# (Ultra-light) Cold Dark Matter from - attractors @ @let ... Swagat Saurav Mishra, IUCAA, Pune...

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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/ 26

Evidences for the Existence of Dark Matter

Observations of the large scale structure, CMB, gravitational lensing in galaxy clusters, missing mass in galaxy clusters, flat rotation curves of galaxies (and luminous mass distribution in the Bullet Cluster) strongly favor the existence of Dark Matter in the universe.

Most of the beyond standard model particle physics theories predict the existence of very weakly interacting MASSIVE and ULTRA-LIGHT particles.

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

Matter.

Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 2/ 26

Coherent Oscillations of a Scalar Field

Action for a canonical scalar field minimally coupled to gravity

S [ϕ] = − ∫

d4x √ −g (

1

2 gµν∂µϕ∂νϕ+ V (ϕ)

) (1)

The equation of state (EOS) parameter is

wϕ = pϕ ρϕ

= 1 2 ϕ̇

2 − V (ϕ) 1 2 ϕ̇

2 + V (ϕ) (2)

The equation of motion of the scalar field is given by

ϕ̈+ 3Hϕ̇+ V ′(ϕ) = 0. (3)

For a coherently oscillations (ϕ̇/ϕ� H) around V (ϕ) ∼ ϕ2p, the time 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).

Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 3/ 26

Basic motivation

Moving past the dominant paradigm of particle-like WIMP DM.

DM from V (ϕ) = 12m 2ϕ2 potential can have a large Jeans

length (a Macroscopic deBroglie Wave Length) (called ’fuzzy’ dark matter) which could resolve the cusp–core and sub-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 initial conditions faced by the m2ϕ2 potential.

α-attractors, originally proposed by [(Kallosh and Linde, 2013a, 2013b)] in the context of cosmic inflation, can have wider 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).

Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 4/ 26

Dark Matter

For the canonical massive scalar field potential

V (ϕ) = 1

2 m2ϕ2 , (5)

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

λJ = π 3/4(Gρ)−1/4m−1/2 . (6)

An oscillating scalar field with an ultra-light mass of 10−22 eV would therefore have a Jeans length of a few kiloparsec (hence called ’fuzzy DM’) which can successfully resolve several key problems faced by the standard cold dark matter scenario including the cusp–core issue, the problem of substructure etc. However such a model of dark matter requires an extreme fine-tuning of initial conditions which we consider to be a serious problem.

Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 5/ 26

Fine-tuning problem associated with V (ϕ) = 12m 2ϕ2

The canonical potential related with scalar field dark matter is given by

V = 1

2 m2ϕ2 . (7)

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

ϕ̈+ 3Hϕ̇+ V ′(ϕ) = 0 . (8)

During the radiation dominated epoch, since Hubble expansion is dominated by the radiation density, the scalar field experiences enormous damping as it attempts to roll down its potential (7) and hence begins to play the role of a cosmological constant until the Hubble parameter H falls below the mass of the scalar field and oscillations begin.

Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 6/ 26

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

−10 lo g 1 0 (ρ

/ G e V

4 ) −→

A

B C

Ω0m = 0.27

Inflation

Transient Inflation

Insufficient Dark Matter

Radiation Scalar Field Λ

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

Fine-tuning problem associated with V (ϕ) = 12m 2ϕ2

The tendency of ρϕ to behave like a cosmological constant deep within the radiative regime enormously influences the kind of initial conditions that need to be imposed on the scalar field in order that the 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 severe fine-tuning problem since, for any given value of m, there is only a very narrow range of ϕi which will lead to currently admissible values of the matter density Ω0m. These results support the earlier findings of [Zlatev and Steinhardt 1999].

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

Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 8/ 26

Relevant α-attractor potentials for dark matter and dark energy

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−

√ 2 3α

ϕ mp

)2 . (10)

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

V (ϕ) = V0 sinh 2

√ 2

3α

ϕ

mp . (11)

Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 9/ 26

Dark Matter from the E-Model

It is interesting that the extreme fine-tuning problem faced by the 1 2m

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

V (ϕ) = V0 (

1− e−λ ϕ mp

)2 , (12)

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

the steep wing with V ∼ e2λ |ϕ| mp that is more useful

Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 10/ 26

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

Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 11/ 26

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 2 m2ϕ2 , λ|ϕ| � mp , (15)

where

m2 = 2V0λ

2

m2p . (16)

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

λ =

√ 2

3α . (17)

Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 12/ 26

Dark Matter from the E-Model

−10 −8 −6 −4 −2 0 log10(a) −→

−50

−45

−40

−35

−30

−25

−20

−15

−10 lo g 1 0 (ρ

/ G e V

4 ) −→

P1

P2

P3

A

Ω0m = 0.27

Radiation Scalar Field Λ

λ = 14.5, V0 = 1.37× 10−28 GeV 4, zosc ' 2.8× 106 Swagat Saurav Mishra, IUCAA, Pune (Ultra-light) Cold Dark Matter from α−attractors 13/ 26

V0, λ parameter space

−30 −29 −28 −27 −26 −25 −24 −23 −22 log10

( V0/GeV

4 ) −→

1.0

1.5

2.0

2.5

3.0 lo g 1 0 (λ

) −→

zosc ≃ 2.8× 106

zosc ≃ 8.7× 106

zosc ≃ 2.8× 107

zosc ≃ 8.7× 107

Ω0 m =

0.3 0

Ω0 m =

0.2 4

m = 10−22 eV m = 10−21 eV m = 10−20 eV m = 10−1

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