 11/21 Field equations:

ρ + 3p − 3u ˙ ρ 2Si 2Πij ∆Φ = 2 , ∆Θ = − 2 , ∆Σi = − 2 , hij = − 2 (21) 2mPl mPl mPl mPl

The tensor perturbations hij are the only gauge-invariant DOFs satisfying a wave equation.

Scalar–vector–tensor (SVT) decomposition

Next: Identify physical DOFs independent of a particular gauge choice, allow for δTµν 6= 0. Decompose δg and δT into functions that transform as irreps (SVT) of spatial rotations:1

δg00 = −2φ (14)

δg0i = ∂iB + Si (15)

δgij = −2 ψ δij + (∂i∂j − 1/3 δij ∆)E + ∂iFj + ∂j Fi + hij (16) δT00 = ρ (17)

δT0i = ∂iu + ui (18)

δTij = p δij + (∂i∂j − 1/3 δij ∆)σ + ∂ivj + ∂j vi + Πij (19)

plus a set of constraint equations. E.g., for the tensor perturbations: ∂ihij = hii = 0. Gauge-invariant linear combinations:

Φ = φ + B˙ − 1/2 E,¨ Θ = −2φ − 1/3 ∆E, Σi = Si − F˙i , hij (20)

1 Caution: New notation, hij → δgij . hij are now the tensor components of the general spatial perturbations δgij .

1. GWs in linerarized gravity 11/21 The tensor perturbations hij are the only gauge-invariant DOFs satisfying a wave equation.

Scalar–vector–tensor (SVT) decomposition

Next: Identify physical DOFs independent of a particular gauge choice, allow for δTµν 6= 0. Decompose δg and δT into functions that transform as irreps (SVT) of spatial rotations:1

δg00 = −2φ (14)

δg0i = ∂iB + Si (15)

δgij = −2 ψ δij + (∂i∂j − 1/3 δij ∆)E + ∂iFj + ∂j Fi + hij (16) δT00 = ρ (17)

δT0i = ∂iu + ui (18)

δTij = p δij + (∂i∂j − 1/3 δij ∆)σ + ∂ivj + ∂j vi + Πij (19)

plus a set of constraint equations. E.g., for the tensor perturbations: ∂ihij = hii = 0. Gauge-invariant linear combinations:

Φ = φ + B˙ − 1/2 E,¨ Θ = −2φ − 1/3 ∆E, Σi = Si − F˙i , hij (20)

Field equations:

ρ + 3p − 3u ˙ ρ 2Si 2Πij ∆Φ = 2 , ∆Θ = − 2 , ∆Σi = − 2 , hij = − 2 (21) 2mPl mPl mPl mPl

1 Caution: New notation, hij → δgij . hij are now the tensor components of the general spatial perturbations δgij .

1. GWs in linerarized gravity 11/21 Scalar–vector–tensor (SVT) decomposition

Next: Identify physical DOFs independent of a particular gauge choice, allow for δTµν 6= 0. Decompose δg and δT into functions that transform as irreps (SVT) of spatial rotations:1

δg00 = −2φ (14)

δg0i = ∂iB + Si (15)

δgij = −2 ψ δij + (∂i∂j − 1/3 δij ∆)E + ∂iFj + ∂j Fi + hij (16) δT00 = ρ (17)

δT0i = ∂iu + ui (18)

δTij = p δij + (∂i∂j − 1/3 δij ∆)σ + ∂ivj + ∂j vi + Πij (19)

plus a set of constraint equations. E.g., for the tensor perturbations: ∂ihij = hii = 0. Gauge-invariant linear combinations:

Φ = φ + B˙ − 1/2 E,¨ Θ = −2φ − 1/3 ∆E, Σi = Si − F˙i , hij (20)

Field equations:

ρ + 3p − 3u ˙ ρ 2Si 2Πij ∆Φ = 2 , ∆Θ = − 2 , ∆Σi = − 2 , hij = − 2 (21) 2mPl mPl mPl mPl

The tensor perturbations hij are the only gauge-invariant DOFs satisfying a wave equation.

1 Caution: New notation, hij → δgij . hij are now the tensor components of the general spatial perturbations δgij .

1. GWs in linerarized gravity 11/21 ¯ GW We now allow for Tµν 6= 0. This includes the contribution from GWs itself, Tµν , which can be calculated in second-order perturbation theory over a general curved background:2

2 GW 1 2 αβ h T = m ∇µh ∇ν h ∼ (23) µν 4 Pl αβ ∆L2

Energy density contained in GWs in transverse-traceless gauge:

GW 1 2 ˙ ˙ ij ρGW = T = m hij h (24) 00 4 Pl

Equation of motion on a curved background without any GW source and in Lorentz gauge:

αβ ¯ λ σ ¯ g¯ ∇α∇β hµν − 2R µν hλσ = 0 (25)

Curved background

Separation of scales: Consider background metric g¯µν that varies on length (time) scales LB (TB ). GWs then deﬁned in terms of perturbations δgµν on scales ∆L LB (∆T TB ):

gµν (x) =g ¯µν (x) + δgµν (x) , |δgµν | |g¯µν | (22)

2 h· · · i denotes an average over length (time) scales ` (τ) with ∆L ` LB (∆T τ TB ).

1. GWs in linerarized gravity 12/21 Energy density contained in GWs in transverse-traceless gauge:

GW 1 2 ˙ ˙ ij ρGW = T = m hij h (24) 00 4 Pl

Equation of motion on a curved background without any GW source and in Lorentz gauge:

αβ ¯ λ σ ¯ g¯ ∇α∇β hµν − 2R µν hλσ = 0 (25)

Curved background

Separation of scales: Consider background metric g¯µν that varies on length (time) scales LB (TB ). GWs then deﬁned in terms of perturbations δgµν on scales ∆L LB (∆T TB ):

gµν (x) =g ¯µν (x) + δgµν (x) , |δgµν | |g¯µν | (22)

¯ GW We now allow for Tµν 6= 0. This includes the contribution from GWs itself, Tµν , which can be calculated in second-order perturbation theory over a general curved background:2

2 GW 1 2 αβ h T = m ∇µh ∇ν h ∼ (23) µν 4 Pl αβ ∆L2

2 h· · · i denotes an average over length (time) scales ` (τ) with ∆L ` LB (∆T τ TB ).

1. GWs in linerarized gravity 12/21 Equation of motion on a curved background without any GW source and in Lorentz gauge:

αβ ¯ λ σ ¯ g¯ ∇α∇β hµν − 2R µν hλσ = 0 (25)

Curved background

Separation of scales: Consider background metric g¯µν that varies on length (time) scales LB (TB ). GWs then deﬁned in terms of perturbations δgµν on scales ∆L LB (∆T TB ):

gµν (x) =g ¯µν (x) + δgµν (x) , |δgµν | |g¯µν | (22)

¯ GW We now allow for Tµν 6= 0. This includes the contribution from GWs itself, Tµν , which can be calculated in second-order perturbation theory over a general curved background:2

2 GW 1 2 αβ h T = m ∇µh ∇ν h ∼ (23) µν 4 Pl αβ ∆L2

Energy density contained in GWs in transverse-traceless gauge:

GW 1 2 ˙ ˙ ij ρGW = T = m hij h (24) 00 4 Pl

2 h· · · i denotes an average over length (time) scales ` (τ) with ∆L ` LB (∆T τ TB ).

