Playing at being a Nobel Prize:

Measuring the Dark Energy in the Universe.

PIIISA 2014

J. Leyva, L. Cabrera, P. Olgoso, C. Munoz, S. Ortiz,

C. Montes, P. Medina, T. Pozo, P. Aranguez, L. Barrios,

A. Molina, I. Marın, N. Finisi, L. Galan, A. Hamod, A. Guindo,

L. Ortiz, J. Vida, I. Vida, C. Martınez, M. Valdivia, D. Jaldo,

A. Molino , J. Caceres and A. Sota (coordinators)

ii

———————————

Scientific Working Groups

organized in this work..

Index

1 Introduction. 1

2 Cosmology. 2

2.1 The Hubble Law and the Expansion of the Universe. . . . . . . . . . . . . . . . . . . . 2

2.2 The Scale Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.3 The Friedman Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.4 The Cosmological Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.5 The Dark Energy and the Fate of the Universe. . . . . . . . . . . . . . . . . . . . . . . 4

3 Astrophysics. 5

3.1 The Main Sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 Stellar Classification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.3 Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.4 Supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.4.1 Measuring distances in the universe . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Observations. 9

4.1 The Astronomical Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.1 Detecting objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2 Observing Remotely with the OSN Telescope. . . . . . . . . . . . . . . . . . . . . . . . 9

4.2.1 Our Observational Proposal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2.2 The Filter Selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Data Analysis. 11

5.1 The SNIa Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.2 The Goodness of fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.3 Data Analysis and Final Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Conclussions. 13

7 Moments. 14

iv

8 Bibliography 15

1Introduction

In this project, the cosmology-group have studied the origin, the evolution and the future of our

Universe, learning the fundamental concepts of cosmology and the basic equations that describe the ex-

pansion of the Universe. The astrophysics-group, after learning the most important concepts about

the stars (such that the formation, the evolution, the spectral-classification, the composition,...), they

understood how and why a specific type of exploiting stars (the SNIa) could be used to calculate

distances to other distant galaxies in the Universe. The observing-group was able to prepare an

observational proposal (using remotely the professional 1.5m telescope from the Sierra Nevada Obser-

vatory (OSN)), and so understand all the complexity behind any astronomical observation. Finally, in

order to calculate the most probable cosmological model for our Universe, the programming-group

created its own Python-based program to analyze real data from scientific publications.

As explained in this manuscript, the results obtained were in agreement with those obtained by the

Nobel Prize Winners, indicating that the Universe is mostly composed by ∼80% of a Dark Energy

which is acceleration its expansion.

2Cosmology

This initial part of the work corresponds to the introduction of several ideas of modern Cosmology,

compressing the study of the origin, the evolution and the eventual fate of the Universe. We will

understand “basic” concepts related with the expansion of the Universe and the physical equations that

describe how the Universe changed with time. These theoretical concepts will help the data analysis

group to interpret their results and to know which is the right future expected for our Universe.

2.1 The Hubble Law and the Expansion of the Universe.

In 1929 the astronomer Edwin Hubble postulated that the Universe was expanding in such a way

that every galaxy moved away from us to a proportional speed to its distance, where the speed of a

galaxy could be deduced from its spectral shifting. This discovery represented the birth of modern

cosmology, enabling to calculate distances to many different galaxies in the Universe. So Hubble found

a relationship (The Hubble’s law) between the redshift and the distance. As seen in Figure 2.1 (left),

this relationship says that galaxies are separated from each other at a speed (V ) proportional to its

distance (D), as indicated in the equation 2.1:

V = H ×D (2.1)

where H is the Hubble Constant and its current value is Ho = 70km/s/Mpc.

2.2 The Scale Factor.

The scale factor explains the relative expansion of the universe. It is related to the increasing distance

between a pair of galaxies caused by the expansion of the space-time. As seen in Figure ?? (right),

this phenomenon can be can be seen as an expanding ballon. As it expands the points on its surface

move away from each other. Therefore the scale factor also relates to the (cosmological) redshift. The

evolution of the scale factor explains the future of the Universe.

2.4 The Friedman Equations. 3

Figure 2.1 Left: This figure represents the Hubble Law. It shows the relation between the distancesof several galaxies (blue dots) and their recessing velocities. It is obtained a linear fit for this data.Right: This figure represents the increment of the space-time in the Universe resembling the Universelike a balloon.

