(Ultra-light) Cold Dark Matter from - attractors @ @let ...€¦ · Swagat Saurav Mishra, IUCAA,...

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(Ultra-light) Cold Dark Matter from α- attractors @ Post-Planck Cosmology 2017 Swagat Saurav Mishra (SRF CSIR) Inter University Centre for Astronomy and Astrophysics —————– Ph.D. supervisor: Prof. Varun Sahni (IUCAA) ————— Other Collaborators: Yuri Shtanov(BITP), Aleksey Toporensky (Moscow State University), Satadru Bag(IUCAA) October, 2017 Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α-attractors 1/ 2

Transcript of (Ultra-light) Cold Dark Matter from - attractors @ @let ...€¦ · Swagat Saurav Mishra, IUCAA,...

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(Ultra-light) Cold Dark Matter from α−attractors

@ Post-Planck Cosmology 2017

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

—————–Ph.D. supervisor: Prof. Varun Sahni (IUCAA)

—————Other Collaborators: Yuri Shtanov(BITP), Aleksey

Toporensky (Moscow State University), SatadruBag(IUCAA)

October, 2017Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 1/ 26

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Evidences for the Existence of Dark Matter

Observations of the large scale structure, CMB, gravitationallensing in galaxy clusters, missing mass in galaxy clusters, flatrotation curves of galaxies (and luminous mass distribution inthe Bullet Cluster) strongly favor the existence of Dark Matterin the universe.

Most of the beyond standard model particle physics theoriespredict the existence of very weakly interacting MASSIVE andULTRA-LIGHT particles.

Standard Scenario CDM : WIMPs : Sub-structure Problem??Alternatives: Warm Dark Matter, CDM from Ultra-lightscalars..Theme: Initial Conditions for scalar field models of Dark

Matter.

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Coherent Oscillations of a Scalar Field

Action for a canonical scalar field minimally coupled to gravity

S [ϕ] = −∫

d4x√−g(

1

2gµν∂µϕ∂νϕ+ V (ϕ)

)(1)

The equation of state (EOS) parameter is

wϕ =pϕρϕ

=12 ϕ

2 − V (ϕ)12 ϕ

2 + V (ϕ)(2)

The equation of motion of the scalar field is given by

ϕ+ 3Hϕ+ V ′(ϕ) = 0. (3)

For a coherently oscillations (ϕ/ϕ H) around V (ϕ) ∼ ϕ2p, thetime average EOS is [Turner 1983]

〈wϕ〉 =p − 1

p + 1(4)

Hence a scalar field oscillating around the minimum of any V (φ)having a ϕ2 asymptote behaves like Dark Matter (DM).

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Basic motivation

Moving past the dominant paradigm of particle-like WIMPDM.

DM from V (ϕ) = 12m

2ϕ2 potential can have a large Jeanslength (a Macroscopic deBroglie Wave Length) (called’fuzzy’ dark matter) which could resolve the cusp–core andsub-structure problems faced by standard cold dark matter.[(Hu, Barkana, Gruzinov 2000), (Sahni and Wang 2000),(Hui, Ostriker, Tremaine and Witten 2017)].

Emphasizing and removing enormous fine-tuning of initialconditions faced by the m2ϕ2 potential.

α-attractors, originally proposed by [(Kallosh and Linde,2013a, 2013b)] in the context of cosmic inflation, can havewider appeal in describing DM [ Mishra, Sahni and Shtanov,JCAP 2017 [arXiv:1703.03295]] (and even DE Bag,Mishra and Sahni 2017 [arXiv:1709.09193] submitted).

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Dark Matter

For the canonical massive scalar field potential

V (ϕ) =1

2m2ϕ2 , (5)

the expression for Jeans length is [Khlopov, Malomed andZeldovich 1985; Hu, Barkana, Gruzinov 2000]

λJ = π3/4(Gρ)−1/4m−1/2 . (6)

An oscillating scalar field with an ultra-light mass of 10−22 eVwould therefore have a Jeans length of a few kiloparsec (hencecalled ’fuzzy DM’) which can successfully resolve several keyproblems faced by the standard cold dark matter scenario includingthe cusp–core issue, the problem of substructure etc.However such a model of dark matter requires an extremefine-tuning of initial conditions which we consider to be a seriousproblem.

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Fine-tuning problem associated with V (ϕ) = 12m

2ϕ2

The canonical potential related with scalar field dark matter isgiven by

V =1

2m2ϕ2 . (7)

For small values, ϕ mp, the scalar field oscillates about theminimum of its potential for m ≥ H and behaves like dark matter.The scalar field equation of motion is

ϕ+ 3Hϕ+ V ′(ϕ) = 0 . (8)

During the radiation dominated epoch, since Hubble expansion isdominated by the radiation density, the scalar field experiencesenormous damping as it attempts to roll down its potential (7) andhence begins to play the role of a cosmological constant until theHubble parameter H falls below the mass of the scalar field andoscillations begin.