1. GWs in linerarized gravity 12/21 Curved background

Separation of scales: Consider background metric g¯µν that varies on length (time) scales LB (TB ). GWs then deﬁned in terms of perturbations δgµν on scales ∆L LB (∆T TB ):

gµν (x) =g ¯µν (x) + δgµν (x) , |δgµν | |g¯µν | (22)

¯ GW We now allow for Tµν 6= 0. This includes the contribution from GWs itself, Tµν , which can be calculated in second-order perturbation theory over a general curved background:2

2 GW 1 2 αβ h T = m ∇µh ∇ν h ∼ (23) µν 4 Pl αβ ∆L2

Energy density contained in GWs in transverse-traceless gauge:

GW 1 2 ˙ ˙ ij ρGW = T = m hij h (24) 00 4 Pl

Equation of motion on a curved background without any GW source and in Lorentz gauge:

αβ ¯ λ σ ¯ g¯ ∇α∇β hµν − 2R µν hλσ = 0 (25)

2 h· · · i denotes an average over length (time) scales ` (τ) with ∆L ` LB (∆T τ TB ).

1. GWs in linerarized gravity 12/21 Repeat SVT decomposition on an expanding background, including T¯µν source of g¯µν . Deﬁnition of GWs then straightforward generalization of the ﬂat-space result. For k = 0:

2 2 2 i j ds = −dt + a (t)(δij + hij ) dx dx , ∂ihij = hii = 0 (27)

◦ The two polarization states encoded in hij are the only propagating DOFs.

◦h hij i = 0, tensor perturbations are gauge-invariant in ﬁrst-order perturbation theory. ◦ No coupling between scalar, vector, tensor modes at ﬁrst order in perturbation theory.

Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) background

Cosmological background solutions:

dr2 ds2 = −dt2 + a2(t) + r2dΩ2 (26) 1 − kr2

Homogeneous, isotropic (cosmological principle!) and expanding solutions of the ﬁeld equations

2. Propagation of GWs in the expanding Universe 13/21 Deﬁnition of GWs then straightforward generalization of the ﬂat-space result. For k = 0:

2 2 2 i j ds = −dt + a (t)(δij + hij ) dx dx , ∂ihij = hii = 0 (27)

◦ The two polarization states encoded in hij are the only propagating DOFs.

◦h hij i = 0, tensor perturbations are gauge-invariant in ﬁrst-order perturbation theory. ◦ No coupling between scalar, vector, tensor modes at ﬁrst order in perturbation theory.

Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) background

Cosmological background solutions:

dr2 ds2 = −dt2 + a2(t) + r2dΩ2 (26) 1 − kr2

Homogeneous, isotropic (cosmological principle!) and expanding solutions of the ﬁeld equations

Repeat SVT decomposition on an expanding background, including T¯µν source of g¯µν .

2. Propagation of GWs in the expanding Universe 13/21 ◦ The two polarization states encoded in hij are the only propagating DOFs.

◦h hij i = 0, tensor perturbations are gauge-invariant in ﬁrst-order perturbation theory. ◦ No coupling between scalar, vector, tensor modes at ﬁrst order in perturbation theory.

Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) background

Cosmological background solutions:

dr2 ds2 = −dt2 + a2(t) + r2dΩ2 (26) 1 − kr2

Homogeneous, isotropic (cosmological principle!) and expanding solutions of the ﬁeld equations

Repeat SVT decomposition on an expanding background, including T¯µν source of g¯µν . Deﬁnition of GWs then straightforward generalization of the ﬂat-space result. For k = 0:

2 2 2 i j ds = −dt + a (t)(δij + hij ) dx dx , ∂ihij = hii = 0 (27)

2. Propagation of GWs in the expanding Universe 13/21 ◦h hij i = 0, tensor perturbations are gauge-invariant in ﬁrst-order perturbation theory. ◦ No coupling between scalar, vector, tensor modes at ﬁrst order in perturbation theory.

Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) background

Cosmological background solutions:

dr2 ds2 = −dt2 + a2(t) + r2dΩ2 (26) 1 − kr2

Homogeneous, isotropic (cosmological principle!) and expanding solutions of the ﬁeld equations

Repeat SVT decomposition on an expanding background, including T¯µν source of g¯µν . Deﬁnition of GWs then straightforward generalization of the ﬂat-space result. For k = 0:

2 2 2 i j ds = −dt + a (t)(δij + hij ) dx dx , ∂ihij = hii = 0 (27)

◦ The two polarization states encoded in hij are the only propagating DOFs.

2. Propagation of GWs in the expanding Universe 13/21 ◦ No coupling between scalar, vector, tensor modes at ﬁrst order in perturbation theory.

Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) background

Cosmological background solutions:

dr2 ds2 = −dt2 + a2(t) + r2dΩ2 (26) 1 − kr2

Homogeneous, isotropic (cosmological principle!) and expanding solutions of the ﬁeld equations

Repeat SVT decomposition on an expanding background, including T¯µν source of g¯µν . Deﬁnition of GWs then straightforward generalization of the ﬂat-space result. For k = 0:

2 2 2 i j ds = −dt + a (t)(δij + hij ) dx dx , ∂ihij = hii = 0 (27)

◦ The two polarization states encoded in hij are the only propagating DOFs.

◦h hij i = 0, tensor perturbations are gauge-invariant in ﬁrst-order perturbation theory.

2. Propagation of GWs in the expanding Universe 13/21 Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) background

Cosmological background solutions:

dr2 ds2 = −dt2 + a2(t) + r2dΩ2 (26) 1 − kr2

Homogeneous, isotropic (cosmological principle!) and expanding solutions of the ﬁeld equations

Repeat SVT decomposition on an expanding background, including T¯µν source of g¯µν . Deﬁnition of GWs then straightforward generalization of the ﬂat-space result. For k = 0:

2 2 2 i j ds = −dt + a (t)(δij + hij ) dx dx , ∂ihij = hii = 0 (27)

◦ The two polarization states encoded in hij are the only propagating DOFs.

◦h hij i = 0, tensor perturbations are gauge-invariant in ﬁrst-order perturbation theory. ◦ No coupling between scalar, vector, tensor modes at ﬁrst order in perturbation theory.