2.3 The Friedman Equations.

The Friedman equations (2.2) express the evolution of the Universe based on its physical constituents:(a

a

)2

=8πG

3

∑i

Ωi (2.2)

Surprisingly, the study of the expansion rate has shown that the universe is so close to the critical

density which can cause its expansion forever. The normal way to express the density is as a fraction of

the density required for the critical condition of the parameter Ω= ρ/ρc so Ω=1 represent the condition

of critical density. It is a common practice add a subscript 0 to these amounts in the present moment.

At the beginning, radiation was dominant the expansion over dark matter. Later on, the (dark)

matter dominated over radiation. Currently, the dark energy (or cosmological constant) completely

dominates the expansion.

2.4 The Cosmological Models.

We know from the General Relativity theory (Einstein 1915), the content of matter-energy of the

universe determines the curvature of space-time, its evolution and eventually its fate. Therefore,

depending on what is the matter-energy content of the Universe there would be several possible

models of the Universe:

• The FLAT Universe: the Universe would expand steadily infinitely.

• The OPEN Universe: the Universe would expand rapidly infinitely.

• The CLOSED Universe: the relative expansion of the Universe would stop now and after

this, the Universe would start to contract. The final of the Universe would be known with the

4 Cosmology. 2.5

Figure 2.2 Left: This figure represents the different Energy Density Parameters: Dark Matter, Radi-ation and Dark Energy, respectively. Right: This figure represents the relation between the quantityof Dark Matter and Dark Matter and the expected expansion of the Universe. The red line corre-sponds to an Universe with both Dark Matter and Dark Energy. In this model, the expansion occursexponentially. Blue and green lines correspond to an Universe without Dark Energy. In these modelsthere is only constant expansion. Orange line corresponds to an Universe without Dark Energy butdominated by Dark Matter. In this model, the Universe will eventually collapse.

name of Big Crunch.

2.5 The Dark Energy and the Fate of the Universe.

We have studied the Cosmological Redshift to understand that we are witnessing a cosmological epoch

where the expansion is dominated by the Dark Energy. Likewise, we have study mathematically the

main cosmological models by means of the Friedman equations and figure out which might be the

right future of our Universe. This part of the work is relevant to understand that the Universe has

never been the same but has pass through different episodes of expansion.

3Astrophysics

In this part of the project, we study the universe and its objects’ laws, the different phenomenons

and their physicochemical nature. We are the theorist basis of this project; we explain how does the

universe work. Moreover, we focus on a special type of Supernova, the type Ia, which has demonstrated

to be very useful in cosmology and especially in this project. When we watch the night sky and the

universe, we can’t help looking at the stars, but... What are those shining points actually?

3.1 The Main Sequence.

The Main Sequence is a continuous band of stars which is divided into zones of different colors, known

as the H-R diagram. In its core, the star generates energy through the fusion of hydrogen into helium.

When a lot of energy is generates lost gravitational pressure. The star expands and it reaches the

hydrostatic equilibrium . If the power production reduces the core is compressed and increases the

fusion . The energy generated in the core is going to the surface by radiation and convection . The

energy pushes the upper layers increasing radius, luminosity and therefore changes position the star in

the diagram. The main sequence is divided into top and lower based on the generation of energy . At

the lower fused with helium and hydrogen and on the top, the fusion occurs with carbon , nitrogen and

oxygen . When hydrogen is consumed energy is lost and the star will die forming white dwarfs or giant

red depending on its mass . The more massive a star is shorter than your life can be calculated by

dividing the total energy produced between brightness ,when massive stars die can form a supernova

if explodes or black holes if compressed . Cold stars : dwarf red, orange and yellow are smaller and

less bright... Hot stars: blues and whites are more highest and lightness

3.2 Stellar Classification.

Stars can be classified in terms of their spectral characteristics. The light from stars can be dispersed

and so observed the rainbow of colours interspersed with absorption lines. The observation of lines

indicate the presence of certain chemical elements, and its width its relative abundance. The relative

abundance of the different ions varies with the temperature of the photosphere, so we can study the

6 Astrophysics. 3.3

Figure 3.1 Figure on the left shows a representation of the different types of stars depending on theirtemperature and their luminosity. Figure on the right represents the evolution for a star during itslife depending on its original mass. Poor-mass stars are represented in the bottom whereas high-massstars are represented in the top.

composition and characteristics from different stars observing these absorption lines. We classify stars

through the MKK System, introduced in 1943, that is a two-dimensional (temperature and luminosity)

scheme based on the spectral lines, sensitive to temperature and surface gravity (related to luminosity).