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Fine-tuning problem associated with V (ϕ) = 12m

2ϕ2

−10 −8 −6 −4 −2 0

log10(a) −→

−50

−45

−40

−35

−30

−25

−20

−15

−10log10(ρ

/GeV

4)−→

A

BC

Ω0m = 0.27

Inflation

Transient Inflation

Insufficient Dark Matter

RadiationScalar FieldΛ

m = 10−22 eVSwagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 7/ 26

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Fine-tuning problem associated with V (ϕ) = 12m

2ϕ2

The tendency of ρϕ to behave like a cosmological constant deepwithin the radiative regime enormously influences the kind of initialconditions that need to be imposed on the scalar field in order thatthe model universe resemble ours (i.e., with Ω0,DE ' 0.69,Ω0m ' 0.27, Ω0b ' 0.04, and Ω0r ' 10−4).

We therefore find that the potential (7) suffers from a severefine-tuning problem since, for any given value of m, there is only avery narrow range of ϕi which will lead to currently admissiblevalues of the matter density Ω0m. These results support the earlierfindings of [Zlatev and Steinhardt 1999].

The same is true for our good-old favourite AXIONS (and in factthe problem is even more severe with String Theory Axions) and infact for any thawing potentials.

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Relevant α-attractor potentials for dark matter and darkenergy

The general form of α-attractor potentials can be written as[Kallosh and Linde 2013b]

V (ϕ) = m4pF

(tanh

ϕ√6αmp

). (9)

In this context, we consider the following two important potentials.

1 The asymmetric E-Model [Kallosh and linde 2013a]

V (ϕ) = V0

(1− e

−√

23α

ϕmp

)2

. (10)

2 The tracker-potential [Sahni and Wang, 2000]

V (ϕ) = V0 sinh2

√2

ϕ

mp. (11)

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Dark Matter from the E-Model

It is interesting that the extreme fine-tuning problem faced by the12m

2ϕ2 potential is easily alleviated if dark matter is based on theE-model. In this case, one can write down the potential (10) in theform

V (ϕ) = V0

(1− e

−λ ϕmp

)2, (12)

which closely resembles the Starobinsky model for inflation[Starobinsky, 1980], where the flat wing of the potential sustainsinflation since the tracker wing is much too steep to causeaccelerated expansion. For dark matter, on the other hand, it is

the steep wing with V ∼ e2λ |ϕ|

mp that is more useful

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Dark Matter from the E-Model

0 2 4 6 8

ϕ/mp −→0

1

2

3

4

5

6

V(ϕ

)/m

4 p−→

Flat Wing (B)

Tracker Wing (A)

Oscillatory Region〈wϕ〉 ≃ 0

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Dark Matter from the E-Model

The potential (12) exhibits three asymptotic branches :

Tracker wing: V (ϕ) ' V0 e2λ|ϕ|/mp , ϕ < 0 , λ|ϕ| mp ,(13)

Flat wing: V (ϕ) ' V0 , λϕ mp , (14)

Oscillatory region: V (ϕ) ' 1

2m2ϕ2 , λ|ϕ| mp , (15)

where

m2 =2V0λ

2

m2p

. (16)

We note that the tracker parameter λ in the potential (12) isrelated to the geometric parameter α in (10) by

λ =

√2

3α. (17)

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Dark Matter from the E-Model

−10 −8 −6 −4 −2 0

log10(a) −→

−50

−45

−40

−35

−30

−25

−20

−15

−10

log10(ρ

/GeV

4)−→

P1

P2

P3

A

Ω0m = 0.27

RadiationScalar FieldΛ

λ = 14.5, V0 = 1.37× 10−28 GeV 4, zosc ' 2.8× 106

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V0, λ parameter space

−30 −29 −28 −27 −26 −25 −24 −23 −22

log10

(V0/GeV4

) −→1.0

1.5

2.0

2.5

3.0log10(λ

)−→

zosc ≃ 2.8× 106

zosc ≃ 8.7× 106

zosc ≃ 2.8× 107

zosc ≃ 8.7× 107

Ω0m=

0.30

Ω0m=

0.24

m = 10−22 eV

m = 10−21 eV

m = 10−20 eV

m = 10−19 eV

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Gravitational Instability

1 For the canonical potential V (ϕ) = 12m

2ϕ2, the Jeanswavenumber is given by [Khlopov, Malomed and Zeldovich1985; Hu, Barkana and Gruzinov 2000]

k2J =√

2ρm

mp. (18)

Where ρ = 12m

2ϕ20 and ϕ0 is the amplitude of coherent

oscillations.2 For the tracker-potential (11), the Jeans scale is given by

[Johnson and Kamionkowski 2008]

k2J = −3

2λ2ρ+

[(3

2λ2ρ

)2

+ ρm2

m2p

]1/2, (19)

where m2 = 2V0λ2

m2p

.

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Gravitational Instability

1 For small oscillations around the minimum of the asymmetricE-Model potential (12), the Jeans scale is given by [Mishra,Sahni and Shtanov JCAP 2017]

k2J =

(5

3

µ2ρ

m4− 9

4

λ0ρ

m2

)+

[2m2

m2p

ρ+

(25

9

µ4

m8+

81

16

λ02

m4− 15

2

λ0µ2

m6

)ρ2] 1

2

.(20)

Where m2 = 2V0λ2

m2p

, µ = 3λ3V0m3

pand λ0 = 7

3λ4V0m4

p.

