Initial Conditions for Inflation in an FRW Universe @ …sriram/professional/events/... · 2019. 1. 29. · Why bother about for Initial Conditions! 1 Traditional Physics aims at

Initial Conditions for Inflation in an FRWUniverse

@ Thursday Seminar, IIT Madras22 November, 2018

Swagat Saurav Mishra (SRF CSIR)Inter-University Centre for Astronomy and

Astrophysics(IUCAA), Pune—————–

In Collaboration with: Prof. Varun Sahni (IUCAA), Prof.Aleksey Toporensky (Moscow State University)

Other Collaborators: Yuri Shtanov (BITP, Ukraine), Armaan Shafieloo (KASI, Korea), Sanil Unnikrishnan

(St. Stephens, Delhi), Satadru Bag (IUCAA), Ujjaini Alam (ISI, Kolkata)

November 22, 2018

Swagat Saurav Mishra, IUCAA, Pune Initial Conditions for Inflation 1/ 27

Published in Phys. Rev. D 98, 083538 [arXiv:1801.04948]


Why bother about for Initial Conditions!

1 Traditional Physics aims at discovering laws of evolution (dynamics)of a system in nature given Initial conditions.

2 Initial conditions are to be given.

3 Cosmology, the study of ORIGIN, evolution and the fate of theuniverse, unfortunately can not afford to ignore initial conditions. Ithas to explain the initial conditions.

4 Standard Hot Big Bang Cosmology seemed (or seems) to besuggesting that universe started out with a set of very finely tunedunnatural initial conditions.

5 Inflationary Cosmology was proposed as an extension to thestandard Hot Expanding Universe Theory in order to set up thestage for the hot Big Bang phase providing the required initialconditions.

So Cosmic Inflation should better happen starting from generic

Initial Conditions!









Initial Conditions!









Initial Conditions!









Initial Conditions!









Initial Conditions!









Initial Conditions!









Initial Conditions!









Initial Conditions!


Need for Cosmic Inflation

1) Flatness Problem :∑

i

Ωi (a)− 1 =K

a2H2

∑i Ωi (a) diverges away from 1 if

a < 0

10−3 10−2 10−1

a

∑Ω

i−

1

Diverging

Away

Diverging

Away

Decelerating Expansion w > −1/3

K = 0

K = +1

K = −1

Long enough period of acceleratedexpansion drives

∑i Ωi (a) −→ 1

10−6 10−5 10−4 10−3 10−2 10−1 100 101

a

∑Ω

i−

1

Inflatio

nw

<−1/3

Inflationw

<−1/3

InflationEnds


K = 0

K = +1

K = −1



1) Flatness Problem :∑

i

Ωi (a)− 1 =K

a2H2

∑i Ωi (a) diverges away from 1 if

a < 0

10−3 10−2 10−1

a

∑Ω

i−

1

Diverging

Away

Diverging

Away


K = 0

K = +1

K = −1

Long enough period of acceleratedexpansion drives

∑i Ωi (a) −→ 1

10−6 10−5 10−4 10−3 10−2 10−1 100 101

a

∑Ω

i−

1

Inflatio

nw

<−1/3

Inflationw

<−1/3

InflationEnds


K = 0

K = +1

K = −1



2) Horizon Problem and Super-HubbleCorrelations:

Why CMB has the same T on angular scalesgreater than 10 ? Why CMB fluctuationsare correlated beyond Hubble scale?

−50 −40 −30 −20 −10 0−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

CM

BZ

≈1100

Horizon entry at CMB

BigBan

g withou

t Inflati

on

dH∼ a

dH∼a2

dH∼ a

3/2

H−1 ∼ const

Particle horizon(dH)

Hubble Horizon(H−1)Modes (λ ∝ a)

3)Generating almost (but notexactly) scale invariantpredominantly Gaussianadiabatic primordial fluctuations

∆2R = As

(K

K∗

)nS−1

As = 2.2× 10−9 and r ≤ 0.06.4)Getting rid of unobservedrelics of phase transitions fromearlier epochs.A period of 50− 60 e-foldingsof accelerated expansion(inflation) is required.How to achieve a brief period ofInflation?



2) Horizon Problem and Super-HubbleCorrelations:Why CMB has the same T on angular scalesgreater than 10 ?

Why CMB fluctuationsare correlated beyond Hubble scale?

−50 −40 −30 −20 −10 0−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

CM

BZ

≈1100


BigBan

g withou

t Inflati

on

dH∼ a

dH∼a2

dH∼ a

3/2

H−1 ∼ const




∆2R = As

(K

K∗

)nS−1




2) Horizon Problem and Super-HubbleCorrelations:Why CMB has the same T on angular scalesgreater than 10 ? Why CMB fluctuationsare correlated beyond Hubble scale?

−50 −40 −30 −20 −10 0−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

CM

BZ

≈1100


BigBan

g withou

t Inflati

on

dH∼ a

dH∼a2

dH∼ a

3/2

H−1 ∼ const




∆2R = As

(K

K∗

)nS−1





−50 −40 −30 −20 −10 0−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

CM

BZ

≈1100


BigBan

g withou

t Inflati

on

dH∼ a

dH∼a2

dH∼ a

3/2

H−1 ∼ const




∆2R = As

(K

K∗

)nS−1

As = 2.2× 10−9 and r ≤ 0.06.

4)Getting rid of unobservedrelics of phase transitions fromearlier epochs.A period of 50− 60 e-foldingsof accelerated expansion(inflation) is required.How to achieve a brief period ofInflation?




−50 −40 −30 −20 −10 0−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

CM

BZ

≈1100


BigBan

g withou

t Inflati

on

dH∼ a

dH∼a2

dH∼ a

3/2

H−1 ∼ const




∆2R = As

(K

K∗

)nS−1

As = 2.2× 10−9 and r ≤ 0.06.4)Getting rid of unobservedrelics of phase transitions fromearlier epochs.

A period of 50− 60 e-foldingsof accelerated expansion(inflation) is required.How to achieve a brief period ofInflation?




