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Computing inflationary predictions in general scalar-tensor theories Héctor Ramírez IFIC - University of Valencia ICG - Portsmouth

Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

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Page 1: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

Computing inflationary predictions in general scalar-tensor theories

Héctor RamírezIFIC - University of Valencia

ICG - Portsmouth

Page 2: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

Slow-roll conditions:

NCMB =

Z �end

�CMB

d�

Mpl

p2✏

⇡ 40� 60Number of e-folds of inflation:

✏H = � H

H2

✏H ⌧ 1

|⌘H | ⌧ 1⌘H =

1

2

d ln✏HdN

� ✏H

V (�)

�end�in

Slow-roll inflation in one slide

Page 3: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

Slow-roll conditions:

NCMB =

Z �end

�CMB

d�

Mpl

p2✏

⇡ 40� 60Number of e-folds of inflation:

✏H = � H

H2

✏H ⌧ 1

|⌘H | ⌧ 1⌘H =

1

2

d ln✏HdN

� ✏H

V (�)

�end�in

Slow-roll inflation in one slide

S(2)� =

X

�=+,⇥

Zd4x

a3

4

✓�2� � k2

a2�2�

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S(2)⇣

=

Zd4xa3✏H

✓⇣2 � k2

a2⇣2◆

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Scalars

Tensors v00k +

✓k2 � z00

z

◆vk = 0

Mukhanov - Sasaki equation:

Page 4: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

Slow-roll conditions:

NCMB =

Z �end

�CMB

d�

Mpl

p2✏

⇡ 40� 60Number of e-folds of inflation:

✏H = � H

H2

✏H ⌧ 1

|⌘H | ⌧ 1⌘H =

1

2

d ln✏HdN

� ✏H

V (�)

�end�in

Slow-roll inflation in one slide

S(2)� =

X

�=+,⇥

Zd4x

a3

4

✓�2� � k2

a2�2�

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S(2)⇣

=

Zd4xa3✏H

✓⇣2 � k2

a2⇣2◆

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Scalars

Tensors v00k +

✓k2 � z00

z

◆vk = 0

Mukhanov - Sasaki equation:

ns � 1 ⌘d ln�2

d lnkr ⌘

�2�

�2⇣

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Inflationary parameters:

Page 5: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

1. Scalar - Tensor theories.

2. Generalized slow-roll (GSR) techniques.

3. G-inflation.

Outline:

Page 6: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

*Borrowed from Hayato Motohashi’s talks

Ostrogradski’s theorem:

“Higher-derivative theories contain extra degrees of freedom, and are usually plagued by negative energies and related instabilities.”

Page 7: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

S =5X

i=2

Zd4x

p�gLi

• G. Horndeski (1974) • A. Nicolis et al.; 0811.2197 • C. Deffayet et al.; 1103.3260

Applications to:

• Late-time Cosmology (dark energy).

• Early-universe Cosmology (inflation).