The figure below shows that k2J in all three models converge to theexpression (18) at late times by z ∼ 103.

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Gravitational Instability

−6 −5 −4 −3 −2 −1 0log10(a) −→

−74

−72

−70

−68

−66

−64

−62log10(k

2 J/GeV

2)−→

m2ϕ2

TrackerE-model

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Discussion and further plans

Canonical scalar field potential for DM V (ϕ) ' 12m

2ϕ2 suffersfrom severe fine-tuning problem of initial conditions. (Thisproblem also appears in axionic dark matter.)

This difficulty is easily avoided if dark matter is sourced byα-attractors possessing at least one tracker wing like theE-model and the tracker potential where one arrives at thelate-time dark matter asymptote from a very wide range ofinitial conditions.

Our analysis of gravitational instability demonstrates that,despite significant differences in dynamics, the Jeans scale inall of our DM models converges to the same late-timeexpression.

Observational Signatures - CMB BAO, Pulsar Timing Array,Gravitational Waves. (Future Work).

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Important References

1 ”Cold and fuzzy dark matter”, W. Hu, R. Barkana and A.Gruzinov, Phys. Rev. Lett. 85 (2000) 1158[astro-ph/0003365].

2 ”Ultralight scalars as cosmological dark matter”, L. Hui,Ostriker, S. Tremaine and Ed. Witten, Phys. Rev. D 95(2017) 043541 [arXiv:1610.08297].

3 ”Sourcing DM and DE from α-attractors”, S.S Mishra, V.Sahni and Y. Shtanov, JCAP 1706 (2017) no.06, 045[arXiv:1703.03295]

4 ”Axion Cosmology”, DJE Marsh, Phys.Rept. 643 (2016) 1-79[arXiv:1510.07633].

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Dark Energy from α-attractors

In addition to the dark matter models alluded to above, we havealso discovered 4 new Tracker Models of Dark Energy (Bag,Mishra and Sahni 2017 [arXiv:1709.09193] submitted) andexplored the possibility that these models give rise to an equationof state close to −1 at the present epoch, as demanded byobservations.

0.0 0.1 0.2 0.3 0.4 0.5

ϕ/mp

V(ϕ

)

Flat Wing

Tracker Wing

V0

L-Model

(a)

−3 −2 −1 0 1 2 3

ϕ/mp

V(ϕ

)Oscillatory Region

Tracker Wing

V0

Oscillatory Tracker Model

(b)

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Dark Energy from α-attractors

V (ϕ) = V0 cothp(ϕmp

).

10−6 10−5 10−4 10−3 10−2 10−1 100

a/a0

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

V (ϕ) ∝ coth(ϕ/mp)

V (ϕ) ∝ coth6(ϕ/mp)

V (ϕ) ∝ 1/ϕ

V (ϕ) ∝ 1/ϕ6

(c)

10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100

a/a0

10−5

10−1

103

107

1011

1015

1019

1023

1027

1031

1035

1039

1043

ρ/ρcr,0

RadiationMatterTrackerOvershootUndershoot

(d)

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Dark Energy from α-attractors

V (ϕ) = V0 cosh(λ ϕmp

).

10−7 10−6 10−5 10−4 10−3 10−2 10−1 100

a/a0

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

0.4

wϕ = 1/3

wϕ = 0

λ = 5

λ = 30

(e)

0.00.51.01.52.02.53.0

z

−1.0 −1.0

−0.9 −0.9

−0.8 −0.8

−0.7 −0.7

−0.6 −0.6

−0.5 −0.5

−0.4 −0.4

−0.3 −0.3

−0.2 −0.2

λ = 5

λ = 10

λ = 30

(f)

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Dark Energy from α-attractors

V (ϕ) = V0 cosh(λ ϕmp

).

10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100

a/a0

10−5

10−1

103

107

1011

1015

1019

1023

1027

1031

1035

1039

1043

1047

1051ρ/ρcr,0

P3

P2

P1 RadiationMatterTrackerOvershootUndershoot

Reference: Bag, Mishra and Sahni 2017 [arXiv:1709.09193](submitted).

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Extra

−15 −14 −13 −12 −11 −10 −9 −8

log10(a) −→−30

−25

−20

−15

−10

−5

0

5

10log10(ρ

/GeV

4)−→

Radiation

ϕi 2=10 30×

V(ϕ

i )

ϕi 2=10 20×

V(ϕ

i )

ϕi 2=

10 10×V (ϕ

i)

ϕi = 0

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Extra

−22 −21 −20 −19 −18

log10(m/eV) −→

−2.2

−2.0

−1.8

−1.6

−1.4

−1.2

log10(ϕ

i/m

p)−→

Ω0m = 0.24

Ω0m = 0.27

Ω0m = 0.30

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−0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0

ϕ/mp

V(ϕ

)

Flat Wing

Tracker Wing

V0

Recliner potential

(g)

−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

(h)

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