2. Propagation of GWs in the expanding Universe 13/21 ◦ Coordinates: Cosmic time t and comoving spatial coordinates x = (xi) ◦ Hubble friction term proportional to Hubble rate H =a/a ˙ −2 ◦ Anisotropic stress tensor Πij = a Tij − p gij of the GW source, with pressure p TT Πij is the transverse-traceless part of Πij . Use projection operator in Fourier space:

Z 3 d k −ikx Πij (t, x) = Πij (t, k) e (29) (2π)3

TT 1 Πij (t, k) = Oijlm(n)Πlm(t, k) = [Pil(n)Pjm(n) − /2 Pij (n)Plm(n)] Πlm(t, k) (30)

Pij (n) = δij − ni nj , n = k/kkk (31)

TT TT TT TT such that kiΠij = Πii = 0 in Fourier space and ∂iΠij = Πii = 0 in position space.

Equation of motion in an FLRW background

EOM for the transverse-traceless (TT) piece of the metric perturbation( ∂ihij = hii = 0):

1 2 h¨ (t, x) + 3 H h˙ (t, x) − ∆ h (t, x) = ΠTT(t, x) (28) ij ij 2 ij 2 ij a mPl

2. Propagation of GWs in the expanding Universe 14/21 ◦ Hubble friction term proportional to Hubble rate H =a/a ˙ −2 ◦ Anisotropic stress tensor Πij = a Tij − p gij of the GW source, with pressure p TT Πij is the transverse-traceless part of Πij . Use projection operator in Fourier space:

Z 3 d k −ikx Πij (t, x) = Πij (t, k) e (29) (2π)3

TT 1 Πij (t, k) = Oijlm(n)Πlm(t, k) = [Pil(n)Pjm(n) − /2 Pij (n)Plm(n)] Πlm(t, k) (30)

Pij (n) = δij − ni nj , n = k/kkk (31)

TT TT TT TT such that kiΠij = Πii = 0 in Fourier space and ∂iΠij = Πii = 0 in position space.

Equation of motion in an FLRW background

EOM for the transverse-traceless (TT) piece of the metric perturbation( ∂ihij = hii = 0):

1 2 h¨ (t, x) + 3 H h˙ (t, x) − ∆ h (t, x) = ΠTT(t, x) (28) ij ij 2 ij 2 ij a mPl

◦ Coordinates: Cosmic time t and comoving spatial coordinates x = (xi)

2. Propagation of GWs in the expanding Universe 14/21 −2 ◦ Anisotropic stress tensor Πij = a Tij − p gij of the GW source, with pressure p TT Πij is the transverse-traceless part of Πij . Use projection operator in Fourier space:

Z 3 d k −ikx Πij (t, x) = Πij (t, k) e (29) (2π)3

TT 1 Πij (t, k) = Oijlm(n)Πlm(t, k) = [Pil(n)Pjm(n) − /2 Pij (n)Plm(n)] Πlm(t, k) (30)

Pij (n) = δij − ni nj , n = k/kkk (31)

TT TT TT TT such that kiΠij = Πii = 0 in Fourier space and ∂iΠij = Πii = 0 in position space.

Equation of motion in an FLRW background

EOM for the transverse-traceless (TT) piece of the metric perturbation( ∂ihij = hii = 0):

1 2 h¨ (t, x) + 3 H h˙ (t, x) − ∆ h (t, x) = ΠTT(t, x) (28) ij ij 2 ij 2 ij a mPl

◦ Coordinates: Cosmic time t and comoving spatial coordinates x = (xi) ◦ Hubble friction term proportional to Hubble rate H =a/a ˙

2. Propagation of GWs in the expanding Universe 14/21 TT Πij is the transverse-traceless part of Πij . Use projection operator in Fourier space:

Z 3 d k −ikx Πij (t, x) = Πij (t, k) e (29) (2π)3

TT 1 Πij (t, k) = Oijlm(n)Πlm(t, k) = [Pil(n)Pjm(n) − /2 Pij (n)Plm(n)] Πlm(t, k) (30)

Pij (n) = δij − ni nj , n = k/kkk (31)

TT TT TT TT such that kiΠij = Πii = 0 in Fourier space and ∂iΠij = Πii = 0 in position space.

Equation of motion in an FLRW background

EOM for the transverse-traceless (TT) piece of the metric perturbation( ∂ihij = hii = 0):

1 2 h¨ (t, x) + 3 H h˙ (t, x) − ∆ h (t, x) = ΠTT(t, x) (28) ij ij 2 ij 2 ij a mPl

◦ Coordinates: Cosmic time t and comoving spatial coordinates x = (xi) ◦ Hubble friction term proportional to Hubble rate H =a/a ˙ −2 ◦ Anisotropic stress tensor Πij = a Tij − p gij of the GW source, with pressure p

2. Propagation of GWs in the expanding Universe 14/21 Use projection operator in Fourier space:

Z 3 d k −ikx Πij (t, x) = Πij (t, k) e (29) (2π)3

TT 1 Πij (t, k) = Oijlm(n)Πlm(t, k) = [Pil(n)Pjm(n) − /2 Pij (n)Plm(n)] Πlm(t, k) (30)

Pij (n) = δij − ni nj , n = k/kkk (31)

TT TT TT TT such that kiΠij = Πii = 0 in Fourier space and ∂iΠij = Πii = 0 in position space.

Equation of motion in an FLRW background

EOM for the transverse-traceless (TT) piece of the metric perturbation( ∂ihij = hii = 0):

1 2 h¨ (t, x) + 3 H h˙ (t, x) − ∆ h (t, x) = ΠTT(t, x) (28) ij ij 2 ij 2 ij a mPl

2. Propagation of GWs in the expanding Universe 14/21 Equation of motion in an FLRW background

EOM for the transverse-traceless (TT) piece of the metric perturbation( ∂ihij = hii = 0):

1 2 h¨ (t, x) + 3 H h˙ (t, x) − ∆ h (t, x) = ΠTT(t, x) (28) ij ij 2 ij 2 ij a mPl

Z 3 d k −ikx Πij (t, x) = Πij (t, k) e (29) (2π)3

TT 1 Πij (t, k) = Oijlm(n)Πlm(t, k) = [Pil(n)Pjm(n) − /2 Pij (n)Plm(n)] Πlm(t, k) (30)

Pij (n) = δij − ni nj , n = k/kkk (31)

TT TT TT TT such that kiΠij = Πii = 0 in Fourier space and ∂iΠij = Πii = 0 in position space.

