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Dean transmigrated into the body of Will, a Night’s Watch ranger. When he opened his eyes in this world, he found himself tied and surrounded by Stark guards, branded a deserter and set to be executed on the spot. Knowing the events of the first episode, Dean realized that the Lord of Winterfell would soon arrive to personally carry out the sentence. Despite his situation, Dean was not without hope. He had been granted an ability called Shadow Extraction, similar to the one possessed by the main lead of Solo Leveling. However, his version was weaker and came with certain limitations. Even so, in the harsh and unforgiving world of Game of Thrones, this ability still held the potential to become a powerful advantage. With death approaching and no time to waste, Dean had to rely on this newfound power and his knowledge of the world to survive a fate that had once been inevitable. ────────────────────── This is NOT a translation. And the Game of Thrones Tv show belongs to their respective creators. All I own is the MC. ────────────────────── For 5+ advance chapters visit my patreon. patreon.com/BlazingInk
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The Big-Bang Theory: Construction, Evolution and Status

Jean-Philippe Uzan

Institut d'Astrophysique de Paris

UMR 7095 du CNRS,

98 bis, bd Arago

75014 Paris.

Abstract. Over the past century, rooted in the theory of general relativity, cos-

mology has developed a very successful physical model of the universe: the

big-bang model. Its construction followed di↵erent stages to incorporate nuclear

processes, the understanding of the matter present in the universe, a description

of the early universe and of the large scale structure. This model has been con-

fronted to a variety of observations that allow one to reconstruct its expansion

history, its thermal history and the structuration of matter. Hence, what we re-

fer to as the big-bang model today is radically di↵erent from what one may have

had in mind a century ago. This construction changed our vision of the universe,

both on observable scales and for the universe as a whole. It o↵ers in particular

physical models for the origins of the atomic nuclei, of matter and of the large

scale structure. This text summarizes the main steps of the construction of the

model, linking its main predictions to the observations that back them up. It

also discusses its weaknesses, the open questions and problems, among which

the need for a dark sector including dark matter and dark energy.

1 Introduction

1.1 From General Relativity to cosmology

A cosmological model is a mathematical representation of our universe that is based

on the laws of nature that have been validated locally in our Solar system and on

their extrapolations (see Refs. [1, 2, 3] for a detailed discussion). It thus seats at the

crossroad between theoretical physics and astronomy. Its basic enterprise is thus to

use tested physical laws to understand the properties and evolution of our universe

and of the matter and the astrophysical objects it contains.

Cosmology is however peculiar among sciences at least on two foundational

aspects. The uniqueness of the universe limits the standard scientific method of

comparing similar objects in order to find regularities and to test for reproductibility;

indeed this limitation depends on the question that is asked. In particular, this will

tend to blur many discussions on chance and necessity. Its historical dimension

forces us to use abduction1 together with deduction (and sometime induction) to

reconstruct the most probable cosmological scenario2. One thus needs to reconstruct

1Abduction is a form of inference which goes from an observation to a theory, ideally looking for the simplest

and most likely explanation. In this reasoning, unlike with deduction, the premises do not guarantee the conclusion, 2 J.-P. Uzan S´eminaire Poincar´e

the conditions in the primordial universe to fit best what is observed at di↵erent

epochs, given a set of physical laws. Again the distinction between laws and initial

conditions may also be subtle.This means that cosmology also involves, whether we

like it or not, some philosophical issues [4].

In particular, one carefully needs to distinguish physical cosmology from the

Cosmology that aims to propose a global picture of the universe [1]. The former

has tremendously progressed during the past decades, both from a theory and an

observation point of view. Its goal is to relate the predictions of a physical theory

of the universe to actual observations. It is thus mostly limited to our observable

universe. The latter is aiming at answering broader questions on the universe as a

whole, such as questions on origins or its finiteness but also on the apparent fine-

tuning of the laws of nature for complexity to emerge or the universe to host a viable

form of life. The boundary between these two approaches is ill-defined and moving,

particularly when it comes to recent developments such as inflation or the multiverse

debate. They are related to the two notions, the universe, i.e. the ensemble of all

what exist, and our observable universe. Both have grown due to the progresses of

our theories, that allow us to conceptualize new continents, and of the technologies,

that have extended the domain of what we can observe and test.

Indeed the physical cosmology sets very strong passive constraints on Cosmol-

ogy. It is then important to evaluate to which extent our observable universe is

representative of the universe as a whole, a question whose answer depends dras-

tically of what is meant by "universe as a whole". Both approaches are legitimate

and the general public is mostly interested by the second. This is why we have the

moral duty to state to which of those approaches we are referring to when we talk

about cosmology.

While a topic of interest for many centuries – since any civilization needs to

be structured by an anthropology and a cosmology, through mythology or science –

we can safely declare [5] that scientific cosmology was born with Albert Einstein's

general relativity a century ago. His theory of gravitation made the geometry of

spacetime dynamical physical fields, gµ⌫, that need to be determined by solving

equations known as Einstein field equations,

Gµ⌫[g↵ L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 3

(c) L. Haddad & G. Duprat

Geological data

O

Figure 1: Astrophysical data are mostly located on our past lightcone and fade away with distance from us so

that we have access to a portion of a 3-dimensional null hypersurface – an object can be observed only when its

worldline (dashed lines) intersects our past lightcone. Some geological data can be extracted on our Solar system

neighborhood. It is important to keep in mind that the interpretation of the observations is not independent of the

spacetime structure, e.g. assigning distances. We are thus looking for compatibility between a universe model and

these observations.

in the universe. Even with perfect data, the fact that (almost)3 all the information

we can extract from the universe is under the form of electromagnetic signal implies

that observations are located on our past lightcone, that is on a 3-dimensional null

hypersurface (see Fig. 1). It can be demonstrated (see e.g. Ref. [6] and Ref. [7] for

a concrete of 2 di↵erent cosmological spacetimes which enjoy the same lightcone

observations) that the 4-dimensional metric cannot be reconstructed from this in-

formation alone. Indeed a further limitation arises from the fact that there is no such

thing as perfect observations. Galaxy catalogs are limited in magnitude or redshift,

evolution e↵ects have to be taken into account, some components of matter (such a

cold di↵use gas or dark matter) cannot be observed electromagnetically. We do not

observe the whole matter distribution but rather classes of particular objects (stars,

galaxies,...) and we need to deal with the variations in the properties of these indi-

vidual objects and evolution e↵ects. A di 4 J.-P. Uzan S´eminaire Poincar´e

them to describe our spacetime from stellar scales to the Hubble scale.