L2 = G2 ,

L3 = G3⇤� ,

L4 = G4R� 2G4,X [(⇤�)2 � �;µ⌫�;µ⌫ ] ,

L5 = G5Gµ⌫�;µ⌫ +

G5,X

3[(⇤�)3 � 3(⇤�)�;µ⌫�

;µ⌫ + 2�;µ⌫�;µ��;⌫

;�] .<latexit 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X ⌘ gµ⌫@µ�@⌫�<latexit sha1_base64="y9IzRzH0Q+dGaXa6gCHn6v1EPw4=">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</latexit><latexit sha1_base64="y9IzRzH0Q+dGaXa6gCHn6v1EPw4=">AAAIfHiclVXdbtMwFA6/68bfgEtuIqpJnRhT2038CCEhEGIXII2KwaQ4qxzHSSwcJ7OdsWKZ5+BpuIVX4GnAaTva2K2AVI2Oz/nOZ3/HJ3ZUUiJkt/vz3PkLFy9dXmmtrl25eu36jfWbt96LouIIH6CCFvwwggJTwvCBJJLiw5JjmEcUf4g+vqjjH04wF6Rg7+SoxGEOU0YSgqA0ruF6/xDg44qc+OmRAnkFWKVBCbkkkA7rcZmR2ZiNx8P1dne7O3581+hNjbY3ffaHN1d2QFygKsdMIgqFCHrdUoaqpkUU6zVQCVxC9BGmODAmgzkWoRqL0/6G8cR+UnDzZ9Ife+czFMyFGOWRQeZQZsKO1c5FsaCSyaNQEVZWEjM0mSipqC8Lv66UHxOOkaQjY0DEiVmrjzLIIZKmnmuNaYTMIR/xuCFFpRyacqFTvXBJCzU08zGbODfmly1KyIjItnJ8SlARqghGmC7GzAVNh9RiIEur8YInACvNqkhzhWwkyOfmZilhdjPDluyEFlBaLsjQKIt504uIxP+AC2gtot78p1Fi2gKWdes2E6Ncmx1h+BMq8hyyWIEU5kwrYEpYcVwXXKXDsw7X2sIKB3u0HKuDfmiD271234VCFzpcBo11sLMQC7KoOFVgS2+1d5w87qgcLFXJHZWDpSq5q3KwRCV3VQ7+qFy1ZDoiwStoYjW1Nkn9BRJTU5ndvyTN1Wh3AQH6D4YnYwa/STFo1g3UbwSpGjiT7WVLoHsOFB8baIRTwpQ5gsfnsQMZY7AxlyEi+IcEUpIy8zHaHHBKsSReShPfmv2scEySxNlgkJhjUAGJT6WKtSnizHYbxDC43dRkOOo3OY4WsswQViwvqVZvhpOosa3wa3aWSZn+YgVfHuugF6qXx9tfOoDjpG6JTYMxB7opan3kqT7KtfUyhw3HMxrQUYDiRHa05d803wdJM7lpB4JJQmD7w2lCqM0127MvVdc46G8/3u693W0/ez69b1veHe+u1/F63kPvmbfn7XsHHvK+et+8796PlV+tjda91v0J9Py5ac5tr/G0HvwG7kIjCQ==</latexit><latexit sha1_base64="y9IzRzH0Q+dGaXa6gCHn6v1EPw4=">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</latexit><latexit sha1_base64="y9IzRzH0Q+dGaXa6gCHn6v1EPw4=">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</latexit>

Horndeski theory:

Page 8: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

Horndeski theory:

S =5X

i=2

Zd4x

p�gLi

L2 = G2 ,

L3 = G3⇤� ,

L4 = G4R� 2G4,X [(⇤�)2 � �;µ⌫�;µ⌫ ] ,

L5 = G5Gµ⌫�;µ⌫ +

G5,X

3[(⇤�)3 � 3(⇤�)�;µ⌫�

;µ⌫ + 2�;µ⌫�;µ��;⌫

;�] .<latexit 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sha1_base64="LM8FLqVILnx7d7L+iRRkPPIbD+A=">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</latexit><latexit sha1_base64="LM8FLqVILnx7d7L+iRRkPPIbD+A=">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</latexit><latexit sha1_base64="LM8FLqVILnx7d7L+iRRkPPIbD+A=">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</latexit>

Canonical inflation:

G2 = �X

2� V (�) ,

G3 = 0 ,<latexit sha1_base64="55iG+JbmREHcqSZHUNuAIXx2CPs=">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</latexit><latexit sha1_base64="55iG+JbmREHcqSZHUNuAIXx2CPs=">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</latexit><latexit sha1_base64="55iG+JbmREHcqSZHUNuAIXx2CPs=">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</latexit><latexit sha1_base64="55iG+JbmREHcqSZHUNuAIXx2CPs=">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</latexit>