−50 −40 −30 −20 −10 0−100

−90

−80

−70

−60

−50

−40

−30

−20

−10

CM

BZ

≈1100


BigBan

g withou

t Inflati

on

dH∼ a

dH∼a2

dH∼ a

3/2

H−1 ∼ const




∆2R = As

(K

K∗

)nS−1



Inflationary Dynamics of a Scalar Field

Action of a scalar field minimally coupled to gravity

S [φ] =

∫d4x√−g L(F , φ)

with F = 12∂µφ ∂

µφ which for a canonical kineticterm leads to

ρφ

=1

2φ2 + V (φ), p

φ=

1

2φ2 − V (φ)

And Einstein’s equations imply

H2 =

(a

a

)2

=

(8πG

3

)ρ

φ=

1

3m2p

(1

2φ2 + V (φ)

),

a

a= −

(4πG

3

)(ρ

φ+ 3 p

φ

)= − 1

3m2p

(φ2 − V (φ)

)

φ −→

V(φ

)−→

Reheating

Slow roll

φend = 0.96mp φQ = 7.9mp

Starobinsky

The dynamics of thescalar field isgoverned by

φ+ 3Hφ+V ′(φ) = 0

while the equation ofstate is

wφ =12 φ

2 − V (φ)12 φ

2 + V (φ)


Inflationary Dynamics

For an extended period of inflation,

εH

= − H

H2< 1 , η

H= − φ

Hφ< 1

Primordial Power-spectra

∆2R = As

(K

K∗

)nS−1

∆2t = At

(K

K∗

)nt−1

with

As =1

8π2

(H∗mp

)21

εH

, At =2

π2

(H∗mp

)2

Scalar Spectral Index

nS = 1 + 2ηH− 4ε

H

Tensor to Scalar Ratio

r = 16εH

Potential slow-roll parameters

εV

(φ) =m2

p

2

(V ′

V

)2

ηV

(φ) = m2p

(V ′′

V

)

Slow-roll conditions corresponds toεH, η

H 1, where

εH' ε

V, η

H' η

V− ε

V




εH

= − H

H2< 1 , η

H= − φ

Hφ< 1


∆2R = As

(K

K∗

)nS−1

∆2t = At

(K

K∗

)nt−1

with

As =1

8π2

(H∗mp

)21

εH

, At =2

π2

(H∗mp

)2


nS = 1 + 2ηH− 4ε

H


r = 16εH


εV

(φ) =m2

p

2

(V ′

V

)2

ηV

(φ) = m2p

(V ′′

V

)


H 1, where

εH' ε

V, η

H' η

V− ε

V




εH

= − H

H2< 1 , η

H= − φ

Hφ< 1


∆2R = As

(K

K∗

)nS−1

∆2t = At

(K

K∗

)nt−1

with

As =1

8π2

(H∗mp

)21

εH

, At =2

π2

(H∗mp

)2


nS = 1 + 2ηH− 4ε

H


r = 16εH


εV

(φ) =m2

p

2

(V ′

V

)2

ηV

(φ) = m2p

(V ′′

V

)


H 1, where

εH' ε

V, η

H' η

V− ε

V


What do we know from CMB Observations

nS ∈ [0.953, 0.977],

r ≤ 0.06

which translates to

εH≤ 0.00375,

|ηH| < 0.02

CMB Observations favour concave potentials over convex

potentials. Tight constraints on the upper bound of r imply

asymptotically flat plateau like potentials are favoured.

Issue of initial conditions for Inflation becomes moreimportant!! In this talk, we are not going to discuss aboutinhomogeneous initial conditions.



nS ∈ [0.953, 0.977],

r ≤ 0.06

which translates to

εH≤ 0.00375,

|ηH| < 0.02







nS ∈ [0.953, 0.977],

r ≤ 0.06

which translates to

εH≤ 0.00375,

|ηH| < 0.02


potentials.

Tight constraints on the upper bound of r imply





nS ∈ [0.953, 0.977],

r ≤ 0.06

which translates to

εH≤ 0.00375,

|ηH| < 0.02







nS ∈ [0.953, 0.977],

r ≤ 0.06

which translates to

εH≤ 0.00375,

|ηH| < 0.02




Issue of initial conditions for Inflation becomes moreimportant!!

In this talk, we are not going to discuss aboutinhomogeneous initial conditions.



nS ∈ [0.953, 0.977],

r ≤ 0.06

which translates to

εH≤ 0.00375,

|ηH| < 0.02






0.94 0.95 0.96 0.97 0.98 0.99 1.00Scalar Spectral Index ns

0.00

0.05

0.10

0.15

0.20

0.25

0.30Ten

sor-to-sca

larratior

BICEP-II + PLANCKr ≤ 0.06

ConvexConcave

Power-law Inflation

α-attractors T-Model

Starobinsky

V ∝ |φ|2/3V ∝ |φ|V ∝ φ2

V ∝ φ4


Initial Conditions for Inflation in an FRW Universe

Inflation ⇒ Accelerated Expansion ⇒ εH < 1. When the energy budgetof the universe is dominated by the inflaton field, this implies,

wφ < −1

3or φ2 < V (φ)

Even though this condition seems restrictive, it is actually quite genericand can be quantified, at least for monotonically increasing potentials.

φ+ 3Hφ+ V ′(φ) = 0

Role of Friction term Hφ and existence of inflationary separatrix(indicating inflationary trajectories are local attractors at least for largefield inflation) [Brandenberger 2016]

Hamilton-Jacobi Formalism: H2,φ − 3

2H2 = − 1

2V (φ), dN = |dφ|√2ε

V

δH(φ) = δH(φi )e−3(N−Ni )


Methodology

We explicitly calculate the set of initial conditions that yieldadequate inflation i.e Ne ≥ 60, where a(te) = a(ti )e

Ne so

Ne =∫ teti

Hdt

For a given fixed initial energy scale ρ(φi ) of inflation (orequivalently Hi ), the first Friedmann equation becomes the equationof a circle

R2 = X 2 + Y 2

where

R =√

6H

mp, X = φ

√2V (φ)

m2p

, Y =1

m2p

dφ

dt

with φ = φ|φ| (this ensures that X and φ have same sign).

So fixing

initial energy scale of inflation is equivalent to fixing the radius ofthe circle R while varying X and Y uniformly. Our analysis[Mishra, Sahni and Toporensky PRD 2018] is a natural andgeneralized extension of the original work by [Belinsky et. al 1985].


Methodology


Ne so

Ne =∫ teti

Hdt


R2 = X 2 + Y 2

where

R =√

6H

mp, X = φ

√2V (φ)

m2p

, Y =1

m2p

dφ

dt

with φ = φ|φ| (this ensures that X and φ have same sign). So fixing

initial energy scale of inflation is equivalent to fixing the radius ofthe circle R while varying X and Y uniformly.

Our analysis[Mishra, Sahni and Toporensky PRD 2018] is a natural andgeneralized extension of the original work by [Belinsky et. al 1985].


Methodology


Ne so

Ne =∫ teti

Hdt


R2 = X 2 + Y 2

where

R =√

6H

mp, X = φ

√2V (φ)

m2p

, Y =1

m2p

dφ

dt

with φ = φ|φ| (this ensures that X and φ have same sign). So fixing

initial energy scale of inflation is equivalent to fixing the radius ofthe circle R while varying X and Y uniformly. Our analysis[Mishra, Sahni and Toporensky PRD 2018] is a natural andgeneralized extension of the original work by [Belinsky et. al 1985].