The first relativistic cosmological model [9] was constructed by Einstein in 1917

and it can be considered as the birthdate of modern cosmology. As we shall see,

most models rely on some hypotheses that are di L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 5

it does not prove that it is the "correct" model of the universe, in the sense that it

is the correct cosmological extrapolation and solution of the local physical laws.

When confronted with an inconsistency in the model, one can either invoke the

need for new physics, i.e. a modification of the laws of physics we have extrapolated

in a regime outside of the domain of validity that has been established so far (e.g.

large cosmological distance, low curvature regimes etc.), or have a more conservative

attitude concerning fundamental physics and modify the cosmological hypotheses.

Let us start by reminding that the construction of any cosmological model relies

on 4 main hypotheses (see Ref. [3] for a detailed description),

(H1) a theory of gravity,

(H2) a description of the matter contained in the universe and their non-gravitational

interactions,

(H3) symmetry hypothesis,

(H4) a hypothesis on the global structure, i.e. the topology, of the universe.

These hypotheses are indeed not on the same footing since H1 and H2 refer to

the local (fundamental) physical theories. These two hypotheses are however not

su 6 J.-P. Uzan S´eminaire Poincar´e

1.2.2 Non-gravitational sector

Einstein equivalence principle, as the heart of general relativity, also implies that the

laws of non-gravitational physics validated locally can be extrapolated. In particular

the constants of nature shall remain constant, a prediction that can also be tested

on astrophysical scales [17, 18]. Our cosmological model assumes (H2) that the

matter and non-gravitational interactions are described by the standard model of

particle physics. As will be discussed later, but this is no breaking news, modern

cosmology requires the universe to contain some dark matter (DM) and a non-

vanishing cosmological constant (⇤). Their existence is inferred from cosmological

observations assuming the validity of general relativity (e.g. flat rotation curves,

large scale structure, dynamics of galaxy clusters for dark matter, accelerated cosmic

expansion for the cosmological constant; see chapters 7 and 12 of Ref. [19]). Dark

matter sets many questions on the standard model of particle physics and its possible

extensions since the physical nature of this new field has to be determined and

integrated consistently in the model. The cosmological constant problem is argued

to be a sign of a multiverse, indeed a very controversial statement. If solved then

one needs to infer some dark energy to be consistently included.

We thus assume that the action of the non-gravitational sector is of the form

S =

Z

L( , gµ⌫)

p L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 7

point of view, it can be shown that it implies that the metric of the universe reduces

to the Friedmann-Lemaˆıtre form (see e.g. chapter 3 of Ref. [19])

ds2 = 8 J.-P. Uzan S´eminaire Poincar´e

since then, for all g, L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 9

This allowed Ralph Alpher, Hans Bethe and George Gamow [31] to predict the ex-

istence and estimate the temperature of a cosmic microwave background (CMB)

radiation and understand the synthesis of light nuclei, the big-bang nucleosynthesis

(BBN), in the early universe. Both have led to theoretical developments compared

successfully to observation. It was understood that the universe is filled with a ther-

mal bath with a black-body spectrum, the temperature of which decreases with the

expansion of the universe. The universe cools down and has a thermal history, and

more important it was concluded that it emerges from a hot and dense phase at ther-

mal equilibrium (see e.g. Ref [19] for the details). This model has however several

problems, such as the fact that the universe is spatially extremely close to Euclidean

(flatness problem), the fact that it has an initial spacelike singularity (known as big-

bang) and the fact that thermal equilibrium, homogeneity and isotropy are set as

initial conditions and not explained (horizon problem). It is also too idealized since

it describes no structure, i.e. does not account for the inhomogeneities of the matter,

which is obviously distributed in galaxies, clusters and voids. The resolution of the

naturalness of the initial conditions was solved by the postulate [32] of the existence

of a primordial accelerated expansion phase, called inflation.

The third and fourth periods were triggered by an analysis of the growth of the

density inhomogeneities by Lifshitz [33], opening the understanding of the evolution

of the large scale structure of the universe, that is of the distribution of the galax-

ies in cluster, filaments and voids. Technically, it opens the way to the theory of

cosmological perturbations [34, 35, 36] in which one considers the FL spacetime as

a background spacetime the geometry and matter content of which are perturbed.

The evolution of these perturbations can be derived from the Einstein equations. For

the mechanism studied by Evgeny Lifshitz to be e 10 J.-P. Uzan S´eminaire Poincar´e

results of the analysis of the Hubble diagram of type Ia supernovae in 1999 [40].

This ⇤CDM model is in agreement with all the existing observations of the large

survey (galaxy catalogs, CMB, weak lensing, Hubble diagram etc.) and its parame-

ters are measured with increasing accuracy. This has opened the era of observational

cosmology with the open question of the physical nature of the dark sector.

This is often advertised as precision cosmology, mostly because of the increase

of the quality of the observations, which allow one to derive sharp constraints on

the cosmological parameters. One has however to be aware, that these parameters

are defined within a very specific model and require many theoretical developments

(and approximations) to compare the predictions of the model to the data. Both

set a limit a on the accuracy of the interpretation of the data; see e.g. Refs. [41, 42]

for an example of the influence of the small scale structure of the universe on the

accuracy of the inference of the cosmological parameters. And indeed, measuring

these parameters with a higher accuracy often does not shed more light on their

physical nature.