This system classify stars using the letters O, B, A, F, G, K and M from the hottest to the coolest.

Each group is then subdivided using a numeric digit with a numeric digit from 0 (the hottest) to 9

(the coolest). In addition, we add a luminosity class using a Roman numeral. Luminosity class I are

supergiants, class II are bright giants, class III are normal giants, class IV are subGiants, class V are

main-sequence stars, class VI are subDwarfs and class VII are white dwarfs.

3.3 Stellar Evolution

Stellar evolution is the process by which a star undergoes a sequence of radical changes during its

lifetime. Nuclear fusion powers a star for most of its life. This process causes the star to gradually

grow in size. The evolution of a stars during its life will depend on its original mass, as it shown in

the Figure 3.1.

• Low-mass stars: What happens after a low-mass star ceases to produce energy through fusion

has or been directly observed.

• Mid-sized stars: Stars of roughly 0.5?10 solar masses become red giants, stars of stellar clas-

sification K or M.

• Red-giant branch phase: The red-giant-branch phase of a star’s life follows the main sequence.

The core reaches hydrostatic equilibrium. As the hydrogen around the core is consumed, the core

absorbs the resulting helium. The energy released by helium fusion causes the core to expand.

3.4 Supernovae 7

• Asymptotic-giant-branch phase: After a star has consumed the helium at the core, fusion

continues in a shell around a hot core of carbon and oxygen. Another well-known class of

asymptotic-giant-branch stars are the Mira variables.

• Massive Stars: When these stars expand and cool, they do not brighten as much as lower-mass

stars but are still brighter than the red giants formed from less massive stars. These stars are

unlikely to survive as red supergiants. Extremely massive stars lose mass so rapidly.

• Supernova: Once the nucleosynthesis process arrives at iron-56, the continuation of this process

consumes energy.

• Stellar remnants: After a star has burned out its fuel supply, its remnants can take one of

three forms, depending on the mass during its lifetime.

• White and Black dwarfs: White dwarfs are stable because the inward pull of gravity is

balanced by the degeneracy pressure of the star’s electrons. White dwarfs of higher mass have

a smaller volume.

• Neutron Stars: When a stellar core collapses, the pressure causes electron capture, thus con-

verting the great majority of the protons into neutrons. These stars, known as neutron stars,

are extremely small. Such neutron stars are called pulsars.

• Black holes: If the mass of the stellar remnant is high enough, the neutron degeneracy pressure

will be insufficient to prevent collapse below the Schwarzschild radius. The stellar remnant thus

becomes a black hole.

3.4 Supernovae

Supernovae are exploding stars. They represent the very final stages of evolution for some stars.

Supernovae, as celestial events, are huge releases of tremendous energy. Currently, supernovae are

only seen in galaxies other than the Milky Way. We know that supernovae have occurred in our

Galaxy in the past. Today, we see remnants of all three supernovae, which appear as expanding

clouds of gas, where each was originally discovered.

The supernovae are classified in two types, I and II, depending on their spectral characteristics.

Type-I supernovae do not present hydrogen lines, and the type-Ia presents a singly ionized silicon

(Si II) line. Why are type-Ia so important in cosmology and in our project? Because they work as

standard candles, what means that they can be used to measure distances in the universe and so the

evolution of the universe. But why are these types of supernovas so useful to determine distances?

Because they all have nearly the same absolute brightness (or luminosity) in their brightest phase, in

their peak of emission. Comparing the apparent brightness of two supernovae, we can thus measure

the distance between them. This property can be seen in Figure 3.2.

8 Astrophysics. 3.4

Figure 3.2 SNIa have all nearly the same absolute magnitude during their brightest phase (right afterthey explode). Therefore, they work as standard candles (like distant light-houses).

3.4.1 Measuring distances in the universe

Defining the luminosity as the amount of light (spherically) emitted by the SNIa in all directions, it

can be expressed as:

F =L

4πR2⇒ DL = R =

√L

4πF⇒ m = −2.5Log10(F ) + cte (3.1)

where F=Flux, L=Luminosity and 4πR2=Area of the sphere, R = Radius (Distance), (m) the

apparent magnitude.

