Predictions in multifield models of inflationJonathan Frazer

University College London

and current work in collaboration withRichard Easther, Hiranya Peiris and Layne Price

arXiv:1303.3611

• The problem with predictions in multifield inflation

• A suggestion for a way forwards

• An analytic example

• Current numerical work

Overview

10 Planck Collaboration: Constraints on inflation

Model Parameter Planck+WP Planck+WP+lensing Planck + WP+high-` Planck+WP+BAO

⇤CDM + tensor ns 0.9624 ± 0.0075 0.9653 ± 0.0069 0.9600 ± 0.0071 0.9643 + 0.0059r0.002 < 0.12 < 0.13 < 0.11 < 0.12

�2� lnLmax 0 0 0 -0.31

Table 4. Constraints on the primordial perturbation parameters in the ⇤CDM+r model from Planck combined with other data sets.The constraints are given at the pivot scale k⇤ = 0.002 Mpc�1.

0.94 0.96 0.98 1.00Primordial Tilt (ns)

0.00

0.05

0.10

0.15

0.20

0.25

Ten

sor-

to-S

cala

rRat

io(r

0.00

2)

ConvexConcave

Planck+WP

Planck+WP+highL

Planck+WP+BAO

Natural Inflation

Power law inflation

Low Scale SSB SUSY

R2 Inflation

V � �2/3

V � �

V � �2

V � �3

N�=50

N�=60

Fig. 1. Marginalized joint 68% and 95% CL regions for ns and r0.002 from Planck in combination with other data sets compared tothe theoretical predictions of selected inflationary models.

reheating priors allowing N⇤ < 50 could reconcile this modelwith the Planck data.

Exponential potential and power law inflation

Inflation with an exponential potential

V(�) = ⇤4 exp

�� �Mpl

!

(35)

is called power law inflation (Lucchin & Matarrese, 1985),because the exact solution for the scale factor is given bya(t) / t2/�2 . This model is incomplete, since inflation wouldnot end without an additional mechanism to stop it. Assumingsuch a mechanism exists and leaves predictions for cosmo-logical perturbations unmodified, this class of models predictsr = �8(ns � 1) and is now outside the joint 99.7% CL contour.

Inverse power law potential

Intermediate models (Barrow, 1990; Muslimov, 1990) with in-verse power law potentials

V(�) = ⇤4

�

Mpl

!��(36)

lead to inflation with a(t) / exp(At f ), with A > 0 and 0 < f < 1,where f = 4/(4 + �) and � > 0. In intermediate inflation thereis no natural end to inflation, but if the exit mechanism leavesthe inflationary predictions on cosmological perturbations un-modified, this class of models predicts r ⇡ �8�(ns � 1)/(� � 2)(Barrow & Liddle, 1993). It is disfavoured, being outside thejoint 95% CL contour for any �.

Hill-top models

In another interesting class of potentials, the inflaton rolls awayfrom an unstable equilibrium as in the first new inflationary mod-els (Albrecht & Steinhardt, 1982; Linde, 1982). We consider

V(�) ⇡ ⇤4

1 � �p

µp + ...

!

, (37)

where the ellipsis indicates higher order terms negligible duringinflation, but needed to ensure the positiveness of the potentiallater on. An exponent of p = 2 is allowed only as a large fieldinflationary model and predicts ns � 1 ⇡ �4M2

pl/µ2 + 3r/8 and

r ⇡ 32�2⇤M2pl/µ

4. This potential leads to predictions in agree-ment with Planck+WP+BAO joint 95% CL contours for super-Planckian values of µ, i.e., µ & 9 Mpl.

Models with p � 3 predict ns � 1 ⇡ �(2/N)(p � 1)/(p � 2)when r ⇠ 0. The hill-top potential with p = 3 lies outside the

Problem: We don’t know how to confront multifield models with observation.

Planck collaboration arXiv: 1303.5082

⌅ o � o(⌅) n ⇥f ⇥ 0o � o(⌅, N) (9)

p(o) =f(⌅)��dod⇥

�� (10)

⌃1 ⌃2 ⇤ = ⇥N ⇥⌃ {P� , ns,� V (⌃) r �⌃ fNL L ⇧ = 3H2

⇥f (11)

2

⌅ o � o(⌅) n ⇥f ⇥ 0o � o(⌅, N) (9)

p(o) =f(⌅)��dod⇥

�� (10)

⌃1 ⌃2 ⇤ = ⇥N ⇥⌃ {P� , ns,� V (⌃) r �⌃ fNL L ⇧ = 3H2

⇥f (11)

2

Figure 1. Sketch showing the horizon crossing surface in a single field inflationary potential and atwo-field model. For a single field model, the horizon crossing surface is a single point, meaning thatobservables are approximately insensitive to inflationary dynamics prior to crossing this point. Forthe case of the two-field model, without knowledge of initial conditions one can not identify a singleinflationary trajectory that corresponds to what we observe. One must therefore consider all possibleinflationary trajectories.

by the value of the inflaton at the moment that scale left the horizon. If we take the largestscales we observe to have left the horizon say 55 e-folds before the end of inflation, ourobservations are generally insensitive to what happened before then, be it another 20 e-foldsof inflation or 200.