The standard history of our universe, according to this model, is summarized in

Fig. 2. Interestingly, the evolution of the universe and of its structures spans a period

ranging from about 1 second after the big-bang to today. This is made possible by

the fact that (1) the relevant microphysics is well-known in the range of energies

reached by the thermal bath during that period (< 100 MeV typically) so that it

involves no speculative physics, and (2) most of the observables can be described by

a linear perturbation theory, which technically simplifies the analysis.

This description is in agreement with all observations performed so far (big-

bang nucleosynthesis abundances, cosmic microwave background temperature and

polarisation anisotropies, distribution of galaxies and galaxy clusters given by large

catalogs and weak lensing observations, supernovae data and their implication for

the Hubble diagram).

This short summary shows that today inflation is a cornerstone of the standard

cosmological model and emphasizes its roles in the development and architecture of

the model,

1. it was postulated in order to explain the required fine-tuning of the initial

conditions of the hot big-bang model,

2. it provides a mechanism for the origin of the large scale structure,

3. it gives a new and unexpected vision of the universe on large scale,

4. it connects, in principle [43], cosmology to high energy physics.

This construction is the endpoint of about one century of theoretical and observa-

tional developments, that we will now detail.

2.2 Relativistic cosmology

2.2.1 Einstein static universe (1917)

The Einstein static universe is a static homogeneous and isotropic universe with

compact spatial sections, hence enjoying the topology of a 3-sphere. It is thus char-

acterized by 3 quantities: the radius of the 3-sphere, the matter density and the L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 11

Figure 2: The standard history of our universe. The local universe provides observations on phenomena from big-

bang nucleosynthesis to today spanning a range between 10 12 J.-P. Uzan S´eminaire Poincar´e

Figure 3: Penrose diagrams of de Sitter space in the flat (left) and static (right) slicings that each cover only part

of the whole de Sitter space, and that are both geodesically incomplete. From Ref. [22].

sections and scale factor a / cosh Ht with H = p⇤/3 constant. It is the only

geodesically complete representation;

2. the flat slicing in which the metric has a FL form (5) with Euclidean spatial

section, in which case a / exp Ht;

3. the hyperbolic slicing in which the metric has a FL form (5) with K = L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 13

to be integrated. It is convenient to use the conformal time defined by dt = a(⌘)d⌘

and the normalized density parameters

⌦i = 8⇡G⇢i/3H2

0 , ⌦⇤ = ⇤/3H2

0 , ⌦K = 14 J.-P. Uzan S´eminaire Poincar´e

Hyperbolic

Spherical

3.0

2.0

1.0

0.0

– 1.0

Bouncing

N

M

F E

I

B

J

A C

H

G

D

Ωm0

ΩΛ0

0.0 1.0 2.0 3.0

– 3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

B I

J

Hyperbolic

A

F E

– 3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

Euclidean

–3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

B A

C

D

Λ = 0

H

D

G

–3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

Spherical

– 3 – 2 –1 0 1 2 3

3.0

2.0

1.0

0.0

K

L F

Hesitating

(Spherical

or

Euclidean)

M

N

Bouncing

(Spherical)

– 3 – 2 –1 0 1 2 3

3.0

2.0

1.0

0.0

a

a0

a

a0

a

a0

a

a0

a

a0

a

a0

H0t H0t H0t

H0t H0t H0t

LK

Figure 5: Depending on the values of the cosmological parameters (upper panel), the scale factor can have very

di↵erent evolutions. It includes bouncing solutions with no big-bang (i.e. no initial singularity), hesitating universes

(with a limiting case in which the universe is asymptotically initially Einstein static), collapsing universes. The

expansion can be accelerating or decelerating. From Ref. [19].

as first established by Lemaˆıtre [27]. This gives the first observational pillar

of the model and the order of magnitudes of the Hubble time and radius are

obtained by expressing the current value of the Hubble parameter as H0 =

100 h km · s L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 15

with E ⌘ H/H0. The use of standard candels, such as type Ia supernovae allow

to constrain the parameter of the models and measured the Hubble constant

(see Fig. 6).

10 16 J.-P. Uzan S´eminaire Poincar´e

2.3 The hot big-bang model

As challenged by Georges Lemaˆıtre [48] in 1931, "une cosmogonie vraiment compl`ete

devrait expliquer les atomes comme les soleils." This is mostly what the hot big-bang

model will achieved.

This next evolution takes into account a better description of the matter content

of the universe. Since for radiation ⇢ / a4 while for pressureless matter ⇢ / a3, it

can be concluded that the universe was dominated by radiation. The density of

radiation today is mostly determined by the temperature of the cosmic microwave

background (see below) so that equality takes place at a redshift

zeq ' 3612 ⇥ L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 17

the given species and, by symmetry, it is a function of t alone, Ti(t). Interacting

species have the same temperature. Among these particles, the universe contains

an electrodynamic radiation with black body spectrum (see below). Any species

interacting with photons will hence have the same temperature as these photons as

long as 18 J.-P. Uzan S´eminaire Poincar´e

Numerically, this amounts to

H(T) ⇠= 1.66g1/2 ⇤

T2

Mp

, t(T) ⇠= 0.3g L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 19

The evolution of the distribution function is obtained from the Boltzmann equa-

tion L[f] = C[f], where C describes the collisions and L = d/ds is the Liouville

operator, with s the length along a worldline. The operator L is a function of eight

variables taking the explicit form

L[f] = p↵ @

@x↵ 20 J.-P. Uzan S´eminaire Poincar´e

n

1

2

5

6

7

8

9

11

12

10

3 4

7Be

3He

1H 2H 3H

4He

7Li

1. p ´ n

2. p (n, g )d

3. d (p, g )3He

4. d (d,

Commons Attribution 4.0 International License 258 IARJSET ISSN (Online) 2393-8021

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Fig. 1 Evolution of big bang theory [4]

II. HISTORY OF BIG BANG THEORY

The history of the Big Bang theory is a journey that spans centuries and involves the contributions of many scientists and thinkers.