Then we define the absolute magnitude M as the magnitude we might measure if the star would

be at a distance of 10 parsecs. 1 If we really know how bright is a star, we can define the Distance

Modulus µ as the difference between the apparent (m). With the Distance Modulus, we can calculate

a distance (D) to the galaxy that hosts the star. Therefore, we can use exploiting stars to calculate

cosmological distances in the Universe.

m−M = log10(D)− 5 ≡ µ ⇒ µ+ 5

5= log10(D) =⇒ D = 10

µ5+1 (3.2)

11 parsec is defined as the distance at which an observer would see the Sun and the Earth separated by an angle ofone degree.

4Observations

The observational-group is responsible for obtaining images of the objects that have been chosen to

be observed remotely in a professional telescope.

4.1 The Astronomical Filters.

In this study, from all the different wavelengths an astronomical object can emit along the electro-

magnetic spectrum (Infrared, UV, X-rays,...), we have observed just in the Visible because it is where

we can see absorption and emission lines from SN remnants. In emission, colors, can be see but not

all are broadcast at the same amount, and that determines us many characteristics of the object that

we want to observe. They are tools that are used in the telescopes to filter a single color and not pass

the rest. Depending on the percent so we get each of the colors, we can perform its spectrum.

4.1.1 Detecting objects

Whereas the telescope serves just to collect photons, it is necessary an instrument to register all this

collected light. The CCD (Charge-Coupled Device) cameras are the detector placed at the focus of the

telescope. Such cameras have photovoltaic chips, ie convert light into electricity. They are sensitive

to different wavelengths.

4.2 Observing Remotely with the OSN Telescope.

Our observation with the professional telescope of Sierra Nevada 4.1 of 1.5 meters took place on

March 12, 2014 remotely. A remote observation means we use the telescope putting us in touch with

an engineer who controls him with our instructions. The conditions to observe him in the first attempt

was pretty bad because of the weather was bad and it did not favor us. There was quite high clouds,

high humidity and allowing engines move the telescope was broken because the weight of snow on

the dome was high. That day we could not take pictures of objects, but days later yes because the

conditions were low humidity, little wind and good seeing. Three filters for observation were used

which are : V, R e I.

10 Observations. 4.2

Figure 4.1 On the left, the image shows the Sierra Nevada Observatory (Granada) from where wetook the images. On the right, the image shows the M82 galaxy. The SNIa remnant (small spot onthe right-side of the galaxy) has a clearly different luminosity in the two epochs, indicating that theSNIa is declining.

Table 4.1 The Filter Selection.

Filters λ Transmission

V-Jonhson 550 nm 65,9%

R1-Cousins 640 nm 79,6%

I1-Cousins 1.030 nm 89,4%

4.2.1 Our Observational Proposal.

We wrote a proposal to observe with the Sierra Nevada Observatory (OSN) and it was accepted. As

shown Figure 4.1, we observed M1 since this is a quite advanced supernovae remnant and, therefore

is a great example for this project. In addition, we observed the galaxy M82, which is an example

of a recently discovery Supernova. In this galaxy can be seen a big point of light on one side of this

galaxy. It is the explosion of a supernova.

4.2.2 The Filter Selection.

Due to the fact that in the spectrum of a Supernova remnant has several emission lines, it was necessary

to select the closer filters to those wavelengths. As indicated in Table 4.1, we selected the following

filters: V Johnson (Green color), R(1) Cousins (Red color) and I(1) Cousins (Yellow color).

5Data Analysis.

This section is dedicated to the data analysis of our research, showing how the SNIa can be used to

determine which Universe (from all those predicted by theory) is best fitting the observed Universe.

5.1 The SNIa Data.

To determine the most probable model of universe, it has been necessary to use data from type-Ia

supernovae, as standard candles used in astrophysics. In this case, the redshift (z) that indicates the

distance to the supernovae; the magnitude of the supernovae itself indicating its brightness and the

associated uncertainties (such that variability due to observational conditions). We extracted the data

from Riess et al. (1998), Perlmutter et al.(1999) and Suzuki et al.(2011).

5.2 The Goodness of fit.

The goodness of fit (Equation 5.1) describes statistically how well a model describes a set of observa-

tions. We used this simple equation to quantify which cosmological model minimized the differences

with the real data; i.e., the model with the lower χ2.