G4 =M2

pl

2,

G5 = 0 .<latexit sha1_base64="joPLPuIQus/XccKdqOcLyJObwoc=">AAAIdHiclVVdb9s2FFW7j7jeR9PtsSggzEiRDIZhOym6oQhQtCiahw3ojGUtICoGRVESUYpSSGqNSzC/ob+mr93f2C/Z665sr4lIG9tkWLi459xDnkuKTGrOlB6P/7xx85NPP/t8p3er/8WXX319e/fON7+pqpGEnpKKV/JVghXlTNBTzTSnr2pJcZlw+jJ5/bTFX/5OpWKV+FUvahqXOBcsYwRrSM13v0cJ7j+fH90/RpnExKCy5mdTa6b2cogQIA/uH48vR31E8Xx3MB6Nl0/oB5N1MAjWz4v5nZ1DlFakKanQhGOlosm41rHBUjPCqe2jRtEak9c4pxGEApdUxWZpyoZ7kEnDrJLwFzpcZq9XGFwqtSgTYJZYF8rF2uQmLGp09kNsmKgbTQVZDZQ1PNRV2HYoTJmkRPMFBJhIBnMNSYGhORr62O8Mo3SJ5UKmHSsml7guGLmwG6e00UO3nopVcu/6tFWNBVPFsKQXjFSxSXBC+WbONRB2RmsGi7xZTnhFcMqcjnRnKBaKve0ullGwmgV1bGe8wtpJYUEWRSq7WcI0/Q+8iLcm2sU/TjLYFrhut2y3MCktrIigb0hVllikBuW4FNYgaGEjadtwk89hVzdINNZah6s87tl2ro2msUseTAZTn4p96nwbNbXR4UYuKpLqwqChHQ4OvTrpuZxtdSk9l7OtLqXvcrbFpfRdzj66vOXY9Eyi5xiwVtpC0XSDxRw6c/QvRdd6dLRBgPwPhUdLhbArMev2DbVvgrmZeYOdFFuoJx6VngM1oTkThp43y3PYoyw5FMJtjAR/FMGc5QI+RlcDryW24LUGfHj1c+CUZZm3wOs7QtMLbVILTbyK/Q0CCv5u6iqcTbsaZxtVrhgOBneVNT/PVyjEDvyT+KeSC3vpgM/ObTSJzbPz0eU+kjRrt8QBcOBAh6a2R56ZktI6LzhsJL2SQfsGcZrpfevkD+D7YHmhD1wgWhVEbj5eF8QWrtmJe6n6wel09ONo8svR4PGT9X3bC+4G3wX7wSR4GDwOToIXwWlAgnfB++BD8MfOX717vUFvb0W9eWNd823QeXqjvwG0jxvh</latexit><latexit sha1_base64="joPLPuIQus/XccKdqOcLyJObwoc=">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</latexit><latexit sha1_base64="joPLPuIQus/XccKdqOcLyJObwoc=">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</latexit><latexit sha1_base64="joPLPuIQus/XccKdqOcLyJObwoc=">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</latexit>

• G. Horndeski (1974) • A. Nicolis et al.; 0811.2197 • C. Deffayet et al.; 1103.3260

Gi = Gi(�, X)<latexit sha1_base64="ykoS9Cc7NxKWWXiXxg5TCmnKbU4=">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</latexit><latexit sha1_base64="ykoS9Cc7NxKWWXiXxg5TCmnKbU4=">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</latexit><latexit sha1_base64="ykoS9Cc7NxKWWXiXxg5TCmnKbU4=">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</latexit><latexit sha1_base64="ykoS9Cc7NxKWWXiXxg5TCmnKbU4=">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</latexit>

X ⌘ gµ⌫@µ�@⌫�<latexit sha1_base64="y9IzRzH0Q+dGaXa6gCHn6v1EPw4=">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</latexit><latexit sha1_base64="y9IzRzH0Q+dGaXa6gCHn6v1EPw4=">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</latexit><latexit sha1_base64="y9IzRzH0Q+dGaXa6gCHn6v1EPw4=">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</latexit><latexit sha1_base64="y9IzRzH0Q+dGaXa6gCHn6v1EPw4=">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</latexit>

Page 9: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

S(2)⇣

=

Zd4x

a3bs✏Hc2s

✓⇣2 � c2

sk2

a2⇣2◆

Inflation

bs = bt = cs = ct = 1

In canonical inflation:

• T. Kobayashi et al.; 1105.5723 • H. Motohashi, W. Hu; 1704.01128

S(2)� =

X

�=+,⇥

Zd4x

a3bt4c2t

✓�2� � c2tk

2

a2�2�

<latexit sha1_base64="IryCGz6hklKfWSXOWL3h2hs+iXE=">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</latexit><latexit sha1_base64="IryCGz6hklKfWSXOWL3h2hs+iXE=">AAAI3niclVXPj9tEFHbKj6bLry0cuVhElRIIq8QbCRBaqWpVsQeQSmBppYxjjcdje7TjsXc8bhNG0ysnxIUD/xr/Ciee7dCsx4kAR7Ge3vveN/O9eZ4XFpyVajb7c3DnjTffevvu8N7JO+++9/4Hp/c//KnMK0noFcl5Lp+HuKScCXqlmOL0eSEpzkJOn4XXj+v4sxdUliwXP6ptQf0MJ4LFjGAFruD09x/WeuxNTIASnGX4ApVVFmjEgSHCF59NkWIZLQ1iQiFFN0pmOjLrxQbFEhON1+dhoIxekECtPYM4jdUYRbnSLZ1Ze8GO6/M2owFeAxZyIaFB7UFIsiRVk+B0NDubNY/bN+Y7Y+TsnqfB/bvnsCqpMioU4bgsV/NZoXyNpWKEU3OCqpIWmFzjhK7AFBhE+bopn3EfgCdy41zCXyi38d7O0Dgry20WAjLDKi3tWO08FFtVKv7S10wUlaKCtAvFFXdV7tZn4UZMUqL4FgxMJIO9uiTFUCYFJ3bSWaZUGZZbGXWk6ETiImVkYw5u6aCGbj4VrfPB7W2XBRasTKcZ3TCS+zrEIeWHMbeC0IO1GCySqtlwC7DSrIp0dyi2Jfu5e1i6hNNMqSU75jlWlgsLsk0j2fUSpuh/wK14LaI+/IswhrbARf1xdBPDzMCJCPqS5NCyImo6XBiNoISVpHXBdQJfTlYhURljLGzZw66PY83K823waD7y+lDchwbHoJFZnR/EojTMNxpNzXR03suTPZXLoyplT+XyqErZV7k8olL2VS5fq7xnyeyJRN80dwzgDSR5ByQmUJnFvyTdqtHiAAH5HwxfNwxul2LZrRuq3wRzvewtdpkegV72oPQGoCFNmND0pmpu/B6kwVAwjyFC/JoEc5YI+BhtDryjOBIvYEKg6f5nhSMWx70DbqdFM3Jg4EAR93a/QYCh301dhrXX5VgfZNkjrFhWcKO/C9oo2Fb4W/FPJhfmlRV8cmNWc18/uTl7NUaSxnVLTAADFzoUtb7ytEcyY73gspF0T4PGup2uxvJP9G5o2oFVm7Cy/f4uwTcwZuf2UO0bV97ZV2fz7xejh49283bofOx84oydufOF89C5dJ46Vw5x/hq4g8ng0yEe/jL8dfhbC70z2OV85HSe4R9/A6dYRwA=</latexit><latexit sha1_base64="IryCGz6hklKfWSXOWL3h2hs+iXE=">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</latexit><latexit sha1_base64="IryCGz6hklKfWSXOWL3h2hs+iXE=">AAAI3niclVXPj9tEFHbKj6bLry0cuVhElRIIq8QbCRBaqWpVsQeQSmBppYxjjcdje7TjsXc8bhNG0ysnxIUD/xr/Ciee7dCsx4kAR7Ge3vveN/O9eZ4XFpyVajb7c3DnjTffevvu8N7JO+++9/4Hp/c//KnMK0noFcl5Lp+HuKScCXqlmOL0eSEpzkJOn4XXj+v4sxdUliwXP6ptQf0MJ4LFjGAFruD09x/WeuxNTIASnGX4ApVVFmjEgSHCF59NkWIZLQ1iQiFFN0pmOjLrxQbFEhON1+dhoIxekECtPYM4jdUYRbnSLZ1Ze8GO6/M2owFeAxZyIaFB7UFIsiRVk+B0NDubNY/bN+Y7Y+TsnqfB/bvnsCqpMioU4bgsV/NZoXyNpWKEU3OCqpIWmFzjhK7AFBhE+bopn3EfgCdy41zCXyi38d7O0Dgry20WAjLDKi3tWO08FFtVKv7S10wUlaKCtAvFFXdV7tZn4UZMUqL4FgxMJIO9uiTFUCYFJ3bSWaZUGZZbGXWk6ETiImVkYw5u6aCGbj4VrfPB7W2XBRasTKcZ3TCS+zrEIeWHMbeC0IO1GCySqtlwC7DSrIp0dyi2Jfu5e1i6hNNMqSU75jlWlgsLsk0j2fUSpuh/wK14LaI+/IswhrbARf1xdBPDzMCJCPqS5NCyImo6XBiNoISVpHXBdQJfTlYhURljLGzZw66PY83K823waD7y+lDchwbHoJFZnR/EojTMNxpNzXR03suTPZXLoyplT+XyqErZV7k8olL2VS5fq7xnyeyJRN80dwzgDSR5ByQmUJnFvyTdqtHiAAH5HwxfNwxul2LZrRuq3wRzvewtdpkegV72oPQGoCFNmND0pmpu/B6kwVAwjyFC/JoEc5YI+BhtDryjOBIvYEKg6f5nhSMWx70DbqdFM3Jg4EAR93a/QYCh301dhrXX5VgfZNkjrFhWcKO/C9oo2Fb4W/FPJhfmlRV8cmNWc18/uTl7NUaSxnVLTAADFzoUtb7ytEcyY73gspF0T4PGup2uxvJP9G5o2oFVm7Cy/f4uwTcwZuf2UO0bV97ZV2fz7xejh49283bofOx84oydufOF89C5dJ46Vw5x/hq4g8ng0yEe/jL8dfhbC70z2OV85HSe4R9/A6dYRwA=</latexit>