Here's a brief overview of key moments and figures in the development of the theory:

A. Early Concepts

⚫ Ancient Cosmological Ideas: Throughout history, various cultures and civilizations developed cosmological ideas about

the origins of the universe. These ranged from creation myths to philosophical speculations.

⚫ Georges Lemaître: A Belgian priest and physicist, Lemaître proposed the idea of an expanding universe in the late 1920s.

He suggested that if the universe is expanding now, it must have been smaller and hotter in the past. He called this initial

state the "primeval atom."

B. Hubble's Discovery

Edwin Hubble: In the 1920s, American astronomer Edwin Hubble made groundbreaking observations of galaxies and their redshifts.

He discovered that galaxies were moving away from us, and the more distant galaxies were moving faster. This observation provided

strong evidence for the expansion of the universe.

C. Formalization of the Big Bang

George Gamow: In the 1940s, physicist George Gamow developed the first detailed model of the Big Bang, incorporating the idea

of a hot, dense early universe and the subsequent expansion. He also predicted the existence of the cosmic microwave background

radiation as a remnant of the initial explosion.

D. Discovery of Cosmic Microwave Background (CMB)

Arno Penzias and Robert Wilson: In 1964, astronomers Penzias and Wilson accidentally discovered the cosmic microwave

background radiation—a faint glow of microwave radiation that fills the universe. This discovery provided strong support for the

Big Bang theory and the early, hot phase of the universe.

E. Confirmation and Refinement

In the following decades, observations of the cosmic microwave background radiation became more precise, matching the

predictions of the Big Bang theory with remarkable accuracy. The theory continued to be refined, incorporating concepts like

inflation to explain the uniformity of the universe on larger scales.

F. Modern Developments

⚫ Advances in Observations: Observational techniques, such as measuring the redshifts of galaxies and mapping the large-scale

structure of the universe, provided further confirmation of the Big Bang theory's predictions.

⚫ Dark Matter and Dark Energy: In the late 20th century, the discovery of discrepancies between observed and predicted galactic

motions led to the proposal of dark matter. Additionally, observations of the accelerated expansion of the universe led to the

concept of dark energy, which drives this acceleration. IARJSET ISSN (Online) 2393-8021

ISSN (Print) 2394-1588

International Advanced Research Journal in Science, Engineering and Technology

6

th National Conference on Science, Technology and Communication Skills – NCSTCS 2K23

Narula Institute of Technology, Agarpara, Kolkata, India

Vol. 10, Special Issue 3, September 2023

©IARJSET This work is licensed under a Creative Commons Attribution 4.0 International License 260

The Big Bang theory has evolved from early philosophical pondering to a well-supported scientific framework explaining the origin

and evolution of the universe. It has been tested through various observations and experiments, and its success in explaining a wide

range of cosmological phenomena has solidified its status as the leading explanation for the universe's history [7-10].

III. WAS THERE THE HOT BIG BANG PHASE?

The Hot Big Bang theory is the prevailing scientific explanation for the origin and evolution of the universe. It suggests that the

universe began as an extremely hot, dense, and infinitely small point known as a singularity. This singularity underwent a rapid

expansion event called cosmic inflation, causing the universe to expand and cool down over time. As the universe expanded, it also

began to evolve and develop into the vast and complex structure we observe today [10,11].

Here is a step-by-step breakdown of the Hot Big Bang theory:

A. Singularity

The universe started as a singularity—an infinitely dense and hot point with all the matter, energy, space, and time compressed

into it.

B. Inflation

In a very short period of time, known as cosmic inflation, the universe underwent a rapid and exponential expansion. This

expansion smoothed out the distribution of matter and energy and set the stage for the universe's subsequent evolution.

C. Particle Formation

As the universe expanded and cooled down, it reached a point where subatomic particles like protons, neutrons, electrons, and

their antiparticles could form. This process occurred within the first fraction of a second after the Big Bang.

D. Nuclear Fusion:

Within the first few minutes after the Big Bang, the universe was still extremely hot and dense. Nuclear reactions occurred,

primarily involving protons and neutrons, leading to the formation of light elements like hydrogen and helium. These reactions were

responsible for creating the initial abundance of these elements.

E. Photon Decoupling

As the universe continued to expand and cool, it reached a point where atoms could form—electrons combined with protons to

create neutral hydrogen atoms. This allowed photons (particles of light) to travel freely without scattering off charged particles,

resulting in the release of the cosmic microwave background radiation.

F. Structure Formation

Over millions of years, the slightly uneven distribution of matter and energy in the early universe led to the formation of cosmic

structures—galaxies, clusters of galaxies, and large-scale cosmic filaments. Gravity played a crucial role in pulling matter together

to form these structures.

G. Cosmic Microwave Background (CMB)

Fig. 2 Big Bang Theory Phase [12] IARJSET ISSN (Online) 2393-8021

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th National Conference on Science, Technology and Communication Skills – NCSTCS 2K23

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The cosmic microwave background radiation is a faint glow of microwave radiation that permeates the universe. It is a remnant of

the extremely hot and dense state of the early universe. The CMB was discovered in 1965 and provides strong evidence for the Big

Bang theory.

The Hot Big Bang theory is supported by a wide range of observations, including the cosmic microwave background radiation, the

observed abundance of light elements in the universe, the distribution of galaxies, and the overall expansion of the universe. These

pieces of evidence collectively provide a robust framework for understanding the origin and evolution of our universe [13-18].

IV. EVIDENCE / PROOF FOR THE BIG BANG

According to the Big Bang theory, there was a phase known as the "hot Big Bang" phase. This phase refers to the early moments of

the universe shortly after the initial singularity, when the universe was extremely hot and dense. During this phase, the universe

expanded and cooled, allowing various fundamental particles to form and interact with one another.