Initial Conditions for Power-law Potentials: QuadraticChaotic

We begin explaining our methodology for power-law potentials throughthe example of simple Quadratic Chaotic Inflation

V (φ) =1

2m2φ2, m = 5.96× 10−6 mp

−80 −60 −40 −20 0 20 40 60 80

X = φ√2V (φ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dφ/dt

Initial EnergyHi = 3 × 10−3mp

×10−4

×10−4

Phase-space Analysis

−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20

X = φ√2V (φ)

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

Y=

dφ/dt

×10−4

×10−4

Zoomed View: AttractorSlow-roll Trajectories




V (φ) =1

2m2φ2, m = 5.96× 10−6 mp

−80 −60 −40 −20 0 20 40 60 80

X = φ√2V (φ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dφ/dt


×10−4

×10−4


−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20

X = φ√2V (φ)

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

Y=

dφ/dt

×10−4

×10−4





V (φ) =1

2m2φ2, m = 5.96× 10−6 mp

−80 −60 −40 −20 0 20 40 60 80

X = φ√2V (φ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dφ/dt


×10−4

×10−4


−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20

X = φ√2V (φ)

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

Y=

dφ/dt

×10−4

×10−4





V (φ) =1

2m2φ2, m = 5.96× 10−6 mp

−80 −60 −40 −20 0 20 40 60 80

X = φ√2V (φ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dφ/dt


×10−4

×10−4


−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20

X = φ√2V (φ)

−0.20

−0.15

−0.10

−0.05

0.00

0.05

0.10

0.15

0.20

Y=

dφ/dt

×10−4

×10−4




Range of Initial Conditions in theField Space

−30 −20 −10 0 10 20 30

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

V(φ

)/m

4 p

×10−8

φA−φA φB−φB

φ > 0φ < 0

Adequate Inflation

Inadequate Inflation

Marginally Adequate Inflation

Defining the ’Measure’

−80 −60 −40 −20 0 20 40 60 80

X = φ√2V (φ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dφ/dt

XA

XB

−XA

−XB

R

∆lB

∆lA

Hi (in mp) φA (in mp) φB (in mp) fA = 2 ∆lAl fB = 2 ∆lB

l

3× 10−3 11.22 19.55 5.80× 10−3 1.01× 10−2

3× 10−2 9.33 21.38 4.83× 10−4 1.11× 10−3

3× 10−1 7.47 23.27 3.86× 10−5 1.20× 10−4




−30 −20 −10 0 10 20 30

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

V(φ

)/m

4 p

×10−8

φA−φA φB−φB

φ > 0φ < 0

Adequate Inflation




−80 −60 −40 −20 0 20 40 60 80

X = φ√2V (φ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dφ/dt

XA

XB

−XA

−XB

R

∆lB

∆lA


l

3× 10−3 11.22 19.55 5.80× 10−3 1.01× 10−2

3× 10−2 9.33 21.38 4.83× 10−4 1.11× 10−3

3× 10−1 7.47 23.27 3.86× 10−5 1.20× 10−4




−30 −20 −10 0 10 20 30

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

V(φ

)/m

4 p

×10−8

φA−φA φB−φB

φ > 0φ < 0

Adequate Inflation




−80 −60 −40 −20 0 20 40 60 80

X = φ√2V (φ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dφ/dt

XA

XB

−XA

−XB

R

∆lB

∆lA


l

3× 10−3 11.22 19.55 5.80× 10−3 1.01× 10−2

3× 10−2 9.33 21.38 4.83× 10−4 1.11× 10−3

3× 10−1 7.47 23.27 3.86× 10−5 1.20× 10−4




−30 −20 −10 0 10 20 30

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

V(φ

)/m

4 p

×10−8

φA−φA φB−φB

φ > 0φ < 0

Adequate Inflation




−80 −60 −40 −20 0 20 40 60 80

X = φ√2V (φ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dφ/dt

XA

XB

−XA

−XB

R

∆lB

∆lA


l

3× 10−3 11.22 19.55 5.80× 10−3 1.01× 10−2

3× 10−2 9.33 21.38 4.83× 10−4 1.11× 10−3

3× 10−1 7.47 23.27 3.86× 10−5 1.20× 10−4


Initial Conditions for Power-law Potentials:Axion-Monodromy

Axion-Monodromy potential is given by[Silverstein, Westphal, McAllister2008-09]

V (φ) = V0

∣∣∣∣φ

mp

∣∣∣∣p

+ Λ4

(cos

φ

f− 1

)

Monotonicity ⇒

b

∣∣∣∣φ

mp

∣∣∣∣1−p

sinφ

f< 1 ,

where b = 1p

Λ4

V0

mp

f < 1. Consistencywith CMB observations imply b 1and f mp, so the oscillatory Axionicterm is insignificant. We haveconsidered two cases namelyp = 2/3, 1

Fractional Monodromy Inflationp = 2/3

−20 −15 −10 −5 0 5 10 15 20

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−9

φA φB−φA−φB

φ > 0φ < 0

Adequate Inflation



Range of Initial field values




V (φ) = V0

∣∣∣∣φ

mp

∣∣∣∣p

+ Λ4

(cos

φ

f− 1

)

Monotonicity ⇒

b

∣∣∣∣φ

mp

∣∣∣∣1−p

sinφ

f< 1 ,

where b = 1p

Λ4

V0

mp

f < 1.

Consistencywith CMB observations imply b 1and f mp, so the oscillatory Axionicterm is insignificant. We haveconsidered two cases namelyp = 2/3, 1


−20 −15 −10 −5 0 5 10 15 20

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−9

φA φB−φA−φB

φ > 0φ < 0

Adequate Inflation







V (φ) = V0

∣∣∣∣φ

mp

∣∣∣∣p

+ Λ4

(cos

φ

f− 1

)

Monotonicity ⇒

b

∣∣∣∣φ

mp

∣∣∣∣1−p

sinφ

f< 1 ,

where b = 1p

Λ4

V0

mp

f < 1. Consistencywith CMB observations imply b 1and f mp, so the oscillatory Axionicterm is insignificant.

We haveconsidered two cases namelyp = 2/3, 1


−20 −15 −10 −5 0 5 10 15 20

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−9

φA φB−φA−φB

φ > 0φ < 0

Adequate Inflation







V (φ) = V0

∣∣∣∣φ

mp

∣∣∣∣p

+ Λ4

(cos

φ

f− 1

)

Monotonicity ⇒

b

∣∣∣∣φ

mp

∣∣∣∣1−p

sinφ

f< 1 ,

where b = 1p

Λ4

V0

mp



−20 −15 −10 −5 0 5 10 15 20

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−9

φA φB−φA−φB

φ > 0φ < 0

Adequate Inflation







V (φ) = V0

∣∣∣∣φ

mp

∣∣∣∣p

+ Λ4

(cos

φ

f− 1

)

Monotonicity ⇒

b

∣∣∣∣φ

mp

∣∣∣∣1−p

sinφ

f< 1 ,

where b = 1p

Λ4

V0

mp



−20 −15 −10 −5 0 5 10 15 20

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−9

φA φB−φA−φB

φ > 0φ < 0

Adequate Inflation





Comparison of Power-law Potentials

fA = 2 ∆lAl

10−2 10−1

Hi/mp

10−4

10−3

2∆l A/l

V (φ) ∼ φ2

V (φ) ∼ |φ|V (φ) ∼ |φ|2/3

fB = 2 ∆lBl

10−2 10−1

Hi/mp

10−4

10−3

10−2

2∆l B

/l

V (φ) ∼ φ2

V (φ) ∼ |φ|V (φ) ∼ |φ|2/3

1 For Concave potentials, Inflation is more likely as compared toConvex potentials.

2 Degree of generality of inflation increases with higher initial energyscale Hi .

3 Results are not dependent on the Measure (small caveat).

But [Planck2018] favours Asymptotically Flat (Plateau-like)

potentials.