Page 10: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

S(2)⇣

=

Zd4x

a3bs✏Hc2s

✓⇣2 � c2

sk2

a2⇣2◆

d2v

d⌧2+

✓c2sk

2 � 1

z

d2z

d⌧2

◆v = 0

v = z⇣

z = a

s2bs✏Hc2s

* Mukhanov - Sasaki equation

• Assume slow-roll approximation.

• Solve numerically.

• Use GSR techniques.

Page 11: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

S(2)⇣

=

Zd4x

a3bs✏Hc2s

✓⇣2 � c2

sk2

a2⇣2◆

d2v

d⌧2+

✓c2sk

2 � 1

z

d2z

d⌧2

◆v = 0

v = z⇣

z = a

s2bs✏Hc2s

• Assume slow-roll approximation.

• Solve numerically.

• Use GSR techniques.

* Mukhanov - Sasaki equation

Page 12: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

Outline:

1. Scalar - Tensor theories.

2. Generalized slow-roll (GSR) techniques.

3. G-inflation.

Page 13: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

• E. Stewart; 0110322 • C. Dvorkin, W. Hu; 0910.2237 • W. Hu; 1104.4500 • W. Hu; 1405.2020 • H. Motohashi, W. Hu; 1503.04810 • H. Motohashi, W. Hu; 1704.01128

… and others.

Generalized Slow-Roll

Page 14: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

1. Write down Mukhanov-Sasaki equation.Generalized slow-roll recipe:

Page 15: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

1. Write down Mukhanov-Sasaki equation.2. Isolate deviations from de Sitter background.

d2y

dx2+

✓1� 2

x2

◆y =

f 00 � 3f 0

f

y

x2

Generalized slow-roll recipe:

ss ⌘Z

csd⌧y ⌘p

2cskv x ⌘ kss

f =

r8⇡2

bs✏Hcs

H2

aHss

cs

�2⇣(k) = lim

x!0

����xy

f

����2

de Sitter background deviations from dS

Page 16: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

3. Apply Green function techniques (GSR).

ln �2(1) = G (ln xm) +

Z 1

xm

d(ln x)W (kx)G0 (ln x)

d2y

dx2+

✓1� 2

x2

◆y =

f 00 � 3f 0

f

y

x2

Generalized slow-roll recipe:

1. Write down Mukhanov-Sasaki equation.2. Isolate deviations from de Sitter background.

G = �2ln f +2

3(ln f)0

W (u) =3 sin(2u)