χ2 =n∑

i=1

(value modeli − real valuei)2

(uncertainty real valuei)2(5.1)

5.3 Data Analysis and Final Results.

In the figures 5.3, we show the Python program that we created to analyze the different cosmological

models. We used our program to calculate the χ2 which represents the differences between the real

SNIa data and the theoretical expectations from cosmological models. As we can see in the table

5.3, our results shows that the χ2 of an Universe dominated by 30% of Dark Matter and 70% of

Dark Energy minimizes the differences. It means that this model is the closest to the real data and

therefore the most likely. The tables shows that this theoretical model is 1.4 times better than an

12 Data Analysis. 5.3

Figure 5.1 Left: The figure shows the Python program that was created to analyze the differentcosmological models using the chi2. Right: The figure represents the comparison between the SNIadata (with their uncertainties) with the expected values for three different Universe models. Whilethe first model (green line) is composed by 30% of Dark Matter and 70% of Dark Energy, the secondmodel (red line) corresponds to a Universe dominated by Dark Matter and the third model (blue line)corresponds to a Universe dominated by Dark Energy.

Universe dominated only by Dark Energy and 2.1 times better than an Universe dominated only by

Dark Matter.

Dark Energy [%] Dark Matter [%] χ2 χ2i /χ

21

Model1 0.7 0.3 1326 1.00

Model2 0.0 1.0 2858 2.15

Model3 1.0 0.0 1845 1.39

Table 5.1 Analysis of the three cosmological models. Model1 minimizes the differences with real data.

This model is composed by 30% of Dark Matter and 70% of Dark Energy.

6Conclussions

Cosmology: Everything in our Universe has a beginning and an ending. In the next billions of

years, due to the effect of the dark energy, first the formation of Superclusters will be retarded,

galaxies and solar systems will be disregarded. Then the rest of planets crashed against the Suns due

to gravitational radiation. Later the remaining matter will become iron through the tunnel effect.

Finally the universe will be dark just formed by isolated particles (photons, electrons, do protons?)

How knows... May quantum fluctuations generate a new universe?

Astrophysics: As the theoretical bases of the project, the astrophysics group had to explain how

the Universe does work or what are its objects that populate it, and provides the research an useful

tool, the type-Ia SN which would let us determine these distances in the Universe in order to find out

how our Universe is actually expanding.

Data Analysis: From our point of view, we are proud of our particular work in this project, since

we think the data analysis is one of the essential parts of a research and the scientific method. This is

because the mathematical data analysis is the empirical basis to build a scientific theory or a result on.

Throw the data analysis we have given this project a fundamental pillar, demonstrating empirically

our theory. We think that it is very important to develop computational tools, because they help us

to work faster, better and with no human mistakes. In this way, human knowledge combined with the

speed and precision of the technology will advance a lot in science, including astronomy!

Observations: In this project we understood how difficult is the direct observation of the Universe.

Likewise, we have learnt how to fill in an observational proposal to get observing time in a professional

telescope. We studied the typical observational problems when observing such that humidity, clouds,

winds or problem occurring in the telescope itself. But also, we discovered how beautiful is to observe

the Universe from our own eye and put into the practice the learnt theory.

7Moments

In this final section, we show several pictures from many memorable moments that we experienced

along these months preparing this project.

Figure 7.1 On May14th, we had a Skype telecom with the nobel-price winner Adam Riess!!

Figure 7.2 Our lessons and Observations at the IAA-CSIC.

8Bibliography

1. Perlmutter et al. (1999)

2. Suzuki et al. (2011)

3. Riess et al. (1998)

4. Brief introduction to astronomy (Material provided by researchers).

5. www.astrocity.es

6. www.astroshop.es

7. www.telescopiomania.com

8. www.osn.iaa.es

9. http://simbad.u-strasbg.fr/simbad/sim-fid

10. http://catserver.ing.iac.es/staralt/

11. http://www.astrofisicayfisica.com/2014/01/supernova-brillante-en-la-galaxia-m82.html

12. Basic Python Manual OSnake Wrangling for KidsO, by Jason R. Briggs

13. http://matplotlib.org/

14. https://www.python.org/

15. http://en.wikipedia.org/wiki/Goodness of fit