Lecture 1:Basics of dark energy

Shinji Tsujikawa

(Tokyo University

of Science)

Outline of lectures

n Letcure 1: Basics of dark energyn Letcure 2: Observational constraints

on dark energy (SN Ia, CMB, BAO)n Lecture 3: Modified matter models of

dark energyn Lecture 4: Modified gravity models of

dark energy

1. E. Copeland, M. Sami, and S. Tsujikawa, ``Dynamics of dark energy’’, IJMPD, 1753 (2006), hep-th/0603057

2. L. Amendola and S. Tsujikawa, ``Dark energy—Theory and observations’’,Cambridge University Press (2010)

3. R. Kase and S. Tsujikawa,``Dark energy in Horndeski theories after

GW170817: A review’’,IJMPD, D28, 1942005 (2006), 1809.08735 [gr-qc]

Suggested readings

Dark energy From the observations of SN Ia, CMB, and BAO etc, about 70 % of the energy density of the Universe is dark energy responsible for cosmic acceleration.

Today’s energy budget

68 %: Dark Energy: Negative pressure (origin is unknowm)

27%: Dark Matter: Pressureless dust (origin is unknown)

Responsible for cosmic acceleration.

Responsible for the growth of large-scale structuresdue to gravitational clustering.

5%: Atoms (baryons)

Responsible for our existence!

0.01 %: Radiation

Remnants of black body radiation

Einstein field equationsIn order to know the expansion history of the Universe, we needto solve the Einstein equation

_____ ____Einsteintensor

Energy momentumtensor

For a given metric

we can evaluate

In the unit

of c = 1

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Geometry Matter

Friedmann-Lemaitre-Robertson-Walker backgroundThe metric in the homogenous and isotropic (FLRW) background is

l The non-vanishing components of the Einstein tensors are

H = a/a

l The energy-momentum tensor for perfect fluid is

is the Hubble parameter

(energy density)

(Pressure)No off-diagonal components on the FLRW background

K: spatial curvature.

K = 0: flat, K > 0: closed, K < 0: open

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Friedmann equationsIn the homogenous and isotropic background we have

Eliminating the curvature term, we obtain

Cosmic deceleration (a < 0) for � + 3P > 0

Cosmic acceleration (a > 0) for � + 3P < 0(negative pressure P)

Combining the above equations, we also have

� + 3H(� + P ) = 0 (continuity equation)

Dark energy: Negative pressureEquation of state :

Friedmann equation:

Continuity equation:

Cosmic acceleration

Exponential expansion:

w � P/� Negative

In the flat Universe (K=0) we have

For constant w, the solutions are

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Cosmological constant: ⇢ =constant

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Observational evidence for dark energy 1. Age of the Universe: early 1990’s

The age of the Universe must be larger than those of globular clusters.2. Number count of galaxies: early 1990’s

Fukugita et al.3. Supernovae type Ia (SN Ia): 1998~

Perlmutter, Riess, Schmidt (2011, Nobel prize)4. Cosmic Microwave Background (CMB):

1992~ (WMAP: 2003~, Planck: 2013~)Mather and Smoot (2006, Nobel prize) : COBE satelliteSpergel et al, Ade et al, … : WMAP, Planck satellites

5. Baryon Acoustic Oscillations (BAO): 2005~Eisenstein et al,..

6. Large-scale structure (LSS), redshift space distortions: 1999~ Tegmark et al,…

7. Weak lensing

Age of the UniverseAs matter components of the Universe, we consider

We define the redshift: 1 + z = a0/a (a0: today’s value)

We assume that the equation of state of dark energy is constant.wDE

They contribute to the right hand side of Friedmann equation:

We introduce the today’s density parameters

Then the Friedmann equation can be written as

On using the relation

the age of the Universe is

wheret0 depends on the matter components in the Universe(especially in the low z regime)

Estimation of the cosmic ageFor the estimation of t0 we can ignore the contribution of radiation.

Dark energy makes the cosmic age larger

The existence of dark energyresolves the cosmic age problem.

The open Universe without dark energy is insufficient to explain the cosmic agebecause large cosmic curvature is required.

11 Gyr

t0 = 13.73 ± 0.12 Gyr

⌦(0)m ' 0.3.

The rest isdark energy!

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Fukugita et al (1990)

From the number count ofgalaxies, they concluded that

‘The best agreement with the data is obtained with a sizable cosmological constant.’

Indirect evidence of dark energy: Number count of galaxies

Larger dark energy density

Number count galaxy data favored the existence of cosmological constant (1990)

Apparent magnitude

SN Ia observationsThe luminosity distance

Ls : Absolute lumonisityF : Observed flux

is related to the Hubble expansion rate H, as

for the flat Universe (K=0)

The absolute magnitude M of SN Ia is related to the observedapparent magnitude m, via

M ⇥ �19 at the peak of brightness for SN Ia (standard candles)

dL =�

Ls/(4�F )

dL(z) = (1 + z)� z

0

dz

H(z)

By measuring m, the expansion rate H(z) is known for z < O(1).

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Comoving distance (flat Universe)．SN Ia Observer

(z=0)Luminosity Ls Luminosity L0

In the flat FLRW background, the light travels along the geodesic satisfying

dr = �dt/a(t)

The comoving distance to SN Ia is given by

at r = r1, t = t1 at r = 0, t = t0

following from the time derivative of the relation 1 + z =a0a(t)

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dt

dz= � 1

(1 + z)H= � a(t)

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Derivation of luminosity distance (flat Universe)

．SN Ia Observer (z=0)

Luminosity Ls Luminosity L0

The observed flux is at z=0 is given by The luminosity distance squared is

We also have

Finally,

Luminosity distance with the spatial curvature Consider the FLRW background with the spatial curvature K:

The luminosity distance yields (includes the K=0 case)

�(0)K = �K/(a0H0)2

E(z) = H(z)/H0

Expansion around z=0 gives

Using the relationwe obtain

�0.0175 < �(0)K < 0.0085 (WMAP 5 yr)

(ii) In the open Universe (�K > 0) the luminosity distance also get larger.But this e⇥ect is limited because the Universe today is close to flat:

Effects of dark energy and spatial curvature on luminosity distance

Luminosity distance with/without dark energy

Flat Universe withoutdark energy

Open Universe withoutdark energy

Flat Universe withdark energy

Perlmutter et al, Riess et al (1998)

(Perlmutter et al, 1998)

High-z SN Ia data started to be obtained in the late 1990s.

Perlmutter et al showed thatthe cosmological constant ( ) is present at the 99 % confidence level, withtoday’s matter density parameter

The rest is dark energy.

⌦(0)m = 0.28+0.09

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M ' �19 at the peak of brightness for SNIa

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Dark energy leads to

the larger apparent

magnitude m.

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w = �1<latexit sha1_base64="bIiD697Snx6iJy+lHK1AWq0TAxw=">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</latexit>

Two groups are competing!Brian Schmidt [Head of HSST (Riess et al) group]

Saul Perlmutter [Head of SCP group]

Observational constraints on the dark energy equation of statefrom SNIa data alone

SN Ia data only

Cosmological constant

From the SNIa data only,

the constraint on w is

not tight, but we see

the evidence that

w < �1/3.<latexit sha1_base64="L0i/dkuka8oTIi7aWqVZ3hqYv9k=">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</latexit>

Cosmicacceleration

Nobel Prize in physics (2011) Nobel committee references

Riess Perlmutter Schmidt