2. Propagation of GWs in the expanding Universe 14/21 p where the polarization tensors eij (n) are constructed such that

p ∗ p p p p p eij (n) = eij (−n) = eij (n) = eji(n) , kieij (n) = eii(n) = 0 (33)

Orthonormal and completeness relations:

X p q X p p eij (n) eij (n) = 2 δpq , eij (n) elm(n) = Pil Pjm + Pim Pjl − Pij Plm (34) ij p

Explicit expressions in terms of {u, v}, which span the 2D vector space perpendicular to n:

+ × eij (n) = uiuj − vivj , eij (n) = uivj + viuj ,Pij = uiuj + vivj (35)

Decomposition into Fourier modes

Decompose TT piece into Fourier modes with deﬁnite momentum k and polarization p:

Z 3 X d k p −ikx hij (t, x) = hp(t, k) e (n) e (32) (2π)3 ij p=+,×

2. Propagation of GWs in the expanding Universe 15/21 Orthonormal and completeness relations:

X p q X p p eij (n) eij (n) = 2 δpq , eij (n) elm(n) = Pil Pjm + Pim Pjl − Pij Plm (34) ij p

Explicit expressions in terms of {u, v}, which span the 2D vector space perpendicular to n:

+ × eij (n) = uiuj − vivj , eij (n) = uivj + viuj ,Pij = uiuj + vivj (35)

Decomposition into Fourier modes

Decompose TT piece into Fourier modes with deﬁnite momentum k and polarization p:

Z 3 X d k p −ikx hij (t, x) = hp(t, k) e (n) e (32) (2π)3 ij p=+,×

p where the polarization tensors eij (n) are constructed such that

p ∗ p p p p p eij (n) = eij (−n) = eij (n) = eji(n) , kieij (n) = eii(n) = 0 (33)

2. Propagation of GWs in the expanding Universe 15/21 Explicit expressions in terms of {u, v}, which span the 2D vector space perpendicular to n:

+ × eij (n) = uiuj − vivj , eij (n) = uivj + viuj ,Pij = uiuj + vivj (35)

Decomposition into Fourier modes

Decompose TT piece into Fourier modes with deﬁnite momentum k and polarization p:

Z 3 X d k p −ikx hij (t, x) = hp(t, k) e (n) e (32) (2π)3 ij p=+,×

p where the polarization tensors eij (n) are constructed such that

p ∗ p p p p p eij (n) = eij (−n) = eij (n) = eji(n) , kieij (n) = eii(n) = 0 (33)

Orthonormal and completeness relations:

X p q X p p eij (n) eij (n) = 2 δpq , eij (n) elm(n) = Pil Pjm + Pim Pjl − Pij Plm (34) ij p

2. Propagation of GWs in the expanding Universe 15/21 Decomposition into Fourier modes

Decompose TT piece into Fourier modes with deﬁnite momentum k and polarization p:

Z 3 X d k p −ikx hij (t, x) = hp(t, k) e (n) e (32) (2π)3 ij p=+,×

p where the polarization tensors eij (n) are constructed such that

p ∗ p p p p p eij (n) = eij (−n) = eij (n) = eji(n) , kieij (n) = eii(n) = 0 (33)

Orthonormal and completeness relations:

X p q X p p eij (n) eij (n) = 2 δpq , eij (n) elm(n) = Pil Pjm + Pim Pjl − Pij Plm (34) ij p

Explicit expressions in terms of {u, v}, which span the 2D vector space perpendicular to n:

+ × eij (n) = uiuj − vivj , eij (n) = uivj + viuj ,Pij = uiuj + vivj (35)

2. Propagation of GWs in the expanding Universe 15/21 EOMs for H (η, k) = a(η) P h (η, k) ep (n) in the presence of a source term: ij p p ij

00 3 00 2 a 2a TT Hij (η, k) + k − Hij (η, k) = 2 Πij (η, k) (37) a mPl

Outside the spacetime volume in which the source is active (or no source at all):

00 00 2 a H (η, k) + k − Hp(η, k) = 0 ,Hp(η, k) = a(η) hp(t, k) (38) p a

General solution in terms of spherical Bessel functions, assuming a(η) ∝ ηn,

η hp(η, k) = [Ap(k) jn−1(kη) + Bp(k) yn−1(kη)] (39) a(η)

Mode equations in Fourier space

Perturbed FLRW metric in terms of conformal time η and comoving spatial coordinates x:

2 2 2 i j 0 dX ds = a (η) −dη + (δij + hij ) dx dx , a(η)dη = dt , X = (36) dη

2. Propagation of GWs in the expanding Universe 16/21 Outside the spacetime volume in which the source is active (or no source at all):

00 00 2 a H (η, k) + k − Hp(η, k) = 0 ,Hp(η, k) = a(η) hp(t, k) (38) p a

General solution in terms of spherical Bessel functions, assuming a(η) ∝ ηn,

η hp(η, k) = [Ap(k) jn−1(kη) + Bp(k) yn−1(kη)] (39) a(η)

Mode equations in Fourier space

Perturbed FLRW metric in terms of conformal time η and comoving spatial coordinates x:

2 2 2 i j 0 dX ds = a (η) −dη + (δij + hij ) dx dx , a(η)dη = dt , X = (36) dη

EOMs for H (η, k) = a(η) P h (η, k) ep (n) in the presence of a source term: ij p p ij

00 3 00 2 a 2a TT Hij (η, k) + k − Hij (η, k) = 2 Πij (η, k) (37) a mPl

2. Propagation of GWs in the expanding Universe 16/21 General solution in terms of spherical Bessel functions, assuming a(η) ∝ ηn,

η hp(η, k) = [Ap(k) jn−1(kη) + Bp(k) yn−1(kη)] (39) a(η)

Mode equations in Fourier space

Perturbed FLRW metric in terms of conformal time η and comoving spatial coordinates x:

2 2 2 i j 0 dX ds = a (η) −dη + (δij + hij ) dx dx , a(η)dη = dt , X = (36) dη

EOMs for H (η, k) = a(η) P h (η, k) ep (n) in the presence of a source term: ij p p ij

00 3 00 2 a 2a TT Hij (η, k) + k − Hij (η, k) = 2 Πij (η, k) (37) a mPl

Outside the spacetime volume in which the source is active (or no source at all):

00 00 2 a H (η, k) + k − Hp(η, k) = 0 ,Hp(η, k) = a(η) hp(t, k) (38) p a

2. Propagation of GWs in the expanding Universe 16/21 Mode equations in Fourier space

Perturbed FLRW metric in terms of conformal time η and comoving spatial coordinates x:

2 2 2 i j 0 dX ds = a (η) −dη + (δij + hij ) dx dx , a(η)dη = dt , X = (36) dη

EOMs for H (η, k) = a(η) P h (η, k) ep (n) in the presence of a source term: ij p p ij

00 3 00 2 a 2a TT Hij (η, k) + k − Hij (η, k) = 2 Πij (η, k) (37) a mPl

Outside the spacetime volume in which the source is active (or no source at all):

00 00 2 a H (η, k) + k − Hp(η, k) = 0 ,Hp(η, k) = a(η) hp(t, k) (38) p a

General solution in terms of spherical Bessel functions, assuming a(η) ∝ ηn,

η hp(η, k) = [Ap(k) jn−1(kη) + Bp(k) yn−1(kη)] (39) a(η)

2. Propagation of GWs in the expanding Universe 16/21 Compare wavenumber k to Hubble rate H = a0/a, or 1/k to Hubble radius 1/H. Sub-Hubble modes, k H ↔ kη 1:

1 ikη −ikη hp(η, k) = Ap(k) e + Bp(k) e (41) a(η) GW amplitude decays like 1/a inside the Hubble radius. Solution in position space:

Z 3 1 X d k p i(kη−kx) hij (η, x) = Ap(k) e (n) e + c.c (42) a(η) (2π)3 ij p=+,×

Super-Hubble modes, k H ↔ kη 1: η Z dη0 hp(η, k) = Ap(k) + Bp(k) (43) a2(η0)

Sum of a constant and a (quickly) decaying contribution outside the Hubble radius.

