Measuring Cosmic acceleration with Type Ia Supernovae

Alex Pagès Ahmad Mel

April 2015

Cover credits to www.astroart.org

Introduction & Motivation

• Big Bang is the prevailing theory about the Universe origin, in agreement with the expansion of the Universe, proposed by Georges Lemaître (1927) and observed by Edwin Hubble (1929), and the discovery of the CMB radiation.

• Before 1990s, expansion was supposed to stop in a future, ending in a big gravitational collapse (Big Crunch) or an steady Universe (Big Freeze), depending on the total density.

• In 1998 Saul Perlmutter from the Supernova Cosmology project, and Adam Riess & Brian P. Schmidt from the High-Z Supernova Search Team discovered an unexpected acceleration of the expansion by measuring distant type Ia supernovae.

Evidence for acceleration expansion

At small redshift (z < 0.1) distance-redshift relation is almost linearly due to Hubble’s Law:

𝑍 ≈ 𝐻𝑙

𝑐

At large distances, since expansion rate has change over time, this relation deviates from linearity, resulting in a larger light-travel time

A Supernova with a Z redshift measured implies it explode when the Universe was a(t) times its present size.

𝑎 𝑡 = 1

1 + 𝑧

Riess found that distances to high redshift supernovae were larger than expected for a decelerating Universe.

Type Ia Supernovae

• A supernova consist in the explosion of a stellar remnant.

• Type Ia supernovae take place in binary systems, with two stars orbiting one another. When one of the stars ends its life becomes a white dwarf. Then this accretes matter from the companion until the Chandrasekhar limit is reached and WD collapse. Before this happens, temperature and density increases inside the core, so carbon ignition starts and the high amount of energy released unbind the outer layers in this supernova explosion.

• Because supernovae are so luminous, their brightness are of the same order to the whole galaxy and they can be observed at faraway distances.

Since all Type Ia Supernovae have a similar mass and similar composition, they all present a characteristic light curve and spectrum after the explosion, with an extremely consistent peak luminosity. This luminosity is generated by the radioactive decay of heavy elements (nickel-56 through cobalt-56 to iron-56). They also characterize by showing a strong ionized silicon absorption line. All this characteristics allow them to be used as standard candles to measure high distances to their host galaxies. Combining its photometry with the cosmological models we can study the expansion of the Universe.

Corrections and systematics

Some corrections and calibrations need to be carried out:

- Color correction: extinction from dust in ISM (higher beyond z ≈ 1) and intrinsic relation between color and luminosity. - Host galaxy properties: SNe Ia from early-type galaxies are brighter than SNe Ia from galaxies of later types (Hicken et al. 2009c).

Early type galaxies contain significantly less dust, so separating SNe Ia offers a way to study the intrinsic component and the relative contribution of dust in a sample.

SNe Ia from low redshift galaxies (most of them corresponding to massive galaxies) have brighter absolute magnitudes. Then a proper host mass is determined adopting a probabilistic approach.

Supernovae lightcurves are fitted by three parameters: an overall normalization to the spectral energy distribution, the deviation from lightcurve shape, and the deviation from the mean SN Ia B-V colour . This parameters and integrated B-band flux are combined with the host mass to form the distance modulus.

𝜇𝐵 = 𝑚𝐵𝑚𝑎𝑥 + 𝛼𝑥1 − 𝛽𝑐 + 𝛿𝑃 𝑚∗

𝑡𝑟𝑢𝑒 < 𝑚∗𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 − 𝑀𝐵

Reproducing the Hubble diagram

We plot the SNe data from Union 2.1 set representing the redshift and distance modulus for each supernova Ia. Then we fit our plot by a theoretical function .

Distance modulus can be defined as:

𝜇 = 𝑚 −𝑀 = 5 log𝐷𝐿 𝑧 + 25

Where 𝐷𝐿 is the luminosity distance, defined as:

𝐷𝐿 = 𝑆𝑘(Χ)

𝑎(𝑡)= Χ(1 + 𝑧)

As we assume a flat Universe, 𝑆𝑘(Χ) is the comoving distance.