As the universe expanded and cooled, it underwent a series of transitions that are described by different eras:

A. Planck Era

This is the earliest phase of the universe, lasting from time zero (the moment of the singularity) to approximately 10-43 seconds.

During this time, the fundamental forces of nature were unified, and the conditions were so extreme that the laws of physics as we

know them today break down.

B. Grand Unified Theory (GUT) Era

From around 10-43 seconds to 10-36 seconds after the Big Bang, the universe entered the GUT era. During this phase, the strong

force, weak force, and electromagnetic force were likely combined into a single unified force.

C. Electroweak Era

Around 10-36 seconds after the Big Bang, the universe cooled down further, and the electroweak force (a combination of the

electromagnetic and weak forces) separated from the strong force.

D. Quark-Gluon Plasma Era

Between about 10-12 seconds and a few microseconds after the Big Bang, the universe was filled with a state of matter known as

quark-gluon plasma. During this era, quarks and gluons, which are usually confined within particles like protons and neutrons, were

free to move independently due to the high temperatures and energies.

E. Hadron Era

As the universe continued to expand and cool, quarks and gluons combined to form protons, neutrons, and other hadrons (particles

composed of quarks). This era lasted from a few microseconds to a few minutes after the Big Bang.

F. Nucleosynthesis Era

Around 3 minutes to 20 minutes after the Big Bang, conditions were such that nuclear reactions could occur, leading to the formation

of light elements like hydrogen, helium, and trace amounts of lithium. This era is responsible for the abundance of these elements

in the universe today.

G. Cosmic Microwave Background Era

After the universe had expanded and cooled sufficiently, around 380,000 years after the Big Bang, atoms formed and photons

(particles of light) were no longer constantly scattering off charged particles. This allowed the universe to become transparent to

light, and the cosmic microwave background radiation—the afterglow of the hot Big Bang—was emitted. This era marks a crucial

turning point in the universe's evolution.

These various eras collectively make up the "hot Big Bang" phase of the universe's history. The observations of the cosmic

microwave background radiation and the abundances of light elements are key pieces of evidence that support the existence of this

hot Big Bang phase and the overall framework of the Big Bang theory.

V. COSMIC MICROWAVE BACKGROUND RADIATION (CMBR)

According to the Big Bang theory, the Universe was initially very hot and dense. As it expanded, it cooled (your refrigerator works

on the same idea, expanding a liquid into a gas to cool the inside). Cosmologists were able to calculate the theoretical temperature

of today's Universe and began to search for evidence of it. It was eventually discovered by accident in 1964 by Arno Penzias and

Robert Wilson as 'noise' in an antenna they had built to research how radio signals could be reflected off orbiting satellites. IARJSET ISSN (Online) 2393-8021

ISSN (Print) 2394-1588

International Advanced Research Journal in Science, Engineering and Technology

6

th National Conference on Science, Technology and Communication Skills – NCSTCS 2K23

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Vol. 10, Special Issue 3, September 2023

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They first thought it was radio interference from nearby New York City, but eventually recognised it as radiation from beyond the

Milky Way. The cosmic microwave background radiation (CMBR) that Penzias and Wilson observed is leftover heat radiation from

the Big Bang. Today, CMBR is very cold due to expansion and cooling of the Universe. It's only 2.725 Kelvin (-270.4 °C), which

is only 2.725 °C above absolute zero. Cosmic microwave background radiation fills the entire Universe and can be detected day and

night in every part of the sky.

A. Studying CMBR

The Cosmic Background Explorer satellite (COBE) was launched in 1992 to look for small variations in CMBR temperature.

The Wilkinson Microwave Anisotropy Probe spacecraft (WMAP) was launched in 2001 to measure variations more accurately.

Cosmologists believe that tiny temperature variations in the CMBR are caused by differences in the density of matter in the early

Universe. Areas of different density led to the formation of galaxies and stars.

B. The WMAP Survey

After nine years of observation the WMAP survey produced a detailed temperature map of the entire sky. Colours indicate tiny

variations in the temperature of background radiation. These correspond to places where galaxies formed. The WMAP survey shows

CMBR is almost the same in all directions. Red spots are slightly warmer and blue spots are slightly cooler, but the difference is

only about ±0.0002 of a degree. The WMAP survey provides strong evidence that supports the Big Bang theory. The pattern of

radiation is similar to what astrophysicists predict it would be if the Universe started from a very dense state and expanded to its

present size.

Fig. 3 Cosmic Microwave Background Radiation [19]

VI. WHAT ARE THE WEAKNESSES OF THE BIG BANG THEORY AND OUR CURRENT CONCEPTION OF THE

ORIGIN OF THE UNIVERSE?

While the Big Bang theory is the leading cosmological model and has successfully explained many observed phenomena, there are

some questions and challenges that it doesn't fully address. Here are some of the weaknesses and unanswered questions associated

with the Big Bang theory and our current understanding of the origin of the universe:

⚫ Initial Singularity

The Big Bang theory describes the expansion of the universe from an initial singularity, but it doesn't explain what caused the

singularity or what conditions were like at that moment. The theory breaks down at the point of singularity, raising questions about

the nature of space and time at that extreme state.

⚫ Horizon Problem

The universe appears to be remarkably uniform on large scales, but different regions of the universe are too far apart to have directly

influenced each other. This poses the "horizon problem"—how did these regions achieve such a uniform temperature and structure

when they shouldn't have had enough time to communicate and equilibrate? IARJSET ISSN (Online) 2393-8021

ISSN (Print) 2394-1588

International Advanced Research Journal in Science, Engineering and Technology

6

th National Conference on Science, Technology and Communication Skills – NCSTCS 2K23

Narula Institute of Technology, Agarpara, Kolkata, India

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©IARJSET This work is licensed under a Creative Commons Attribution 4.0 International License 263

⚫ Flatness Problem

The universe is very close to being "flat" on a large scale, meaning that the angles of a triangle add up to 180 degrees and parallel

lines don't diverge. However, the theory suggests that the universe's curvature should have changed over time due to the effects of

matter and energy. The question of why the universe is so close to flatness is known as the "flatness problem."