fA = 2 ∆lAl

10−2 10−1

Hi/mp

10−4

10−3

2∆l A/l

V (φ) ∼ φ2

V (φ) ∼ |φ|V (φ) ∼ |φ|2/3

fB = 2 ∆lBl

10−2 10−1

Hi/mp

10−4

10−3

10−2

2∆l B

/l

V (φ) ∼ φ2

V (φ) ∼ |φ|V (φ) ∼ |φ|2/3





potentials.



fA = 2 ∆lAl

10−2 10−1

Hi/mp

10−4

10−3

2∆l A/l

V (φ) ∼ φ2

V (φ) ∼ |φ|V (φ) ∼ |φ|2/3

fB = 2 ∆lBl

10−2 10−1

Hi/mp

10−4

10−3

10−2

2∆l B

/l

V (φ) ∼ φ2

V (φ) ∼ |φ|V (φ) ∼ |φ|2/3



3 Results are not dependent on the Measure

(small caveat).


potentials.



fA = 2 ∆lAl

10−2 10−1

Hi/mp

10−4

10−3

2∆l A/l

V (φ) ∼ φ2

V (φ) ∼ |φ|V (φ) ∼ |φ|2/3

fB = 2 ∆lBl

10−2 10−1

Hi/mp

10−4

10−3

10−2

2∆l B

/l

V (φ) ∼ φ2

V (φ) ∼ |φ|V (φ) ∼ |φ|2/3





potentials.Swagat Saurav Mishra, IUCAA, Pune Initial Conditions for Inflation 15/ 27

Asymptotically Flat (Plateau-like) Potentials

They typically possess one or two plateauwings.

Because of the extreme flatness of theplateau, they produce low tensor to scalarratio r .

Energy scale of observable inflation is alsocomparatively lower (Since scalar power isfixed, lower εH implies lower HI ).

Typical height of the plateauV0 ≤ 10−10 m4

p. And this is the mainsource of the recent intense debates relatedto the initial conditions for inflation.

Most of them are Physically wellmotivating along with a proper theory ofreheating. And they satisfy observationsextremely well !!

−5 0 5

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p

T-Model α-attractors, Higgs Inflation

0 2 4 6 8 10

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p

Starobinsky R + αR2









−5 0 5

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p


0 2 4 6 8 10

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p










−5 0 5

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p


0 2 4 6 8 10

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p








p.

And this is the mainsource of the recent intense debates relatedto the initial conditions for inflation.


−5 0 5

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p


0 2 4 6 8 10

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p










−5 0 5

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p


0 2 4 6 8 10

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p










−5 0 5

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p


0 2 4 6 8 10

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p










−5 0 5

φ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p


0 2 4 6 8 10

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

V0 ≃ 10−10 m4p



Standard Model Higgs Inflation

Attractive to use minimal degrees of freedom of nature.

Standard Model Higgs potential (Mexican Hat)

U(φ) =λ

4

(φ2 − σ2

)2

where λ = 0.1 and VEV of Higgs, σ = 246 GeV, is negligible duringinflation. So U(φ) ' λ

4φ4.

Higgs coupling λ is too high to produce small quantum fluctuationsduring inflation. And any way φ4 is ruled our by observation.

Modifications are required. In general there have been two types ofmodification of the inflationary scenario to get Inflation from SMHiggs.

1 Higgs field is non-minimally coupled to Gravity [Bezrukov,Shaposhnokov 2008].

2 Higgs field has a non-canonical Kinetic term [Unnikrishnan,Sahni JCAP 2012], [Mishra, Sahni and Toporensky PRD 2018].

In both cases Standard Model Higgs can act like the inflaton fieldgenerating necessary initial conditions for structure formation.





U(φ) =λ

4

(φ2 − σ2

)2


4φ4.










U(φ) =λ

4

(φ2 − σ2

)2


4φ4.










U(φ) =λ

4

(φ2 − σ2

)2


4φ4.










U(φ) =λ

4

(φ2 − σ2

)2


4φ4.


Modifications are required.

In general there have been two types ofmodification of the inflationary scenario to get Inflation from SMHiggs.








U(φ) =λ

4

(φ2 − σ2

)2


4φ4.










U(φ) =λ

4

(φ2 − σ2

)2


4φ4.










U(φ) =λ

4

(φ2 − σ2

)2


4φ4.










U(φ) =λ

4

(φ2 − σ2

)2


4φ4.







Non-minimal Higgs Inflation

Action in the Jordan frame

SJ =

∫d4x√−g[f (φ)R − 1

2gµν∂µφ∂νφ− U(φ)

]

where U(φ) = λ4

(φ2 − σ2

)2

, f (φ) = 12 (m2 + ξφ2) and m2 = m2

p − ξσ2

leads to

f (φ) ' 1

2(m2

p + ξφ2) =m2

p

2

(1 +

ξφ2

m2p

).

After doing a conformal transformation gµν −→ gµν = Ω2gµν with

Ω2 = 2m2

pf (φ) = 1 + ξφ2

m2p

and a field redefinition φ→ χ we end up at the

Einstein frame where action is given by

SE =

∫d4x

√−g[m2

p

2R − 1

2gµν∂µχ∂νχ− V (χ)

],

where

V (χ) =U[φ(χ)]

Ω4and

∂χ

∂φ= ± 1

Ω2

√Ω2 +

6ξ2φ2

m2p




SJ =

∫d4x√−g[f (φ)R − 1


]

where U(φ) = λ4

(φ2 − σ2

)2

, f (φ) = 12 (m2 + ξφ2) and m2 = m2

p − ξσ2

leads to

f (φ) ' 1

2(m2

p + ξφ2) =m2

p

2

(1 +

ξφ2

m2p

).


Ω2 = 2m2

pf (φ) = 1 + ξφ2

m2p



SE =

∫d4x

√−g[m2

p

2R − 1


],

where

V (χ) =U[φ(χ)]

Ω4and

∂χ

∂φ= ± 1

Ω2

√Ω2 +

6ξ2φ2

m2p




SJ =

∫d4x√−g[f (φ)R − 1


]

where U(φ) = λ4

(φ2 − σ2

)2

, f (φ) = 12 (m2 + ξφ2) and m2 = m2

p − ξσ2

leads to

f (φ) ' 1

2(m2

p + ξφ2) =m2

p

2

(1 +

ξφ2

m2p

).