Solutions far inside / outside the Hubble radius

For a(η) ∝ ηn, we have (n = 1 in RD, n = 2 in MD, n = −1 in dS):

00 00 0 00 2 a a n − 1 2 a n H (η, k) + k − Hp(η, k) = 0 , = H , H = = (40) p a a n a η

2. Propagation of GWs in the expanding Universe 17/21 Sub-Hubble modes, k H ↔ kη 1:

1 ikη −ikη hp(η, k) = Ap(k) e + Bp(k) e (41) a(η) GW amplitude decays like 1/a inside the Hubble radius. Solution in position space:

Z 3 1 X d k p i(kη−kx) hij (η, x) = Ap(k) e (n) e + c.c (42) a(η) (2π)3 ij p=+,×

Super-Hubble modes, k H ↔ kη 1: η Z dη0 hp(η, k) = Ap(k) + Bp(k) (43) a2(η0)

Sum of a constant and a (quickly) decaying contribution outside the Hubble radius.

Solutions far inside / outside the Hubble radius

For a(η) ∝ ηn, we have (n = 1 in RD, n = 2 in MD, n = −1 in dS):

00 00 0 00 2 a a n − 1 2 a n H (η, k) + k − Hp(η, k) = 0 , = H , H = = (40) p a a n a η

Compare wavenumber k to Hubble rate H = a0/a, or 1/k to Hubble radius 1/H.

2. Propagation of GWs in the expanding Universe 17/21 GW amplitude decays like 1/a inside the Hubble radius. Solution in position space:

Z 3 1 X d k p i(kη−kx) hij (η, x) = Ap(k) e (n) e + c.c (42) a(η) (2π)3 ij p=+,×

Super-Hubble modes, k H ↔ kη 1: η Z dη0 hp(η, k) = Ap(k) + Bp(k) (43) a2(η0)

Sum of a constant and a (quickly) decaying contribution outside the Hubble radius.

Solutions far inside / outside the Hubble radius

For a(η) ∝ ηn, we have (n = 1 in RD, n = 2 in MD, n = −1 in dS):

00 00 0 00 2 a a n − 1 2 a n H (η, k) + k − Hp(η, k) = 0 , = H , H = = (40) p a a n a η

Compare wavenumber k to Hubble rate H = a0/a, or 1/k to Hubble radius 1/H. Sub-Hubble modes, k H ↔ kη 1:

1 ikη −ikη hp(η, k) = Ap(k) e + Bp(k) e (41) a(η)

2. Propagation of GWs in the expanding Universe 17/21 Super-Hubble modes, k H ↔ kη 1: η Z dη0 hp(η, k) = Ap(k) + Bp(k) (43) a2(η0)

Sum of a constant and a (quickly) decaying contribution outside the Hubble radius.

Solutions far inside / outside the Hubble radius

For a(η) ∝ ηn, we have (n = 1 in RD, n = 2 in MD, n = −1 in dS):

00 00 0 00 2 a a n − 1 2 a n H (η, k) + k − Hp(η, k) = 0 , = H , H = = (40) p a a n a η

Compare wavenumber k to Hubble rate H = a0/a, or 1/k to Hubble radius 1/H. Sub-Hubble modes, k H ↔ kη 1:

1 ikη −ikη hp(η, k) = Ap(k) e + Bp(k) e (41) a(η) GW amplitude decays like 1/a inside the Hubble radius. Solution in position space:

Z 3 1 X d k p i(kη−kx) hij (η, x) = Ap(k) e (n) e + c.c (42) a(η) (2π)3 ij p=+,×

2. Propagation of GWs in the expanding Universe 17/21 Sum of a constant and a (quickly) decaying contribution outside the Hubble radius.

Solutions far inside / outside the Hubble radius

For a(η) ∝ ηn, we have (n = 1 in RD, n = 2 in MD, n = −1 in dS):

00 00 0 00 2 a a n − 1 2 a n H (η, k) + k − Hp(η, k) = 0 , = H , H = = (40) p a a n a η

Compare wavenumber k to Hubble rate H = a0/a, or 1/k to Hubble radius 1/H. Sub-Hubble modes, k H ↔ kη 1:

1 ikη −ikη hp(η, k) = Ap(k) e + Bp(k) e (41) a(η) GW amplitude decays like 1/a inside the Hubble radius. Solution in position space:

Z 3 1 X d k p i(kη−kx) hij (η, x) = Ap(k) e (n) e + c.c (42) a(η) (2π)3 ij p=+,×

Super-Hubble modes, k H ↔ kη 1: η Z dη0 hp(η, k) = Ap(k) + Bp(k) (43) a2(η0)

2. Propagation of GWs in the expanding Universe 17/21 Solutions far inside / outside the Hubble radius

For a(η) ∝ ηn, we have (n = 1 in RD, n = 2 in MD, n = −1 in dS):

00 00 0 00 2 a a n − 1 2 a n H (η, k) + k − Hp(η, k) = 0 , = H , H = = (40) p a a n a η

Compare wavenumber k to Hubble rate H = a0/a, or 1/k to Hubble radius 1/H. Sub-Hubble modes, k H ↔ kη 1:

1 ikη −ikη hp(η, k) = Ap(k) e + Bp(k) e (41) a(η) GW amplitude decays like 1/a inside the Hubble radius. Solution in position space:

Z 3 1 X d k p i(kη−kx) hij (η, x) = Ap(k) e (n) e + c.c (42) a(η) (2π)3 ij p=+,×

Super-Hubble modes, k H ↔ kη 1: η Z dη0 hp(η, k) = Ap(k) + Bp(k) (43) a2(η0)

Sum of a constant and a (quickly) decaying contribution outside the Hubble radius.

2. Propagation of GWs in the expanding Universe 17/21 ◦ Causality: GW signal only correlated within causally connected patches. ◦ Homogeneity, isotropy: Same processes everywhere, across all causal patches.

→ The GW signal we receive today is composed of N ≫ 1 individual contributions. 3 → Stochastic GW signal; hij = random variable; ensemble = spatial / temporal average. Typical properties of cosmological GWs: ◦ Statistical homogeneity and isotropy, inherited from the FLRW background geometry ◦ In the absence of parity violation, no preferred polarization:

hh+(η, k)h×(η, k)i = 0 ⇔ hhL(η, k)hL(η, k)i = hhR(η, k)hR(η, k)i (44)

with the left- and right-handed tensor modes hL and hR in the helicity basis: 1 eL,R = e+ ∓ i e× (45) ij 2 ij ij

◦ Mostly Gaussian statistics, in consequence of N ≫ 1 (central limit theorem).