⚫ Matter-Antimatter Asymmetry

According to the theory, equal amounts of matter and antimatter should have been produced in the early universe. However, we

observe a universe dominated by matter, and the asymmetry between matter and antimatter isn't fully explained.

⚫ Dark Matter and Dark Energy

The Big Bang theory doesn't provide a direct explanation for dark matter and dark energy, which together make up the majority

of the universe's mass-energy content. While these components are included in cosmological models, their nature and origins are

still unknown.

⚫ Inflation's Cause

While the concept of cosmic inflation helps explain certain observations, such as the uniformity of the cosmic microwave

background radiation, the exact cause of inflation and the physical mechanism driving it remain theoretical and unproven.

⚫ Cosmic Structures and Formation

While the Big Bang theory explains the overall expansion and large-scale structure of the universe, the details of how galaxies,

galaxy clusters, and other cosmic structures formed from the initial conditions are complex and not fully understood.

⚫ Dark Matter Particle

While dark matter is postulated to exist based on its gravitational effects, the nature of the dark matter particle itself remains

unknown. Efforts to directly detect dark matter particles have not yet been successful.

⚫ Quantum Gravity

The Big Bang theory doesn't account for the effects of quantum mechanics and gravity at extremely high energies and

temperatures. A complete theory of quantum gravity is still an active area of research.

⚫ Multiverse Hypotheses

Some theories suggest the possibility of a multiverse—a collection of multiple universes, each with its own set of physical laws.

While these hypotheses are interesting, they are currently speculative and difficult to test.

Despite these weaknesses and unanswered questions, the Big Bang theory remains the most widely accepted framework for

understanding the origin and evolution of the universe. Ongoing research, advancements in observational technology, and theoretical

developments continue to shed light on these challenges and may lead to a more comprehensive understanding in the future.

Fig. 4 Current concept of big bang theory [20] IARJSET ISSN (Online) 2393-8021

ISSN (Print) 2394-1588

International Advanced Research Journal in Science, Engineering and Technology

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VII. HOW DO YOU PREDICT THIS THEORY WILL EVOLVE, OR BE ADJUSTED, WITH TIME?

The evolution of scientific theories, including the Big Bang theory, is a dynamic process that is shaped by new observations,

technological advancements, and theoretical developments. While I can't predict the exact future of the theory, I can offer some

general trends and possibilities for how the Big Bang theory might evolve or be adjusted over time:

⮚ Fine-Tuning and Precision

As observational technology improves and cosmological measurements become more precise, scientists may refine various

parameters and constants associated with the Big Bang theory. This could lead to a more accurate understanding of the early

universe's conditions and evolution.

⮚ Early Universe Physics

The current theory doesn't fully account for the interactions of fundamental forces and particles at extremely high energies and

temperatures, where both quantum mechanics and gravity play significant roles. Efforts to develop a comprehensive theory of

quantum gravity could provide insights into the earliest moments of the universe.

➢ Unifying Fundamental Forces

The Big Bang theory is based on the assumption that the four fundamental forces (gravity, electromagnetism, weak force, and strong

force) were once unified. Future developments might involve finding deeper connections between these forces and explaining their

unification.

➢ Dark Matter and Dark Energy

As research progresses, we might gain a better understanding of the nature of dark matter and dark energy. Identifying the particles

responsible for dark matter and elucidating the properties of dark energy could lead to adjustments in our understanding of the

universe's evolution.

➢ Multiverse and String Theories

Concepts like the multiverse and string theory propose that our universe might be one of many, and that extra dimensions could

exist beyond the ones we currently perceive. These ideas could influence our understanding of the early universe and the conditions

The Big-Bang Theory: Construction, Evolution and Status

Jean-Philippe Uzan

Institut d'Astrophysique de Paris

UMR 7095 du CNRS,

98 bis, bd Arago

75014 Paris.

Abstract. Over the past century, rooted in the theory of general relativity, cos-

mology has developed a very successful physical model of the universe: the

big-bang model. Its construction followed di↵erent stages to incorporate nuclear

processes, the understanding of the matter present in the universe, a description

of the early universe and of the large scale structure. This model has been con-

fronted to a variety of observations that allow one to reconstruct its expansion

history, its thermal history and the structuration of matter. Hence, what we re-

fer to as the big-bang model today is radically di↵erent from what one may have

had in mind a century ago. This construction changed our vision of the universe,

both on observable scales and for the universe as a whole. It o↵ers in particular

physical models for the origins of the atomic nuclei, of matter and of the large

scale structure. This text summarizes the main steps of the construction of the

model, linking its main predictions to the observations that back them up. It

also discusses its weaknesses, the open questions and problems, among which

the need for a dark sector including dark matter and dark energy.

1 Introduction

1.1 From General Relativity to cosmology

A cosmological model is a mathematical representation of our universe that is based

on the laws of nature that have been validated locally in our Solar system and on

their extrapolations (see Refs. [1, 2, 3] for a detailed discussion). It thus seats at the

crossroad between theoretical physics and astronomy. Its basic enterprise is thus to

use tested physical laws to understand the properties and evolution of our universe

and of the matter and the astrophysical objects it contains.

Cosmology is however peculiar among sciences at least on two foundational

aspects. The uniqueness of the universe limits the standard scientific method of

comparing similar objects in order to find regularities and to test for reproductibility;

indeed this limitation depends on the question that is asked. In particular, this will

tend to blur many discussions on chance and necessity. Its historical dimension

forces us to use abduction1 together with deduction (and sometime induction) to

reconstruct the most probable cosmological scenario2. One thus needs to reconstruct

1Abduction is a form of inference which goes from an observation to a theory, ideally looking for the simplest

and most likely explanation. In this reasoning, unlike with deduction, the premises do not guarantee the conclusion, 2 J.-P. Uzan S´eminaire Poincar´e

the conditions in the primordial universe to fit best what is observed at di↵erent

epochs, given a set of physical laws. Again the distinction between laws and initial

conditions may also be subtle.This means that cosmology also involves, whether we

like it or not, some philosophical issues [4].