After doing a conformal transformation gµν −→ gµν = Ω2gµν

with

Ω2 = 2m2

pf (φ) = 1 + ξφ2

m2p



SE =

∫d4x

√−g[m2

p

2R − 1


],

where

V (χ) =U[φ(χ)]

Ω4and

∂χ

∂φ= ± 1

Ω2

√Ω2 +

6ξ2φ2

m2p




SJ =

∫d4x√−g[f (φ)R − 1


]

where U(φ) = λ4

(φ2 − σ2

)2

, f (φ) = 12 (m2 + ξφ2) and m2 = m2

p − ξσ2

leads to

f (φ) ' 1

2(m2

p + ξφ2) =m2

p

2

(1 +

ξφ2

m2p

).


Ω2 = 2m2

pf (φ) = 1 + ξφ2

m2p



SE =

∫d4x

√−g[m2

p

2R − 1


],

where

V (χ) =U[φ(χ)]

Ω4and

∂χ

∂φ= ± 1

Ω2

√Ω2 +

6ξ2φ2

m2p




SJ =

∫d4x√−g[f (φ)R − 1


]

where U(φ) = λ4

(φ2 − σ2

)2

, f (φ) = 12 (m2 + ξφ2) and m2 = m2

p − ξσ2

leads to

f (φ) ' 1

2(m2

p + ξφ2) =m2

p

2

(1 +

ξφ2

m2p

).


Ω2 = 2m2

pf (φ) = 1 + ξφ2

m2p

and a field redefinition φ→ χ

we end up at the


SE =

∫d4x

√−g[m2

p

2R − 1


],

where

V (χ) =U[φ(χ)]

Ω4and

∂χ

∂φ= ± 1

Ω2

√Ω2 +

6ξ2φ2

m2p




SJ =

∫d4x√−g[f (φ)R − 1


]

where U(φ) = λ4

(φ2 − σ2

)2

, f (φ) = 12 (m2 + ξφ2) and m2 = m2

p − ξσ2

leads to

f (φ) ' 1

2(m2

p + ξφ2) =m2

p

2

(1 +

ξφ2

m2p

).


Ω2 = 2m2

pf (φ) = 1 + ξφ2

m2p



SE =

∫d4x

√−g[m2

p

2R − 1


],

where

V (χ) =U[φ(χ)]

Ω4and

∂χ

∂φ= ± 1

Ω2

√Ω2 +

6ξ2φ2

m2p




SJ =

∫d4x√−g[f (φ)R − 1


]

where U(φ) = λ4

(φ2 − σ2

)2

, f (φ) = 12 (m2 + ξφ2) and m2 = m2

p − ξσ2

leads to

f (φ) ' 1

2(m2

p + ξφ2) =m2

p

2

(1 +

ξφ2

m2p

).


Ω2 = 2m2

pf (φ) = 1 + ξφ2

m2p



SE =

∫d4x

√−g[m2

p

2R − 1


],

where

V (χ) =U[φ(χ)]

Ω4and

∂χ

∂φ= ± 1

Ω2

√Ω2 +

6ξ2φ2

m2p


Higgs Potential in the Einstein frame

Approximate form of the potential is givenby [Mishra, Sahni and Toporensky PRD2018]

V (χ) ' V0

(1− exp

[−√

2

3

|χ|mp

])2

where V0 =λm4

p

4ξ2 = 9.6× 10−11 m4p,

⇒ ξ = 1.6× 104

−20 −15 −10 −5 0 5 10 15 20

χ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

V(χ

)/m

4 p

×10−10

Exact Higgs Potential

Approximate Higgs Potential

−1.0 −0.5 0.0 0.5 1.0

X = χ√2V (χ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dχ/dt

XA−XA

×10−4

×10−4


Slanted trajectories in the

phase-space indicate that

starting from φ = 0, it is possible

to converge on to the inflationary

separatrix to obtain adequate

inflation.




V (χ) ' V0

(1− exp

[−√

2

3

|χ|mp

])2

where V0 =λm4

p

4ξ2 = 9.6× 10−11 m4p,

⇒ ξ = 1.6× 104

−20 −15 −10 −5 0 5 10 15 20

χ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

V(χ

)/m

4 p

×10−10



−1.0 −0.5 0.0 0.5 1.0

X = χ√2V (χ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dχ/dt

XA−XA

×10−4

×10−4







inflation.




V (χ) ' V0

(1− exp

[−√

2

3

|χ|mp

])2

where V0 =λm4

p

4ξ2 = 9.6× 10−11 m4p,

⇒ ξ = 1.6× 104

−20 −15 −10 −5 0 5 10 15 20

χ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

V(χ

)/m

4 p

×10−10



−1.0 −0.5 0.0 0.5 1.0

X = χ√2V (χ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dχ/dt

XA−XA

×10−4

×10−4







inflation.




V (χ) ' V0

(1− exp

[−√

2

3

|χ|mp

])2

where V0 =λm4

p

4ξ2 = 9.6× 10−11 m4p,

⇒ ξ = 1.6× 104

−20 −15 −10 −5 0 5 10 15 20

χ/mp

0.0

0.2

0.4

0.6

0.8

1.0

1.2

V(χ

)/m

4 p

×10−10



−1.0 −0.5 0.0 0.5 1.0

X = χ√2V (χ)

−80

−60

−40

−20

0

20

40

60

80

Y=

dχ/dt

XA−XA

×10−4

×10−4







inflation.Swagat Saurav Mishra, IUCAA, Pune Initial Conditions for Inflation 19/ 27

Range of initial field values

−20 −15 −10 −5 0 5 10 15 20

χ/mp

0.0

0.2

0.4

0.6

0.8

1.0

V(χ

)/m

4 p

×10−10

χB−χB

Adequate Inflation


−0.4 −0.2 0.0 0.2 0.40.0

0.2

0.4

0.6

0.8

1.0

1.2×10−11

χA−χA

Also applicable to the T -Model ofα-attractors [Kallosh, Linde andRoest JCAP(2013)]

V (φ) = V0 tanh2

(λφ

mp

)

with λ = 2√6α

Starobinsky Inflation: [Starobinsky 1980,Phys. Lett 91B]Jordan frame action

S =

∫d4x√−g

m2p

2

[R +

1

6m2R2

]

takes the form

SE =

∫d4x√−g[m2

p

2R−1

2gµν∂µφ∂νφ−V (φ)

]

with V (φ) = 34m

2m2p

(1− e

−√

23

φmp

)2

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

φCφB

A

B−

B+

C+C− D



Adequate Inflation



−20 −15 −10 −5 0 5 10 15 20

χ/mp

0.0

0.2

0.4

0.6

0.8

1.0

V(χ

)/m

4 p

×10−10

χB−χB

Adequate Inflation


−0.4 −0.2 0.0 0.2 0.40.0

0.2

0.4

0.6

0.8

1.0

1.2×10−11

χA−χA


V (φ) = V0 tanh2

(λφ

mp

)

with λ = 2√6α


S =

∫d4x√−g

m2p

2

[R +

1

6m2R2

]

takes the form

SE =

∫d4x√−g[m2

p

2R−1


]

with V (φ) = 34m

2m2p

(1− e

−√

23

φmp

)2

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

φCφB

A

B−

B+

C+C− D



Adequate Inflation



−20 −15 −10 −5 0 5 10 15 20

χ/mp

0.0

0.2

0.4

0.6

0.8

1.0

V(χ

)/m

4 p

×10−10

χB−χB

Adequate Inflation


−0.4 −0.2 0.0 0.2 0.40.0

0.2

0.4

0.6

0.8

1.0

1.2×10−11

χA−χA


V (φ) = V0 tanh2

(λφ

mp

)

with λ = 2√6α

Starobinsky Inflation: [Starobinsky 1980,Phys. Lett 91B]