2u3� 3 cos(2u)

u2� 3 sin(2u)

2u

Source function

Page 17: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

4. Taylor expand GSR formula and write down analytic equations (OSR).

ln �2(1) = G (ln xm) +

Z 1

xm

d(ln x)W (kx)G0 (ln x)

d2y

dx2+

✓1� 2

x2

◆y =

f 00 � 3f 0

f

y

x2

Generalized slow-roll recipe:

1. Write down Mukhanov-Sasaki equation.2. Isolate deviations from de Sitter background.

3. Apply Green function techniques (GSR).

Page 18: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

Optimized SR for Horndeski (leading order):

ln�2⇣⇡ ln

✓H

2

8⇡2bscs✏H

◆� 10

3✏H � 2

3�1 �

7

3�s1 �

1

3⇠s1

���x=x1

ns � 1 ⇡ �4✏H � 2�1 � �s1 � ⇠s1 �2

3�2 �

7

3�s2 �

1

3⇠s2

���x=x1

↵s ⇡ �2�2 � �s2 � ⇠s2 �2

3�3 �

7

3�s3 �

1

3⇠s3 � 8✏2

H� 10✏H�1 + 2�21

���x=x1

ln�2�⇡ ln

✓H

2

2⇡2btct

◆� 8

3✏H � 7

3�t1 �

1

3⇠t1

���x=x1

nt ⇡ �2✏H � �t1 � ⇠t1 �7

3�t2 �

1

3⇠t2

���x=x1

↵t ⇡ ��t2 � ⇠t2 �7

3�t3 �

1

3⇠t3 � 4✏2

H� 4✏H�1

���x=x1

Tensors

Scalars

• H. Motohashi, W. Hu; 1704.01128

ln x1 ⌘ 7

3� ln 2� �E

Page 19: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

Optimized SR for Horndeski (leading order):

r ⌘4�2

�2⇣

⇡ 16✏Hbscsbtct

⇡ �8bscsbtct

nt,

Deviations from the standard consistency relation in the observations could be checked in this context.

• H. Motohashi, W. Hu; 1704.01128

Page 20: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

Outline:

1. Scalar - Tensor theories.

2. Generalized slow-roll (GSR) techniques.

3. G-inflation.

J. Ohashi, S. Tsujikawa; 1207.4879

K. Kamada, T. Kobayashi, M. Yamaguchi, J. Yokoyama; 1012.4238

G3 = const<latexit sha1_base64="ftaUMJt4IaBmC7JTjgcxALdoZuA=">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</latexit><latexit sha1_base64="ftaUMJt4IaBmC7JTjgcxALdoZuA=">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</latexit><latexit sha1_base64="ftaUMJt4IaBmC7JTjgcxALdoZuA=">AAAIW3iclVVdb9MwFM02YF0ZsIF44iWimrRJ09SPSSChSRNo2h5AGhXbkJKschwnseY4me2MFsv7JbzCf+KB/4LTlm2xWwGpGl3dc+6xz7VjhwXBXLTbPxcWl+7df7DcWGk+XH30+Mna+tNTnpcMohOYk5x9DgFHBFN0IrAg6HPBEMhCgs7Ci3cVfnaFGMc5/SRGBQoykFAcYwiETg3W1g4HvT1foKGQMKdcqMFaq73THj+uHXSmQcuZPseD9eWeH+WwzBAVkADOvU67EIEETGBIkGr6JUcFgBcgQZ4OKcgQD+R46srd0JnIjXOm/1S44+zdCgkyzkdZqJkZECk3sSo5C/NKEb8OJKZFKRCFk4Hikrgid6s+uBFmCAoy0gGADOu5ujAFDEChu9WsDcNFBtiIRTUrMmGgSDEcqplTmumhXo/oJLlxd9q8ABTzdDtDQwzzQIYgRGQ25w6o178yA2hSjic8IRhlRkfqM6Qjjr/WF0tyvZopMmzHJAfCSAEKR2nE6lmIBfoHnkcqE9Xi74Wx3hagqDZmvTDMlF4Rir7APMsAjaSfgIwq6esWlgxVDZfJQPpZ6dNSKWVwucU9n89VXjcwya1Oq2tTgU0dzKNGyuvN5PppmA+lv622Wz2rjlku+3NdMstlf65LZrvsz3HJbJf9G5crhk3LpH8INFZJK13UnWEx0Z3Z/UvRnR7tzhCA/6HwZqzg1iX69b751RsCIvvWYEfpHOqRRUWXmhqiBFOJLsvxaWtRxhykw3mMENyIAIITqj9GUwNMJebghdD49u3PgCMcx9YC+7E+BuXkUoiUbuJtbG8QrWDvprrCebeucT5T5ZZhYFlBlPwwmKA6NuD39E8loeraAA8uldcJ5MHlzvWmz1BcbYktzdEHum5qdeTJLsyU8dKHDUO3Mv6m9AmKxaYy8lv6+8BJKrZMwJsUeGY+mBYE1TXbMS9VOzjt7nTaO52Pu639t9MLt+G8cF46m07HeeXsO0fOsXPiQOfK+eZ8d34s/2osNZqN1Ql1cWFa88ypPY3nvwGkQhP4</latexit><latexit sha1_base64="ftaUMJt4IaBmC7JTjgcxALdoZuA=">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</latexit>