Generation of GWs in the early Universe

Consider GW source in the early Universe and assume:

3Primordial GWs from inﬂation are sourced by quantum ﬂuctuations and hence intrinsically stochastic.

3. Stochastic backgrounds of cosmological GWs 18/21 ◦ Homogeneity, isotropy: Same processes everywhere, across all causal patches.

hh+(η, k)h×(η, k)i = 0 ⇔ hhL(η, k)hL(η, k)i = hhR(η, k)hR(η, k)i (44)

with the left- and right-handed tensor modes hL and hR in the helicity basis: 1 eL,R = e+ ∓ i e× (45) ij 2 ij ij

◦ Mostly Gaussian statistics, in consequence of N ≫ 1 (central limit theorem).

Generation of GWs in the early Universe

Consider GW source in the early Universe and assume: ◦ Causality: GW signal only correlated within causally connected patches.

3Primordial GWs from inﬂation are sourced by quantum ﬂuctuations and hence intrinsically stochastic.

3. Stochastic backgrounds of cosmological GWs 18/21 → The GW signal we receive today is composed of N ≫ 1 individual contributions. 3 → Stochastic GW signal; hij = random variable; ensemble = spatial / temporal average. Typical properties of cosmological GWs: ◦ Statistical homogeneity and isotropy, inherited from the FLRW background geometry ◦ In the absence of parity violation, no preferred polarization:

hh+(η, k)h×(η, k)i = 0 ⇔ hhL(η, k)hL(η, k)i = hhR(η, k)hR(η, k)i (44)

with the left- and right-handed tensor modes hL and hR in the helicity basis: 1 eL,R = e+ ∓ i e× (45) ij 2 ij ij

◦ Mostly Gaussian statistics, in consequence of N ≫ 1 (central limit theorem).

Generation of GWs in the early Universe

Consider GW source in the early Universe and assume: ◦ Causality: GW signal only correlated within causally connected patches. ◦ Homogeneity, isotropy: Same processes everywhere, across all causal patches.

3Primordial GWs from inﬂation are sourced by quantum ﬂuctuations and hence intrinsically stochastic.

3. Stochastic backgrounds of cosmological GWs 18/21 3 → Stochastic GW signal; hij = random variable; ensemble = spatial / temporal average. Typical properties of cosmological GWs: ◦ Statistical homogeneity and isotropy, inherited from the FLRW background geometry ◦ In the absence of parity violation, no preferred polarization:

hh+(η, k)h×(η, k)i = 0 ⇔ hhL(η, k)hL(η, k)i = hhR(η, k)hR(η, k)i (44)

with the left- and right-handed tensor modes hL and hR in the helicity basis: 1 eL,R = e+ ∓ i e× (45) ij 2 ij ij

◦ Mostly Gaussian statistics, in consequence of N ≫ 1 (central limit theorem).

Generation of GWs in the early Universe

→ The GW signal we receive today is composed of N ≫ 1 individual contributions.

3Primordial GWs from inﬂation are sourced by quantum ﬂuctuations and hence intrinsically stochastic.

3. Stochastic backgrounds of cosmological GWs 18/21 Typical properties of cosmological GWs: ◦ Statistical homogeneity and isotropy, inherited from the FLRW background geometry ◦ In the absence of parity violation, no preferred polarization:

hh+(η, k)h×(η, k)i = 0 ⇔ hhL(η, k)hL(η, k)i = hhR(η, k)hR(η, k)i (44)

with the left- and right-handed tensor modes hL and hR in the helicity basis: 1 eL,R = e+ ∓ i e× (45) ij 2 ij ij

◦ Mostly Gaussian statistics, in consequence of N ≫ 1 (central limit theorem).

Generation of GWs in the early Universe

→ The GW signal we receive today is composed of N ≫ 1 individual contributions. 3 → Stochastic GW signal; hij = random variable; ensemble = spatial / temporal average.

3Primordial GWs from inﬂation are sourced by quantum ﬂuctuations and hence intrinsically stochastic.

3. Stochastic backgrounds of cosmological GWs 18/21 ◦ Statistical homogeneity and isotropy, inherited from the FLRW background geometry ◦ In the absence of parity violation, no preferred polarization:

hh+(η, k)h×(η, k)i = 0 ⇔ hhL(η, k)hL(η, k)i = hhR(η, k)hR(η, k)i (44)

with the left- and right-handed tensor modes hL and hR in the helicity basis: 1 eL,R = e+ ∓ i e× (45) ij 2 ij ij

◦ Mostly Gaussian statistics, in consequence of N ≫ 1 (central limit theorem).

Generation of GWs in the early Universe

3Primordial GWs from inﬂation are sourced by quantum ﬂuctuations and hence intrinsically stochastic.

3. Stochastic backgrounds of cosmological GWs 18/21 ◦ In the absence of parity violation, no preferred polarization:

hh+(η, k)h×(η, k)i = 0 ⇔ hhL(η, k)hL(η, k)i = hhR(η, k)hR(η, k)i (44)

with the left- and right-handed tensor modes hL and hR in the helicity basis: 1 eL,R = e+ ∓ i e× (45) ij 2 ij ij

◦ Mostly Gaussian statistics, in consequence of N ≫ 1 (central limit theorem).

Generation of GWs in the early Universe

3Primordial GWs from inﬂation are sourced by quantum ﬂuctuations and hence intrinsically stochastic.

3. Stochastic backgrounds of cosmological GWs 18/21 ◦ Mostly Gaussian statistics, in consequence of N ≫ 1 (central limit theorem).

Generation of GWs in the early Universe

hh+(η, k)h×(η, k)i = 0 ⇔ hhL(η, k)hL(η, k)i = hhR(η, k)hR(η, k)i (44)

with the left- and right-handed tensor modes hL and hR in the helicity basis: 1 eL,R = e+ ∓ i e× (45) ij 2 ij ij

3Primordial GWs from inﬂation are sourced by quantum ﬂuctuations and hence intrinsically stochastic.

3. Stochastic backgrounds of cosmological GWs 18/21 Generation of GWs in the early Universe

hh+(η, k)h×(η, k)i = 0 ⇔ hhL(η, k)hL(η, k)i = hhR(η, k)hR(η, k)i (44)

with the left- and right-handed tensor modes hL and hR in the helicity basis: 1 eL,R = e+ ∓ i e× (45) ij 2 ij ij

◦ Mostly Gaussian statistics, in consequence of N ≫ 1 (central limit theorem).

3Primordial GWs from inﬂation are sourced by quantum ﬂuctuations and hence intrinsically stochastic.

3. Stochastic backgrounds of cosmological GWs 18/21 In position space: Z ∞ dk 2 1 hhij (η, x) hij (η, x)i = 2 h (η, k) , hc(η, k) ∝ (47) k c a(η) 0

Characteristic strain hc describes typical GW amplitude per logarithmic k interval.