Jordan frame action

S =

∫d4x√−g

m2p

2

[R +

1

6m2R2

]

takes the form

SE =

∫d4x√−g[m2

p

2R−1


]

with V (φ) = 34m

2m2p

(1− e

−√

23

φmp

)2

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

φCφB

A

B−

B+

C+C− D



Adequate Inflation



−20 −15 −10 −5 0 5 10 15 20

χ/mp

0.0

0.2

0.4

0.6

0.8

1.0

V(χ

)/m

4 p

×10−10

χB−χB

Adequate Inflation


−0.4 −0.2 0.0 0.2 0.40.0

0.2

0.4

0.6

0.8

1.0

1.2×10−11

χA−χA


V (φ) = V0 tanh2

(λφ

mp

)

with λ = 2√6α


S =

∫d4x√−g

m2p

2

[R +

1

6m2R2

]

takes the form

SE =

∫d4x√−g[m2

p

2R−1


]

with V (φ) = 34m

2m2p

(1− e

−√

23

φmp

)2

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

φCφB

A

B−

B+

C+C− D



Adequate Inflation



−20 −15 −10 −5 0 5 10 15 20

χ/mp

0.0

0.2

0.4

0.6

0.8

1.0

V(χ

)/m

4 p

×10−10

χB−χB

Adequate Inflation


−0.4 −0.2 0.0 0.2 0.40.0

0.2

0.4

0.6

0.8

1.0

1.2×10−11

χA−χA


V (φ) = V0 tanh2

(λφ

mp

)

with λ = 2√6α


S =

∫d4x√−g

m2p

2

[R +

1

6m2R2

]

takes the form

SE =

∫d4x√−g[m2

p

2R−1


]

with V (φ) = 34m

2m2p

(1− e

−√

23

φmp

)2

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

φ/mp

0.0

0.5

1.0

1.5

2.0

2.5

V(φ

)/m

4 p

×10−10

φCφB

A

B−

B+

C+C− D



Adequate Inflation


Asymptotically flat Potentials

Favoured by Observations.

Inflation can happenstarting from close to theminimum of the potential.

Large range of initial fieldvalues yield adequateinflation.

Difficulty in defining’measure’.

Dynamics is very differentfrom Power-law potentials.WHY!!

0 2 4 6 8 10 12 14

ϕi/mp

0

20

40

60

80

100

Ne

Worst Initial

Conditions

(ϕi < 0)

V (ϕ) = 12m2ϕ2

Asymptotically Flat

However Positive Spatial Curvature is a serious problem forAsymptotically Flat potentials [Steinhardt, Ijjas and Loeb 2014]








0 2 4 6 8 10 12 14

ϕi/mp

0

20

40

60

80

100

Ne

Worst Initial

Conditions

(ϕi < 0)

V (ϕ) = 12m2ϕ2

Asymptotically Flat









0 2 4 6 8 10 12 14

ϕi/mp

0

20

40

60

80

100

Ne

Worst Initial

Conditions

(ϕi < 0)

V (ϕ) = 12m2ϕ2

Asymptotically Flat









0 2 4 6 8 10 12 14

ϕi/mp

0

20

40

60

80

100

Ne

Worst Initial

Conditions

(ϕi < 0)

V (ϕ) = 12m2ϕ2

Asymptotically Flat









0 2 4 6 8 10 12 14

ϕi/mp

0

20

40

60

80

100

Ne

Worst Initial

Conditions

(ϕi < 0)

V (ϕ) = 12m2ϕ2

Asymptotically Flat








Dynamics is very differentfrom Power-law potentials.

WHY!!

0 2 4 6 8 10 12 14

ϕi/mp

0

20

40

60

80

100

Ne

Worst Initial

Conditions

(ϕi < 0)

V (ϕ) = 12m2ϕ2

Asymptotically Flat









0 2 4 6 8 10 12 14

ϕi/mp

0

20

40

60

80

100

Ne

Worst Initial

Conditions

(ϕi < 0)

V (ϕ) = 12m2ϕ2

Asymptotically Flat









0 2 4 6 8 10 12 14

ϕi/mp

0

20

40

60

80

100

Ne

Worst Initial

Conditions

(ϕi < 0)

V (ϕ) = 12m2ϕ2

Asymptotically Flat









0 2 4 6 8 10 12 14

ϕi/mp

0

20

40

60

80

100

Ne

Worst Initial

Conditions

(ϕi < 0)

V (ϕ) = 12m2ϕ2

Asymptotically Flat



Effect of Large Positive Spatial Curvature K = +1

Friedmann Equation

H2 =1

3m2p

(1

2φ2 + V (φ)

)− 1

a2

gives an upper bound on the potential

V (ϕ)

m4p

≤ 3

a2m2p

.

Turn over point is described by H = 0 and a = ab. Collapse happenswhen a < ab ( a

a < 0) while bounce happens when a > ab ( aa > 0) where

abmp =

√2

V (φ)/m4p

So starting from Hi = 0 at the Planck boundary ρφ = m4p, aimp =

√3,

we haveCollapse: V (ϕ)

m4p< 2

a2m2p

Bounce: 2a2m2

p< V (ϕ)

m4p≤ 3

a2m2p



Friedmann Equation

H2 =1

3m2p

(1

2φ2 + V (φ)

)− 1

a2


V (ϕ)

m4p

≤ 3

a2m2p

.



abmp =

√2

V (φ)/m4p


√3,


m4p< 2

a2m2p

Bounce: 2a2m2

p< V (ϕ)

m4p≤ 3

a2m2p



Friedmann Equation

H2 =1

3m2p

(1

2φ2 + V (φ)

)− 1

a2


V (ϕ)

m4p

≤ 3

a2m2p

.



abmp =

√2

V (φ)/m4p


√3,


m4p< 2

a2m2p

Bounce: 2a2m2

p< V (ϕ)

m4p≤ 3

a2m2p



Friedmann Equation

H2 =1

3m2p

(1

2φ2 + V (φ)

)− 1

a2


V (ϕ)

m4p

≤ 3

a2m2p

.



abmp =

√2

V (φ)/m4p


√3,


m4p< 2

a2m2p

Bounce: 2a2m2

p< V (ϕ)

m4p≤ 3

a2m2p



Friedmann Equation

H2 =1

3m2p

(1

2φ2 + V (φ)

)− 1

a2


V (ϕ)

m4p

≤ 3

a2m2p

.