And the comoving distance to a given redshift is:

Χ 𝑧 = 𝑐

𝐻0

𝑑𝑎

Ω𝑚𝑎 + Ω𝑟 + Ω𝑘𝑎2 + ΩΛ𝑎

4

1

𝑎

For a Universe without curvature Ω𝑘 is 0, whereas we neglect the term Ω𝑟 since is only important in the very early Universe. Then we have only one free parameter:

Χ 𝑧 = 𝑐

𝐻0

𝑑𝑧

(1 − ΩΛ)(1 + 𝑧)3+ΩΛ

𝑧

0

Taking h = 0.7 we have obtained ΩΛ ≈ 0.75 & Ω𝑚 ≈ 0.25

Can we measure cosmic acceleration?

Cosmic acceleration can be expressed in terms of the deceleration parameter, defined as:

𝑞 ≡ −𝑎 𝑎

𝑎 2= (1 + 3𝜔)

4𝜋𝐺𝜌(𝑎)

3𝐻2(𝑎)

We consider ρ(a) as the total density with Ω𝑚 =𝜌𝑚

𝜌

and analogous expressions for the other cosmological parameters at a given age, and also the different values of ω, which are (in the ΛCDM cosmology model):

ω = 0 for matter & ω = −1 for dark energy

Finally, by the definition of H(a):

𝐻2 = 𝐻02 Ω𝑚𝑎

−3 + Ω𝑟𝑎−4 + Ω𝑘𝑎

−2 + ΩΛ

We can determine which is the deceleration parameter nowadays using the current values of the cosmological parameters. For our flat Universe we have:

𝑞0 = 1

2Ω𝑚 − ΩΛ =

1

21 − 3ΩΛ

With our ΩΛ we obtain q₀ ≈ -0.625 As q is the deceleration parameter, a negative value means a positive acceleration.

Can we measure H₀? H₀ is the Hubble constant parameter nowadays. Hubble parameter H relates the velocity of the Universe expansion at a given distance. So this parameter can be determined by a linear regression if radial velocities (redshift) and distances of a sample of objects are known.

𝑧 ≡ 𝑣

𝑐=

𝐻𝑑

𝑐

Because H is not constant, its current value needs to be found out considering only small redshift objects.

From the Union 2.1 sample and considering distances up to z = 0.1 we obtain as the Hubble constant:

H₀ = 73.2 ± 4.8 km/s/Mpc

The most accepted value nowadays is H₀ = 100h km/s/Mpc H₀ = 72 ± 8 km/s/Mpc

Can we measure ω(DE equation of state)?

If we consider a ωCDM model, then ω can be different to -1 for the dark energy. So the Hubble parameter becomes:

𝐻2 = 𝐻02 Ω𝑚𝑎

−3 + Ω𝑟𝑎−4 + Ω𝑘𝑎

−2 + Ω𝐷𝐸𝑎−3(1+𝜔)

Cosmological parameters can be also measured by the anisotropies of the CMB temperatures and the baryon acoustic oscillations (BAO), contained in the CMB and in the galaxies distribution, which allow to constrain Ω𝑚ℎ factor and also Ω𝑘 in a curved Universe. If we know H₀, then we can estimate ω, whose value is usually around -1 (Suzuki et al. 2011).

Do peculiar velocities alter these results?

Some galaxies may have peculiar movement between them, what implies a peculiar velocity which alters the observed redshift:

𝑐𝑧𝑜 = 𝑎 Χ + 𝑎Χ ≈ 𝐻𝑙 + 𝑣𝑝

1 + 𝑧𝑜 = (1 + 𝑧𝑐)(1 + 𝑧𝐷)

These peculiar velocities are small comparing to light speed and can be considered as random noise in the Hubble’s Law. They can be both positive or negative and their mean value tends to be <𝑣𝑝> = 0 for many measurements.

Do redshift errors alter the results? From the Hubble diagram

we have 𝜎𝑧 ∝𝑑𝑧

𝑑𝜇𝜎𝜇

At small redshifts: dz/dμ is small → good precision required 𝜎𝑧 ≈ (0.005 – 0.01) Fortunately, spectra are easier to detect

At high redshifts: dz/dμ is large → Redshift error is not so significant.

Assuming a flat Universe: ΩΛ = 0.705 ± 0.043

(Suzuki et al. 2011).

Conclusions

• Type Ia Supernovae can be used as standard candles as a tool to measure the Universe expansion and other cosmological parameters.

• Dark energy is the dominant type of matter in the current Universe (ΩΛ ≈ 0.75).

• The cosmic expansion is accelerating nowadays (q₀ < 0).

• At small redshifts we can estimate the Hubble’s constant by the linear Hubble’s Law.

• Peculiar velocities have very small effect on the observed redshift.

• Redshift measurement precision is more important at low redshifts.