Related to strain power spectrum Sh:

h2(f) S (f) = c (48) h f

hc(f) = hc(η0, f) and f = 1/(2π) k/a0. 2 Note: Sh = hc /(2f) in some references. [Christopher Berry on cplberry.com] [gwplotter.com]

Characteristic strain

Statistical properties described by power spectrum of the mode functions in Fourier space:

5 ∗ 8π (3) 2 hp(η, k) h (η, `) = δ (k − `) δpq h (η, k) (46) q k3 c

for a homogeneous, isotropic, unpolarized, an Gaussian background.

3. Stochastic backgrounds of cosmological GWs 19/21 Characteristic strain hc describes typical GW amplitude per logarithmic k interval.

Related to strain power spectrum Sh:

h2(f) S (f) = c (48) h f

hc(f) = hc(η0, f) and f = 1/(2π) k/a0. 2 Note: Sh = hc /(2f) in some references. [Christopher Berry on cplberry.com] [gwplotter.com]

Characteristic strain

Statistical properties described by power spectrum of the mode functions in Fourier space:

5 ∗ 8π (3) 2 hp(η, k) h (η, `) = δ (k − `) δpq h (η, k) (46) q k3 c

for a homogeneous, isotropic, unpolarized, an Gaussian background. In position space: Z ∞ dk 2 1 hhij (η, x) hij (η, x)i = 2 h (η, k) , hc(η, k) ∝ (47) k c a(η) 0

3. Stochastic backgrounds of cosmological GWs 19/21 Related to strain power spectrum Sh:

h2(f) S (f) = c (48) h f

hc(f) = hc(η0, f) and f = 1/(2π) k/a0. 2 Note: Sh = hc /(2f) in some references.

Characteristic strain

Statistical properties described by power spectrum of the mode functions in Fourier space:

5 ∗ 8π (3) 2 hp(η, k) h (η, `) = δ (k − `) δpq h (η, k) (46) q k3 c

for a homogeneous, isotropic, unpolarized, an Gaussian background. In position space: Z ∞ dk 2 1 hhij (η, x) hij (η, x)i = 2 h (η, k) , hc(η, k) ∝ (47) k c a(η) 0

Characteristic strain hc describes typical GW amplitude per logarithmic k interval.

[Christopher Berry on cplberry.com] [gwplotter.com]

3. Stochastic backgrounds of cosmological GWs 19/21 Characteristic strain

Statistical properties described by power spectrum of the mode functions in Fourier space:

5 ∗ 8π (3) 2 hp(η, k) h (η, `) = δ (k − `) δpq h (η, k) (46) q k3 c

for a homogeneous, isotropic, unpolarized, an Gaussian background. In position space: Z ∞ dk 2 1 hhij (η, x) hij (η, x)i = 2 h (η, k) , hc(η, k) ∝ (47) k c a(η) 0

Characteristic strain hc describes typical GW amplitude per logarithmic k interval.

Related to strain power spectrum Sh:

h2(f) S (f) = c (48) h f

hc(f) = hc(η0, f) and f = 1/(2π) k/a0. 2 Note: Sh = hc /(2f) in some references. [Christopher Berry on cplberry.com] [gwplotter.com]

3. Stochastic backgrounds of cosmological GWs 19/21 For sub-Hubble modes, k H, the energy density of GWs thus redshifts like radiation:

dρ m2 1 GW = Pl k2h2(η, k) ∝ (50) d ln k 2 a2(η) c a4(η)

Evaluate dρGW/d ln k today, in units of the 2 2 critical energy density ρc = 3 mPl H0 :

1 dρGW ΩGW(f) = (51) ρc d ln k 2 2 2π 2 2 2π 3 = 2 f hc (f) = 2 f Sh(f) (52) 3 H0 3 H0

2 Multiply by h , where H0 = 100 h km/s/Mpc, [Christopher Berry on cplberry.com] [gwplotter.com] to remove dependence on Hubble parameter.

Energy density spectrum

GW energy density per logarithmic k interval dρGW/d ln k: Z ∞ 1 2 ˙ ˙ ij 1 2 0 0ij dk dρGW ρGW = m hij h = m h h = (49) 4 Pl 4 a2(η) Pl ij k d ln k 0

3. Stochastic backgrounds of cosmological GWs 20/21 Evaluate dρGW/d ln k today, in units of the 2 2 critical energy density ρc = 3 mPl H0 :

1 dρGW ΩGW(f) = (51) ρc d ln k 2 2 2π 2 2 2π 3 = 2 f hc (f) = 2 f Sh(f) (52) 3 H0 3 H0

2 Multiply by h , where H0 = 100 h km/s/Mpc, [Christopher Berry on cplberry.com] [gwplotter.com] to remove dependence on Hubble parameter.

Energy density spectrum

GW energy density per logarithmic k interval dρGW/d ln k: Z ∞ 1 2 ˙ ˙ ij 1 2 0 0ij dk dρGW ρGW = m hij h = m h h = (49) 4 Pl 4 a2(η) Pl ij k d ln k 0

For sub-Hubble modes, k H, the energy density of GWs thus redshifts like radiation:

dρ m2 1 GW = Pl k2h2(η, k) ∝ (50) d ln k 2 a2(η) c a4(η)

3. Stochastic backgrounds of cosmological GWs 20/21 2 Multiply by h , where H0 = 100 h km/s/Mpc, to remove dependence on Hubble parameter.

Energy density spectrum

GW energy density per logarithmic k interval dρGW/d ln k: Z ∞ 1 2 ˙ ˙ ij 1 2 0 0ij dk dρGW ρGW = m hij h = m h h = (49) 4 Pl 4 a2(η) Pl ij k d ln k 0

For sub-Hubble modes, k H, the energy density of GWs thus redshifts like radiation:

dρ m2 1 GW = Pl k2h2(η, k) ∝ (50) d ln k 2 a2(η) c a4(η)

Evaluate dρGW/d ln k today, in units of the 2 2 critical energy density ρc = 3 mPl H0 :

1 dρGW ΩGW(f) = (51) ρc d ln k 2 2 2π 2 2 2π 3 = 2 f hc (f) = 2 f Sh(f) (52) 3 H0 3 H0

[Christopher Berry on cplberry.com] [gwplotter.com]

3. Stochastic backgrounds of cosmological GWs 20/21 Energy density spectrum

GW energy density per logarithmic k interval dρGW/d ln k: Z ∞ 1 2 ˙ ˙ ij 1 2 0 0ij dk dρGW ρGW = m hij h = m h h = (49) 4 Pl 4 a2(η) Pl ij k d ln k 0

For sub-Hubble modes, k H, the energy density of GWs thus redshifts like radiation:

dρ m2 1 GW = Pl k2h2(η, k) ∝ (50) d ln k 2 a2(η) c a4(η)

Evaluate dρGW/d ln k today, in units of the 2 2 critical energy density ρc = 3 mPl H0 :

1 dρGW ΩGW(f) = (51) ρc d ln k 2 2 2π 2 2 2π 3 = 2 f hc (f) = 2 f Sh(f) (52) 3 H0 3 H0

2 Multiply by h , where H0 = 100 h km/s/Mpc, [Christopher Berry on cplberry.com] [gwplotter.com] to remove dependence on Hubble parameter.