abmp =

√2

V (φ)/m4p


√3,


m4p< 2

a2m2p

Bounce: 2a2m2

p< V (ϕ)

m4p≤ 3

a2m2p


Trouble with asymptotically flat potentials for K = +1

a/m−1p

φ/m

p

BounceCollapse

V (ϕ) = 12m2ϕ2

At Hi = 0, for any given initial valueof ρφ (hence a), there is a range ofvalues of φ for which universebounces back and inflation ispossible. For higher Hi > 0,probability of bounce increases(quantifiable).

a/m−1p

φ/m

p

Bounce

Colla

pse Asymptotically Flat Potential

amab

am =√

3V0/m4

p≃ √

3 × 105 m−1p

ab =√

2V0/m4

p≃ √

2 × 105 m−1p

For Hi = 0, Collapse is inevitable.While for Hi > 0, bounce becomespossible only if ai > 1600 ai whichputs Hi close to its flat universevalue upto 6 decimal places. VeryBad for Inflation

This is a serious issue and needs to be resolved! [Mishra, Sahni and

Toporensky (in preparation)]



a/m−1p

φ/m

p

BounceCollapse

V (ϕ) = 12m2ϕ2

At Hi = 0, for any given initial valueof ρφ (hence a), there is a range ofvalues of φ for which universebounces back and inflation ispossible.

For higher Hi > 0,probability of bounce increases(quantifiable).

a/m−1p

φ/m

p

Bounce

Colla


amab

am =√

3V0/m4

p≃ √

3 × 105 m−1p

ab =√

2V0/m4

p≃ √

2 × 105 m−1p






a/m−1p

φ/m

p

BounceCollapse

V (ϕ) = 12m2ϕ2


a/m−1p

φ/m

p

Bounce

Colla


amab

am =√

3V0/m4

p≃ √

3 × 105 m−1p

ab =√

2V0/m4

p≃ √

2 × 105 m−1p






a/m−1p

φ/m

p

BounceCollapse

V (ϕ) = 12m2ϕ2


a/m−1p

φ/m

p

Bounce

Colla


amab

am =√

3V0/m4

p≃ √

3 × 105 m−1p

ab =√

2V0/m4

p≃ √

2 × 105 m−1p






a/m−1p

φ/m

p

BounceCollapse

V (ϕ) = 12m2ϕ2


a/m−1p

φ/m

p

Bounce

Colla


amab

am =√

3V0/m4

p≃ √

3 × 105 m−1p

ab =√

2V0/m4

p≃ √

2 × 105 m−1p

For Hi = 0, Collapse is inevitable.While for Hi > 0, bounce becomespossible only if ai > 1600 ai whichputs Hi close to its flat universevalue upto 6 decimal places.

VeryBad for Inflation





a/m−1p

φ/m

p

BounceCollapse

V (ϕ) = 12m2ϕ2


a/m−1p

φ/m

p

Bounce

Colla


amab

am =√

3V0/m4

p≃ √

3 × 105 m−1p

ab =√

2V0/m4

p≃ √

2 × 105 m−1p





Possible Resolution

Consider the Margarita Potential[Bag, Mishra and Sahni JCAP 2017]

V (φ) = V0 tanh

(λ1φmp

)cosh

(λ2φ

mp

), λ1 > λ2

which has three asymptotes

Exponential wing V (ϕ) ' V0

2exp (λ2|ϕ|/mp) ,

|ϕ|mp 1

λ2,

Flat wing: V (ϕ) ' V0 +1

2m2

2ϕ2 ,

1

λ1 |ϕ|

mp 1

λ2,

Oscillatory region: V (ϕ) ' 1

2m2

1ϕ2 ,

|ϕ|mp 1

λ1,

where m21 =

2V0λ21

m2p

and m22 =

V0λ22

m2p

with m1 m2.

We need λ1 > λ2

and λ2 <√

2

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

ϕ/mp

V(ϕ

)

Flat Wing

Tracker Wing

OscillatoryRegion

V0

Margarita potential


Possible Resolution

Consider the Margarita Potential[Bag, Mishra and Sahni JCAP 2017]

V (φ) = V0 tanh

(λ1φmp

)cosh

(λ2φ

mp

), λ1 > λ2

which has three asymptotes

Exponential wing V (ϕ) ' V0

2exp (λ2|ϕ|/mp) ,

|ϕ|mp 1

λ2,

Flat wing: V (ϕ) ' V0 +1

2m2

2ϕ2 ,

1

λ1 |ϕ|

mp 1

λ2,

Oscillatory region: V (ϕ) ' 1

2m2

1ϕ2 ,

|ϕ|mp 1

λ1,

where m21 =

2V0λ21

m2p

and m22 =

V0λ22

m2p

with m1 m2. We need λ1 > λ2

and λ2 <√

2

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

ϕ/mp

V(ϕ

)

Flat Wing

Tracker Wing

OscillatoryRegion

V0

Margarita potential


Margarita Potential r vs nS

[Mishra, Sahni and Toporensky (in preparation)]


0.00

0.05

0.10

0.15

0.20

0.25

Ten

sor-to-sca

larratior

V (φ) = V0 tanh2 λ1φ

mpcosh λ2φ

mp


ConvexConcave

Power-law InflationMargarita λ2 = 0.04


Starobinsky

V ∝ |φ|2/3V ∝ |φ|V ∝ φ2

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02Scalar Spectral Index ns

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

Ten

sor-to-sca

larratior

Ne = 60V (φ) = V0 tanh

2 λ1φmp

cosh λ2φmp

BICEP-II + PLANCK

r ≤ 0.06

Power-law Inflationλ2 = 0.08

λ2 = 0.04

λ2 = 0.01



Margarita Potential r vs nS

[Mishra, Sahni and Toporensky (in preparation)]


0.00

0.05

0.10

0.15

0.20

0.25

Ten

sor-to-sca

larratior

V (φ) = V0 tanh2 λ1φ

mpcosh λ2φ

mp


ConvexConcave

Power-law InflationMargarita λ2 = 0.04


Starobinsky

V ∝ |φ|2/3V ∝ |φ|V ∝ φ2

0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02Scalar Spectral Index ns

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

Ten

sor-to-sca

larratior

Ne = 60V (φ) = V0 tanh

2 λ1φmp

cosh λ2φmp

BICEP-II + PLANCK

r ≤ 0.06

Power-law Inflationλ2 = 0.08

λ2 = 0.04

λ2 = 0.01



Concluding Remarks: Take Office MessagesInitial Conditions for inflation is an important issue to beconsidered.

Inflationary phase (slow-roll trajectory) is a local attractor for largefield models.Planck 2018 data prefers/favours concave potentials which areplateau-like.For Plateau-like potentials, inflation can happen starting from largerset of initial conditions as compared to power-law potentials.Inflation can happen even starting from close to the minimum of thepotential.Positive spatial curvature is a serious problem for plateau-likepotentials and needs close attention.Possible modifications of the potential is required for large fieldvalues (or multi-field inflation is desirable.).Plateau-potentials with an initial exponential-type modification hasadvantages both theoretically and observationally.Initial conditions should also be thoroughly analysed for alternativesto Inflation like String Gas Cosmology, Matter Bounce Models,Ekpyrotic Models, Emergent Scenarios etc.