Page 21: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

G3 + tanh + chaotic inflation = transient G-inflation

1 5 10 50 100 500

e-foldings N

0.00

0.01

0.02

0.03

0.04

0.05

✏ H

Chaotic inflation

G-model

Step model

m2�2

G3

Step size ~ 4 e-folds

Ideal for OSR

HR, S. Passaglia, H. Motohashi, W. Hu, O. Mena; JCAP 1804 (2018) no.04, 039

L2 = G2 ,

L3 = G3⇤� ,

L4 = G4R� 2G4,X [(⇤�)2 � �;µ⌫�;µ⌫ ] ,

L5 = G5Gµ⌫�;µ⌫ +

G5,X

3[(⇤�)3 � 3(⇤�)�;µ⌫�

;µ⌫ + 2�;µ⌫�;µ��;⌫

;�] .<latexit 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sha1_base64="LM8FLqVILnx7d7L+iRRkPPIbD+A=">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</latexit><latexit sha1_base64="LM8FLqVILnx7d7L+iRRkPPIbD+A=">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</latexit>

G2 =1

2�2 � 1

2m2�2 ,

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G3 = M�3

1 + tanh

✓�� �r

d

◆�,

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G5 = 0<latexit sha1_base64="j4brtU8w6Ntb2Bk26qMzR/T603k=">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</latexit><latexit sha1_base64="j4brtU8w6Ntb2Bk26qMzR/T603k=">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</latexit><latexit sha1_base64="j4brtU8w6Ntb2Bk26qMzR/T603k=">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</latexit><latexit sha1_base64="j4brtU8w6Ntb2Bk26qMzR/T603k=">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</latexit>

G4 =1

2M2

pl ,<latexit 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Page 22: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

G3 + tanh + chaotic inflation = transient G-inflation

54.5 55.0 55.5 56.0 56.5 57.0

e-foldings N

�0.5

0.0

0.5

1.0

1.5

2.0

c2 s

d = 45

d = 20

d = 10

d = 0.081

No gradient instabilities for any mass scale .M

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M = 1.3⇥ 10�4

�r = 13<latexit sha1_base64="eN8mnAVNGLzhUTueAiGZVyYE/yU=">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</latexit><latexit sha1_base64="eN8mnAVNGLzhUTueAiGZVyYE/yU=">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</latexit><latexit sha1_base64="eN8mnAVNGLzhUTueAiGZVyYE/yU=">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</latexit><latexit sha1_base64="eN8mnAVNGLzhUTueAiGZVyYE/yU=">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</latexit>

L2 = G2 ,

L3 = G3⇤� ,

L4 = G4R� 2G4,X [(⇤�)2 � �;µ⌫�;µ⌫ ] ,

L5 = G5Gµ⌫�;µ⌫ +

G5,X

3[(⇤�)3 � 3(⇤�)�;µ⌫�

;µ⌫ + 2�;µ⌫�;µ��;⌫

;�] .<latexit 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sha1_base64="LM8FLqVILnx7d7L+iRRkPPIbD+A=">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</latexit><latexit sha1_base64="LM8FLqVILnx7d7L+iRRkPPIbD+A=">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</latexit>