In particular, one carefully needs to distinguish physical cosmology from the

Cosmology that aims to propose a global picture of the universe [1]. The former

has tremendously progressed during the past decades, both from a theory and an

observation point of view. Its goal is to relate the predictions of a physical theory

of the universe to actual observations. It is thus mostly limited to our observable

universe. The latter is aiming at answering broader questions on the universe as a

whole, such as questions on origins or its finiteness but also on the apparent fine-

tuning of the laws of nature for complexity to emerge or the universe to host a viable

form of life. The boundary between these two approaches is ill-defined and moving,

particularly when it comes to recent developments such as inflation or the multiverse

debate. They are related to the two notions, the universe, i.e. the ensemble of all

what exist, and our observable universe. Both have grown due to the progresses of

our theories, that allow us to conceptualize new continents, and of the technologies,

that have extended the domain of what we can observe and test.

Indeed the physical cosmology sets very strong passive constraints on Cosmol-

ogy. It is then important to evaluate to which extent our observable universe is

representative of the universe as a whole, a question whose answer depends dras-

tically of what is meant by "universe as a whole". Both approaches are legitimate

and the general public is mostly interested by the second. This is why we have the

moral duty to state to which of those approaches we are referring to when we talk

about cosmology.

While a topic of interest for many centuries – since any civilization needs to

be structured by an anthropology and a cosmology, through mythology or science –

we can safely declare [5] that scientific cosmology was born with Albert Einstein's

general relativity a century ago. His theory of gravitation made the geometry of

spacetime dynamical physical fields, gµ⌫, that need to be determined by solving

equations known as Einstein field equations,

Gµ⌫[g↵ L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 3

(c) L. Haddad & G. Duprat

Geological data

O

Figure 1: Astrophysical data are mostly located on our past lightcone and fade away with distance from us so

that we have access to a portion of a 3-dimensional null hypersurface – an object can be observed only when its

worldline (dashed lines) intersects our past lightcone. Some geological data can be extracted on our Solar system

neighborhood. It is important to keep in mind that the interpretation of the observations is not independent of the

spacetime structure, e.g. assigning distances. We are thus looking for compatibility between a universe model and

these observations.

in the universe. Even with perfect data, the fact that (almost)3 all the information

we can extract from the universe is under the form of electromagnetic signal implies

that observations are located on our past lightcone, that is on a 3-dimensional null

hypersurface (see Fig. 1). It can be demonstrated (see e.g. Ref. [6] and Ref. [7] for

a concrete of 2 di↵erent cosmological spacetimes which enjoy the same lightcone

observations) that the 4-dimensional metric cannot be reconstructed from this in-

formation alone. Indeed a further limitation arises from the fact that there is no such

thing as perfect observations. Galaxy catalogs are limited in magnitude or redshift,

evolution e↵ects have to be taken into account, some components of matter (such a

cold di↵use gas or dark matter) cannot be observed electromagnetically. We do not

observe the whole matter distribution but rather classes of particular objects (stars,

galaxies,...) and we need to deal with the variations in the properties of these indi-

vidual objects and evolution e↵ects. A di 4 J.-P. Uzan S´eminaire Poincar´e

them to describe our spacetime from stellar scales to the Hubble scale.

The first relativistic cosmological model [9] was constructed by Einstein in 1917

and it can be considered as the birthdate of modern cosmology. As we shall see,

most models rely on some hypotheses that are di L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 5

it does not prove that it is the "correct" model of the universe, in the sense that it

is the correct cosmological extrapolation and solution of the local physical laws.

When confronted with an inconsistency in the model, one can either invoke the

need for new physics, i.e. a modification of the laws of physics we have extrapolated

in a regime outside of the domain of validity that has been established so far (e.g.

large cosmological distance, low curvature regimes etc.), or have a more conservative

attitude concerning fundamental physics and modify the cosmological hypotheses.

Let us start by reminding that the construction of any cosmological model relies

on 4 main hypotheses (see Ref. [3] for a detailed description),

(H1) a theory of gravity,

(H2) a description of the matter contained in the universe and their non-gravitational

interactions,

(H3) symmetry hypothesis,

(H4) a hypothesis on the global structure, i.e. the topology, of the universe.

These hypotheses are indeed not on the same footing since H1 and H2 refer to

the local (fundamental) physical theories. These two hypotheses are however not

su 6 J.-P. Uzan S´eminaire Poincar´e

1.2.2 Non-gravitational sector

Einstein equivalence principle, as the heart of general relativity, also implies that the

laws of non-gravitational physics validated locally can be extrapolated. In particular

the constants of nature shall remain constant, a prediction that can also be tested

on astrophysical scales [17, 18]. Our cosmological model assumes (H2) that the

matter and non-gravitational interactions are described by the standard model of

particle physics. As will be discussed later, but this is no breaking news, modern

cosmology requires the universe to contain some dark matter (DM) and a non-

vanishing cosmological constant (⇤). Their existence is inferred from cosmological

observations assuming the validity of general relativity (e.g. flat rotation curves,

large scale structure, dynamics of galaxy clusters for dark matter, accelerated cosmic

expansion for the cosmological constant; see chapters 7 and 12 of Ref. [19]). Dark

matter sets many questions on the standard model of particle physics and its possible

extensions since the physical nature of this new field has to be determined and

integrated consistently in the model. The cosmological constant problem is argued

to be a sign of a multiverse, indeed a very controversial statement. If solved then

one needs to infer some dark energy to be consistently included.