When there is more than one active field the situation changes. As illustrated in Fig. 1,if the model involves Nf inflationary fields, instead of there being only one possible valueof the inflaton at horizon crossing, now there is an infinite set of possible locations in fieldspace forming an Nf � 1 dimentional hypersurface.3 Without specifying initial conditions,it is not possible to say which inflationary trajectory corresponds to our observable universeand since di�erent trajectories will in general give rise to di�erent values for observables, themodel is only as predictive as the volume in the space of observables permitted by the model.If we are to seriously confront multifield models of inflation with observation, then it is ofparamount importance that this problem is overcome.

The obvious question then is whether a description of initial conditions can be derivedwithin the framework of the model. One can hope that an ultraviolet complete theory ofinflation will provide information on the initial state but calculations along this line areclearly well beyond our current understanding of fundamental physics. A more promisingapproach is to consider how chaotic inflation populates the potential. Although this issuehas received some attention in the past for the case of single field inflation (see for instance

3Discussed in more detail in section §2, this assumes only one trajectory passes through a given point infield space.

– 2 –

Multifield models are sensitive to initial conditions⌅ o � o(⌅) n ⇥f ⇥ 0

o � o(⌅, N) (9)

p(o) =f(⌅)��dod⇥

�� (10)

⌃1 ⌃2 ⇤ = ⇥N ⇥⌃ {P� , ns,� V (⌃) r �⌃ fNL L ⇧ = 3H2

⇥f (11)

2

⌅ o � o(⌅) n ⇥f ⇥ 0o � o(⌅, N) (9)

p(o) =f(⌅)��dod⇥

�� (10)

⌃1 ⌃2 ⇤ = ⇥N ⇥⌃ {P� , ns,� V (⌃) r �⌃ fNL L ⇧ = 3H2

⇥f (11)

2

Figure 1. Sketch showing the horizon crossing surface in a single field inflationary potential and atwo-field model. For a single field model, the horizon crossing surface is a single point, meaning thatobservables are approximately insensitive to inflationary dynamics prior to crossing this point. Forthe case of the two-field model, without knowledge of initial conditions one can not identify a singleinflationary trajectory that corresponds to what we observe. One must therefore consider all possibleinflationary trajectories.

by the value of the inflaton at the moment that scale left the horizon. If we take the largestscales we observe to have left the horizon say 55 e-folds before the end of inflation, ourobservations are generally insensitive to what happened before then, be it another 20 e-foldsof inflation or 200.

When there is more than one active field the situation changes. As illustrated in Fig. 1,if the model involves Nf inflationary fields, instead of there being only one possible valueof the inflaton at horizon crossing, now there is an infinite set of possible locations in fieldspace forming an Nf � 1 dimentional hypersurface.3 Without specifying initial conditions,it is not possible to say which inflationary trajectory corresponds to our observable universeand since di�erent trajectories will in general give rise to di�erent values for observables, themodel is only as predictive as the volume in the space of observables permitted by the model.If we are to seriously confront multifield models of inflation with observation, then it is ofparamount importance that this problem is overcome.

The obvious question then is whether a description of initial conditions can be derivedwithin the framework of the model. One can hope that an ultraviolet complete theory ofinflation will provide information on the initial state but calculations along this line areclearly well beyond our current understanding of fundamental physics. A more promisingapproach is to consider how chaotic inflation populates the potential. Although this issuehas received some attention in the past for the case of single field inflation (see for instance

3Discussed in more detail in section §2, this assumes only one trajectory passes through a given point infield space.

– 2 –

Prediction is single valued

Prediction is a PDF

V (�)

� �1

�2

V (�1,�2)⌃f⌃f

Split the problem into 2 parts:

1. Compute the PDF of initial conditions

2. Use PDF of initial conditions to compute PDF for observables (a.k.a the prediction)

f(✓)

p(o)

Strategy: Use conservation of probability to map PDF of initial conditions to a PDF for observables .f(✓) p(o)

�1

�2

fp(o)

o(✓)

⌃f

✓

Parameterise with variables .

p(o)|do| = f(✓)|d✓|

o

o(✓)Strategy: Turning points in enables robust predictions without detailed knowledge of initial conditions.