Concluding Remarks: Take Office MessagesInitial Conditions for inflation is an important issue to beconsidered.Inflationary phase (slow-roll trajectory) is a local attractor for largefield models.

Planck 2018 data prefers/favours concave potentials which areplateau-like.For Plateau-like potentials, inflation can happen starting from largerset of initial conditions as compared to power-law potentials.Inflation can happen even starting from close to the minimum of thepotential.Positive spatial curvature is a serious problem for plateau-likepotentials and needs close attention.Possible modifications of the potential is required for large fieldvalues (or multi-field inflation is desirable.).Plateau-potentials with an initial exponential-type modification hasadvantages both theoretically and observationally.Initial conditions should also be thoroughly analysed for alternativesto Inflation like String Gas Cosmology, Matter Bounce Models,Ekpyrotic Models, Emergent Scenarios etc.


Concluding Remarks: Take Office MessagesInitial Conditions for inflation is an important issue to beconsidered.Inflationary phase (slow-roll trajectory) is a local attractor for largefield models.Planck 2018 data prefers/favours concave potentials which areplateau-like.

For Plateau-like potentials, inflation can happen starting from largerset of initial conditions as compared to power-law potentials.Inflation can happen even starting from close to the minimum of thepotential.Positive spatial curvature is a serious problem for plateau-likepotentials and needs close attention.Possible modifications of the potential is required for large fieldvalues (or multi-field inflation is desirable.).Plateau-potentials with an initial exponential-type modification hasadvantages both theoretically and observationally.Initial conditions should also be thoroughly analysed for alternativesto Inflation like String Gas Cosmology, Matter Bounce Models,Ekpyrotic Models, Emergent Scenarios etc.


Concluding Remarks: Take Office MessagesInitial Conditions for inflation is an important issue to beconsidered.Inflationary phase (slow-roll trajectory) is a local attractor for largefield models.Planck 2018 data prefers/favours concave potentials which areplateau-like.For Plateau-like potentials, inflation can happen starting from largerset of initial conditions as compared to power-law potentials.Inflation can happen even starting from close to the minimum of thepotential.

Positive spatial curvature is a serious problem for plateau-likepotentials and needs close attention.Possible modifications of the potential is required for large fieldvalues (or multi-field inflation is desirable.).Plateau-potentials with an initial exponential-type modification hasadvantages both theoretically and observationally.Initial conditions should also be thoroughly analysed for alternativesto Inflation like String Gas Cosmology, Matter Bounce Models,Ekpyrotic Models, Emergent Scenarios etc.


Concluding Remarks: Take Office MessagesInitial Conditions for inflation is an important issue to beconsidered.Inflationary phase (slow-roll trajectory) is a local attractor for largefield models.Planck 2018 data prefers/favours concave potentials which areplateau-like.For Plateau-like potentials, inflation can happen starting from largerset of initial conditions as compared to power-law potentials.Inflation can happen even starting from close to the minimum of thepotential.Positive spatial curvature is a serious problem for plateau-likepotentials and needs close attention.

Possible modifications of the potential is required for large fieldvalues (or multi-field inflation is desirable.).Plateau-potentials with an initial exponential-type modification hasadvantages both theoretically and observationally.Initial conditions should also be thoroughly analysed for alternativesto Inflation like String Gas Cosmology, Matter Bounce Models,Ekpyrotic Models, Emergent Scenarios etc.


Concluding Remarks: Take Office MessagesInitial Conditions for inflation is an important issue to beconsidered.Inflationary phase (slow-roll trajectory) is a local attractor for largefield models.Planck 2018 data prefers/favours concave potentials which areplateau-like.For Plateau-like potentials, inflation can happen starting from largerset of initial conditions as compared to power-law potentials.Inflation can happen even starting from close to the minimum of thepotential.Positive spatial curvature is a serious problem for plateau-likepotentials and needs close attention.Possible modifications of the potential is required for large fieldvalues (or multi-field inflation is desirable.).

Plateau-potentials with an initial exponential-type modification hasadvantages both theoretically and observationally.Initial conditions should also be thoroughly analysed for alternativesto Inflation like String Gas Cosmology, Matter Bounce Models,Ekpyrotic Models, Emergent Scenarios etc.


Concluding Remarks: Take Office MessagesInitial Conditions for inflation is an important issue to beconsidered.Inflationary phase (slow-roll trajectory) is a local attractor for largefield models.Planck 2018 data prefers/favours concave potentials which areplateau-like.For Plateau-like potentials, inflation can happen starting from largerset of initial conditions as compared to power-law potentials.Inflation can happen even starting from close to the minimum of thepotential.Positive spatial curvature is a serious problem for plateau-likepotentials and needs close attention.Possible modifications of the potential is required for large fieldvalues (or multi-field inflation is desirable.).Plateau-potentials with an initial exponential-type modification hasadvantages both theoretically and observationally.

Initial conditions should also be thoroughly analysed for alternativesto Inflation like String Gas Cosmology, Matter Bounce Models,Ekpyrotic Models, Emergent Scenarios etc.


Concluding Remarks: Take Office MessagesInitial Conditions for inflation is an important issue to beconsidered.Inflationary phase (slow-roll trajectory) is a local attractor for largefield models.Planck 2018 data prefers/favours concave potentials which areplateau-like.For Plateau-like potentials, inflation can happen starting from largerset of initial conditions as compared to power-law potentials.Inflation can happen even starting from close to the minimum of thepotential.Positive spatial curvature is a serious problem for plateau-likepotentials and needs close attention.Possible modifications of the potential is required for large fieldvalues (or multi-field inflation is desirable.).Plateau-potentials with an initial exponential-type modification hasadvantages both theoretically and observationally.Initial conditions should also be thoroughly analysed for alternativesto Inflation like String Gas Cosmology, Matter Bounce Models,Ekpyrotic Models, Emergent Scenarios etc.


References

1 Initial Conditions for Inflation in an FRW Universe, Mishra,Sahni and Toporensky, PRD 98, 083538(2018)[arXiv:1801:04948]

2 Initial Conditions for inflation: A short Review, RobertBrandenberger, Int. J. Mod. Phys. D 26, 1740002(2017)

3 Beginning of Inflation in an inhomogeneous universe, East,Kleban, Linde, Senatore, JCAP 09, (2016) 010

4 Guth and Nomura, Phys. Lett. B 733, 112 (2014)

5 Carrasco, Kallosh and Linde, PRD 92, 063519(2015)

6 Initial Conditions for inflation in an FRW Universe - II Mishra,Sahni and Toporensky (in preparation).


Documents

Initial Conditions for Inflation in an FRW Universe @ …sriram/professional/events/... · 2019. 1. 29. · Why bother about for Initial Conditions! 1 Traditional Physics aims at