HR, S. Passaglia, H. Motohashi, W. Hu, O. Mena; JCAP 1804 (2018) no.04, 039

G2 =1

2�2 � 1

2m2�2 ,

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G3 = M�3

1 + tanh

✓�� �r

d

◆�,

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G5 = 0<latexit sha1_base64="j4brtU8w6Ntb2Bk26qMzR/T603k=">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</latexit><latexit sha1_base64="j4brtU8w6Ntb2Bk26qMzR/T603k=">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</latexit><latexit sha1_base64="j4brtU8w6Ntb2Bk26qMzR/T603k=">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</latexit><latexit sha1_base64="j4brtU8w6Ntb2Bk26qMzR/T603k=">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</latexit>

G4 =1

2M2

pl ,<latexit 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Page 23: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

0.00

0.01

0.02

0.03

0.04

✏ Hd = 0.5

d = 0.3

d = 0.15

d = 0.02

�5 0 5 10 15

e-foldings N

0.0

0.2

0.4

0.6

0.8

1.0

1.2

✏SR

H/✏

H�

1

✏H = � H

H2

• : CMB scales.

• Vertical lines: where the transition occurs.

• SR violation is maximal around the transition.

N = 0

Slow - roll violation:

Page 24: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

1

2

109 �

2 ⇣

Exact

SR

GSR

OSR

10�4 10�3 10�2 10�1 100 101 102 103

k [Mpc�1]

�0.6

�0.5

�0.4

�0.3

�0.2

�0.1

0.0

0.1

0.2

�2 ⇣a

pp/�

2 ⇣�

1

GSR

SR

OSR

M = 1.3⇥ 10�4

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Different approximations

XX

Page 25: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

0.95 0.96 0.97 0.98

Primordial tilt (ns)

10�2

10�1

Ten

sor-

to-s

cala

rra

tio

(r)

�N = 50

�N = 60

Planck TT + lowP + BKP

Planck TT + lowP + BKP + BAO

Chaotic inflation

G – model

Step model (�N = 55)

Chaotic inflation

G – model

Step model (�N = 55)

• and fixed. • Find a set of values for and . • This places lower and upper bounds on .

ns = 9.66↵s = �0.011

Msmaller

ns ↵s

rd �r

Observables

Page 26: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

0.95 0.96 0.97 0.98

Primordial tilt (ns)

10�2

10�1

Ten

sor-

to-s

cala

rra

tio

(r)

�N = 50

�N = 60

Planck TT + lowP + BKP

Planck TT + lowP + BKP + BAO

Chaotic inflation

G – model

Step model (�N = 55)

Chaotic inflation

G – model

Step model (�N = 55)

• A smaller would shift the line upwards because the step gets wider.

• A larger would be in tension with measurements.

r & 0.005

Msmaller

↵s

↵s

ns = 9.66↵s = �0.011

Observables

Potentially observable with next-generation surveys.

Page 27: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

1

2

10

9�

2 ⇣

Exact

SRH

10�4 10�3 10�2 10�1 100 101 102 103

k [Mpc�1]

�0.06

�0.04

�0.02

0.00

0.02

0.04

�2

(SR

H)

⇣/�

2 ⇣�

1

�2(SRH)

⇣ (k) = As

✓k

k⇤

◆ns�1+12↵s ln(k/k⇤)

Deviations of less than 1%

Slow - roll hierarchy formula:

Using OSR parameters

Page 28: Computing inflationary predictions in general scalar-tensor ...k + k2 z00 z v k =0 Mukhanov - Sasaki equation: Slow-roll conditions: N CMB = Z end plCMB d M p 2

Inflation in the Horndeski framework is viable and can cure some popular models.

The transient G-inflation model allows us to compute observables during a G-inflation period and to end inflation as canonical.

Generalized slow-roll and Optimized slow-roll techniques are efficient tools for this type of models to compute the power spectra an also the bispectrum.

Summary