⌅ o � o(⌅) n ⇥f ⇥ 0o � o(⌅, N) (9)

p(o) =f(⌅)��dod⇥

�� (10)

⌃1 ⌃2 ⇤ = ⇥N ⇥⌃ {P� , ns,� V (⌃) r �⌃ fNL L ⇧ = 3H2

⇥f (11)

p(o)f(⌅)o(⌅)

2

⌅ o � o(⌅) n ⇥f ⇥ 0o � o(⌅, N) (9)

p(o) =f(⌅)��dod⇥

�� (10)

⌃1 ⌃2 ⇤ = ⇥N ⇥⌃ {P� , ns,� V (⌃) r �⌃ fNL L ⇧ = 3H2

⇥f (11)

p(o)f(⌅)o(⌅)

2

⌅ o � o(⌅) n ⇥f ⇥ 0o � o(⌅, N) (9)

p(o) =f(⌅)��dod⇥

�� (10)

⌃1 ⌃2 ⇤ = ⇥N ⇥⌃ {P� , ns,� V (⌃) r �⌃ fNL L ⇧ = 3H2

⇥f (11)

p(o) f(⌅) o(⌅)

2

⌅ o � o(⌅) n ⇥f ⇥ 0o � o(⌅, N) (9)

p(o) =f(⌅)��dod⇥

�� (10)

⌃1 ⌃2 ⇤ = ⇥N ⇥⌃ {P� , ns,� V (⌃) r �⌃ fNL L ⇧ = 3H2

⇥f (11)

p(o) f(⌅) o(⌅)

2

Figure 4. Sketch (Rotring pens on Bristol Board) of why a stationary point in the functional formof o(�) will in general give rise to a spike in the density function p(o). The spacing of the red linesrepresents the density function f(�) and how it is distorted under a change of variable to give p(o).

[2] A. A. Starobinsky, JETP Lett., 42, 152 (1985); D. H. Lyth, Phys.Rev., D31, 1792 (1985); M.Sasaki and E. D. Stewart, Prog.Theor.Phys., 95, 71 (1996), arXiv:astro-ph/9507001; D. S.Salopek and J. R. Bond, Phys. Rev., D42, 3936 (1990); M. Sasaki and T. Tanaka,Prog.Theor.Phys., 99, 763 (1998), arXiv:gr-qc/9801017 [gr-qc]; D. Wands, K. A. Malik, D. H.24Lyth, and A. R. Liddle, Phys.Rev., D62, 043527 (2000), arXiv:astro-ph/0003278 [astro-ph].

[3] D. H. Lyth and Y. Rodrguez, Phys.Rev.Lett., 95, 121302 (2005), arXiv:astro-ph/0504045.

[4] G. I. Rigopoulos and E. P. S. Shellard, JCAP, 0510, 006 (2005), arXiv:astro-ph/0405185; G. I.Rigopoulos, E. P. S. Shellard, and B. J. W. van Tent, Phys.Rev., D73, 083521 (2006),arXiv:astro-ph/0504508.

[5] S. Yokoyama, T. Suyama, and T. Tanaka, JCAP, 0707, 013 (2007), arXiv:0705.3178 [astro-ph];Phys.Rev., D77, 083511 (2008), arXiv:0711.2920 [astro-ph]; JCAP, 0902, 012 (2009),arXiv:0810.3053 [astro-ph].

[6] D. J. Mulryne, D. Seery, and D. Wesley, JCAP, 1001, 024 (2010), arXiv:0909.2256[astro-ph.CO]; 1104, 030 (2011), arXiv:1008.3159 [astro-ph.CO].

[7] L. Amendola, C. Gordon, D. Wands, and M. Sasaki, Phys.Rev.Lett., 88, 211302 (2002),arXiv:astro-ph/0107089 [astro-ph]. [10] S. Groot Nibbelink and B. van Tent,Class.Quant.Grav., 19, 613 (2002), arXiv:hep-ph/0107272 [hep-ph].

[8] Z. Lalak, D. Langlois, S. Pokorski, and K. Turzynski, JCAP, 0707, 014 (2007),arXiv:0704.0212 [hep-th].

[9] C. M. Peterson and M. Tegmark, Phys.Rev., D83, 023522 (2011), arXiv:1005.4056[astro-ph.CO]; D84, 023520 (2011), arXiv:1011.6675 [astro-ph.CO]; (2011), arXiv:1111.0927[astro-ph.CO].