3. Stochastic backgrounds of cosmological GWs 20/21 ◦ GWs open a new window onto the Universe, unique tool for fundamental physics.

◦ GWs are radiative and gauge-invariant tensor perturbations of the spacetime metric.

◦ The physical DOFs correspond to two transverse polarization states h+ and h×.

◦ Manifest after complete gauge ﬁxing (TT gauge in vacuum) or SVT decomposition.

◦ GWs also well-deﬁned on curved backgrounds, in particular, in FLRW cosmology.

◦ Source-free EOM in FLRW can be solved exactly / inside / outside Hubble radius.

◦ Cosmological GWs are typically homogeneous, isotropic, unpolarized, and Gaussian.

2 ◦ Signal today can be represented in terms of hc, Sh, or h ΩGW as function of f.

End of Lecture 1A. Thanks a lot for your attention!

Summary Lecture 1A

Take-home messages:

4. Summary 21/21 ◦ GWs are radiative and gauge-invariant tensor perturbations of the spacetime metric.

◦ The physical DOFs correspond to two transverse polarization states h+ and h×.

◦ Manifest after complete gauge ﬁxing (TT gauge in vacuum) or SVT decomposition.

◦ GWs also well-deﬁned on curved backgrounds, in particular, in FLRW cosmology.

◦ Source-free EOM in FLRW can be solved exactly / inside / outside Hubble radius.

◦ Cosmological GWs are typically homogeneous, isotropic, unpolarized, and Gaussian.

2 ◦ Signal today can be represented in terms of hc, Sh, or h ΩGW as function of f.

End of Lecture 1A. Thanks a lot for your attention!

Summary Lecture 1A

Take-home messages:

◦ GWs open a new window onto the Universe, unique tool for fundamental physics.

4. Summary 21/21 ◦ The physical DOFs correspond to two transverse polarization states h+ and h×.

◦ Manifest after complete gauge ﬁxing (TT gauge in vacuum) or SVT decomposition.

◦ GWs also well-deﬁned on curved backgrounds, in particular, in FLRW cosmology.

◦ Source-free EOM in FLRW can be solved exactly / inside / outside Hubble radius.

◦ Cosmological GWs are typically homogeneous, isotropic, unpolarized, and Gaussian.

2 ◦ Signal today can be represented in terms of hc, Sh, or h ΩGW as function of f.

End of Lecture 1A. Thanks a lot for your attention!

Summary Lecture 1A

Take-home messages:

◦ GWs open a new window onto the Universe, unique tool for fundamental physics.

◦ GWs are radiative and gauge-invariant tensor perturbations of the spacetime metric.

4. Summary 21/21 ◦ Manifest after complete gauge ﬁxing (TT gauge in vacuum) or SVT decomposition.

◦ GWs also well-deﬁned on curved backgrounds, in particular, in FLRW cosmology.

◦ Source-free EOM in FLRW can be solved exactly / inside / outside Hubble radius.

◦ Cosmological GWs are typically homogeneous, isotropic, unpolarized, and Gaussian.

2 ◦ Signal today can be represented in terms of hc, Sh, or h ΩGW as function of f.

End of Lecture 1A. Thanks a lot for your attention!

Summary Lecture 1A

Take-home messages:

◦ GWs open a new window onto the Universe, unique tool for fundamental physics.

◦ GWs are radiative and gauge-invariant tensor perturbations of the spacetime metric.

◦ The physical DOFs correspond to two transverse polarization states h+ and h×.

4. Summary 21/21 ◦ GWs also well-deﬁned on curved backgrounds, in particular, in FLRW cosmology.

◦ Source-free EOM in FLRW can be solved exactly / inside / outside Hubble radius.

◦ Cosmological GWs are typically homogeneous, isotropic, unpolarized, and Gaussian.

2 ◦ Signal today can be represented in terms of hc, Sh, or h ΩGW as function of f.

End of Lecture 1A. Thanks a lot for your attention!

Summary Lecture 1A

Take-home messages:

◦ GWs open a new window onto the Universe, unique tool for fundamental physics.

◦ GWs are radiative and gauge-invariant tensor perturbations of the spacetime metric.

◦ The physical DOFs correspond to two transverse polarization states h+ and h×.

◦ Manifest after complete gauge ﬁxing (TT gauge in vacuum) or SVT decomposition.

4. Summary 21/21 ◦ Source-free EOM in FLRW can be solved exactly / inside / outside Hubble radius.

◦ Cosmological GWs are typically homogeneous, isotropic, unpolarized, and Gaussian.

2 ◦ Signal today can be represented in terms of hc, Sh, or h ΩGW as function of f.

End of Lecture 1A. Thanks a lot for your attention!

Summary Lecture 1A

Take-home messages:

◦ GWs open a new window onto the Universe, unique tool for fundamental physics.

◦ GWs are radiative and gauge-invariant tensor perturbations of the spacetime metric.

◦ The physical DOFs correspond to two transverse polarization states h+ and h×.

◦ Manifest after complete gauge ﬁxing (TT gauge in vacuum) or SVT decomposition.

◦ GWs also well-deﬁned on curved backgrounds, in particular, in FLRW cosmology.

4. Summary 21/21 ◦ Cosmological GWs are typically homogeneous, isotropic, unpolarized, and Gaussian.

2 ◦ Signal today can be represented in terms of hc, Sh, or h ΩGW as function of f.

End of Lecture 1A. Thanks a lot for your attention!

Summary Lecture 1A

Take-home messages:

◦ GWs open a new window onto the Universe, unique tool for fundamental physics.

◦ GWs are radiative and gauge-invariant tensor perturbations of the spacetime metric.

◦ The physical DOFs correspond to two transverse polarization states h+ and h×.

◦ Manifest after complete gauge ﬁxing (TT gauge in vacuum) or SVT decomposition.

◦ GWs also well-deﬁned on curved backgrounds, in particular, in FLRW cosmology.

◦ Source-free EOM in FLRW can be solved exactly / inside / outside Hubble radius.

4. Summary 21/21 2 ◦ Signal today can be represented in terms of hc, Sh, or h ΩGW as function of f.

End of Lecture 1A. Thanks a lot for your attention!

Summary Lecture 1A

Take-home messages:

◦ GWs open a new window onto the Universe, unique tool for fundamental physics.

◦ GWs are radiative and gauge-invariant tensor perturbations of the spacetime metric.

◦ The physical DOFs correspond to two transverse polarization states h+ and h×.

◦ Manifest after complete gauge ﬁxing (TT gauge in vacuum) or SVT decomposition.

◦ GWs also well-deﬁned on curved backgrounds, in particular, in FLRW cosmology.

◦ Source-free EOM in FLRW can be solved exactly / inside / outside Hubble radius.

◦ Cosmological GWs are typically homogeneous, isotropic, unpolarized, and Gaussian.

4. Summary 21/21 End of Lecture 1A. Thanks a lot for your attention!