– 11 –

p(o)|do| = f(✓)|d✓|

f(✓)

p(o)

o(✓)

✓

Since , move to polar coordinates:

0 ⇤

2⇤ 3 ⇤

22 ⇤

�0.40

�0.35

�0.30

�0.25

�0.20

�0.15

�0.10

�0.05

⌅

n s�1

0 ⇤

2⇤ 3 ⇤

22 ⇤

�0.05

0.00

0.05

0.10

0.15

⌅

⇥

Figure 2. Example plots of ns(⌅) and �(⌅) for scales leaving the horizon 55 e-folds before the end ofinflation in double quadratic inflation with masses m2/m1 = 9.

where here and what follows, unless explicitly written otherwise, fields only lie on the surface�f and so the “*” label has been dropped. For quadratic inflation, the observables (2.4),(2.8) and (2.10) become

P�(⌅, N) =H2

4⇧2N, (4.5)

ns(⌅, N)� 1 = �2⇥� 1

N, (4.6)

�(⌅, N) = �8⇥2 � 2

N2+ 4 (⇥1⇤1 + ⇥2⇤2) . (4.7)

The amplitude of the power spectrum is in a sense less interesting than the other observablessince it may always be adjusted by a pre-factor on the potential which does not a⇥ect theinflationary dynamics. Other observables one might want to consider are the non-Gaussianityparameter fNL and the tensor-to-scalar ratio r. For the case of double quadratic inflation, itwas shown in Ref. [42] that �6

5fNL = 1/N . Since �f is defined as being at a fixed numberof e-folds before the end of inflation, fNL is single valued over �f which is why it has notbeen part of the discussion until now (the same is true for the tensor-to-scalar ratio). Hence,for this example, the objective is to calculate p(ns,�). Fig. 2 shows plots of the spectralindex and running evaluated 55 e-folds before the end of inflation. As mentioned previously,an important characteristic to bear in mind for what follows is that these observables areperiodic in ⌅.

4.1 Regarding the field space density function f�f

It is necessary to specify the density function f�f . As already mentioned, this is dependanton the details of the model as well as the choice of measure. For the purposes of this paperthe distribution is chosen to be flat

f�f = c =1�

�fd�f

. (4.8)

Following Ref. [47], this choice is a statement of ignorance. We simply adopt the distributionrequiring the least additional assumptions. No further justification will be given at this stage,however in §5.1 the implications of this choice will be discussed.

– 9 –

Express observables as

Example: Double quadratic inflation

N = 14 (�

21 + �2

2)

V = 12m

21�

21 +

12m

22�

22

�1 = 2

pN cos ✓ �2 = 2

pN sin ✓

o(✓, N)

�0.06

�0.40

ns�

1

0 2⇡✓

0.15

�0.06

↵

0 2⇡✓

Example: Double quadratic inflation V = 12m

21�

21 +

12m

22�

22

Try a flat distribution over the horizon crossing surface.

p(ns,↵) =2

⇡

1q�dnsd✓

�2+�d↵d✓

�2

Result!

-0.40 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05

-0.05

0.00

0.05

0.10

0.15

ns-1

a

-0.4-0.3

-0.2-0.1ns-1

-0.050.00

0.050.100.15

a

0

20

40pHns-1,aL

ns � 1

ns � 1

↵

↵

↵

Is there something misleading about the use of iso -fold slicing?

Plots taken from Easther and Price arXiv: 1304.4244

0 .012 .024 .036 .0480

.05

.1

.15

.2

y0 êM Pl

f0êM

Pl

0 .05 .1 .15 .20

.05

.1

.15

.2

y 0 ê M Pl

f0êM

Pl

0 .05 .1 .15 .20

.05

.1

.15

.2

y0 êM Pl

f0êM

Pl

0 .05 .1 .15 .20

.05

.1

.15

.2

y0 êM Pl

f0êM

Pl

Iso velocity Iso energy

e

See also Clesse, Ringeval, Rocher arXiv: 0909.0402

2-field 3-field

m1

m2= 7, 9

m1

m2= 7

ns � 1

p(ns,↵)p(ns,↵)

↵

↵

PDFs of observables for -quadratic inflation Nf

m1

m3= 9

ns � 1

2-field 9-field

m1

m2= 7, 9

ns � 1

p(ns,↵) p(ns,↵)

↵

↵

PDFs of observables for -quadratic inflation Nf

ns � 1

m1

m2= 2

m1

m9= 9...

• Expect sharp spike in PDF to be a common characteristic

• Prediction is surprisingly insensitive to initial conditions

-30 -20 -10 0 10 20 30-30

-20

-10

0

10

20

30

f1

f 2

-20 -10 0 10 20-20

-10

0

10

20

f1

f 2

Is there a better way to study this?

• What are the criteria for a predictive model?

• Is it sensible to consider initial conditions and model parameters separately?

• Is there a better way to study multifield inflation than evolving individual classical trajectories?

p(o)|do| = f(✓)|d✓|

• Expect this property in a broad class of models but still WIP.

• Spike in PDF is robust to different slicings.

• Numerical tools extendable to arbitrary potentials.

Conclusions