We thus assume that the action of the non-gravitational sector is of the form

S =

Z

L( , gµ⌫)

p L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 7

point of view, it can be shown that it implies that the metric of the universe reduces

to the Friedmann-Lemaˆıtre form (see e.g. chapter 3 of Ref. [19])

ds2 = 8 J.-P. Uzan S´eminaire Poincar´e

since then, for all g, L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 9

This allowed Ralph Alpher, Hans Bethe and George Gamow [31] to predict the ex-

istence and estimate the temperature of a cosmic microwave background (CMB)

radiation and understand the synthesis of light nuclei, the big-bang nucleosynthesis

(BBN), in the early universe. Both have led to theoretical developments compared

successfully to observation. It was understood that the universe is filled with a ther-

a

a0

H0t H0t H0t

H0t H0t H0t

LK

Figure 5: Depending on the values of the cosmological parameters (upper panel), the scale factor can have very

di↵erent evolutions. It includes bouncing solutions with no big-bang (i.e. no initial singularity), hesitating universes

(with a limiting case in which the universe is asymptotically initially Einstein static), collapsing universes. The

expansion can be accelerating or decelerating. From Ref. [19].

as first established by Lemaˆıtre [27]. This gives the first observational pillar

of the model and the order of magnitudes of the Hubble time and radius are

obtained by expressing the current value of the Hubble parameter as H0 =

100 h km · s L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 15

with E ⌘ H/H0. The use of standard candels, such as type Ia supernovae allow

to constrain the parameter of the models and measured the Hubble constant

(see Fig. 6).

10 16 J.-P. Uzan S´eminaire Poincar´e

2.3 The hot big-bang model

As challenged by Georges Lemaˆıtre [48] in 1931, "une cosmogonie vraiment compl`ete

devrait expliquer les atomes comme les soleils." This is mostly what the hot big-bang

model will achieved.

This next evolution takes into account a better description of the matter content

of the universe. Since for radiation ⇢ / a4 while for pressureless matter ⇢ / a3, it

can be concluded that the universe was dominated by radiation. The density of

radiation today is mostly determined by the temperature of the cosmic microwave

background (see below) so that equality takes a

a0

H0t H0t H0t

H0t H0t H0t

LK

Figure 5: Depending on the values of the cosmological parameters (upper panel), the scale factor can have very

di↵erent evolutions. It includes bouncing solutions with no big-bang (i.e. no initial singularity), hesitating universes

(with a limiting case in which the universe is asymptotically initially Einstein static), collapsing universes. The

expansion can be accelerating or decelerating. From Ref. [19].

as first established by Lemaˆıtre [27]. This gives the first observational pillar

of the model and the order of magnitudes of the Hubble time and radius are

obtained by expressing the current value of the Hubble parameter as H0 =

100 h km · s L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 15

with E ⌘ H/H0. The use of standard candels, such as type Ia supernovae allow

to constrain the parameter of the models and measured the Hubble constant

(see Fig. 6).

10 16 J.-P. Uzan S´eminaire Poincar´e

2.3 The hot big-bang model

As challenged by Georges Lemaˆıtre [48] in 1931, "une cosmogonie vraiment compl`ete

devrait expliquer les atomes comme les soleils." This is mostly what the hot big-bang

model will achieved.

This next evolution takes into account a better description of the matter content

of the universe. Since for radiation ⇢ / a4 while for pressureless matter ⇢ / a3, it

can be concluded that the universe was dominated by radiation. The density of

radiation today is mostly determined by the temperature of the cosmic microwave

background (see below) so that equality takes a

a0

H0t H0t H0t

H0t H0t H0t

LK

Figure 5: Depending on the values of the cosmological parameters (upper panel), the scale factor can have very

di↵erent evolutions. It includes bouncing solutions with no big-bang (i.e. no initial singularity), hesitating universes

(with a limiting case in which the universe is asymptotically initially Einstein static), collapsing universes. The

expansion can be accelerating or decelerating. From Ref. [19].

as first established by Lemaˆıtre [27]. This gives the first observational pillar

of the model and the order of magnitudes of the Hubble time and radius are

obtained by expressing the current value of the Hubble parameter as H0 =

100 h km · s L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 15

with E ⌘ H/H0. The use of standard candels, such as type Ia supernovae allow

to constrain the parameter of the models and measured the Hubble constant

(see Fig. 6).

10 16 J.-P. Uzan S´eminaire Poincar´e

2.3 The hot big-bang model

As challenged by Georges Lemaˆıtre [48] in 1931, "une cosmogonie vraiment compl`ete

devrait expliquer les atomes comme les soleils." This is mostly what the hot big-bang

model will achieved.

This next evolution takes into account a better description of the matter content

of the universe. Since for radiation ⇢ / a4 while for pressureless matter ⇢ / a3, it

can be concluded that the universe was dominated by radiation. The density of

radiation today is mostly determined by the temperature of the cosmic microwave

background (see below) so that equality takes place at a redshift

zeq ' 3612 ⇥ L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 17

the given species and, by symmetry, it is a function of t alone, Ti(t). Interacting

species have the same temperature. Among these particles, the un at a redshift

zeq ' 3612 ⇥ L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 17

the given species and, by symmetry, it is a function of t alone, Ti(t). Interacting

species have the same temperature. Among these particles, the un at a redshift

zeq ' 3612 ⇥ L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 17

the given species and, by symmetry, it is a function of t alone, Ti(t). Interacting

species have the same temperature. Among these particles, the un

mal bath with a black-body spectrum, the temperature of which decreases with the

expansion of the universe. The universe cools down and has a thermal history, and

more important it was concluded that it emerges from a hot and dense phase at ther-

mal equilibrium (see e.g. Ref [19] for the details). This model has however several

problems, such as the fact that the universe is spatially extremely close to Euclidean

(flatness problem), the fact that it has an initial spacelike singularity (known as big-

bang) and the fact that thermal equilibrium, homogeneity and isotropy are set as

initial conditions and not explained (horizon problem). It is also too idealized since

it describes no structure, i.e. does not account for the inhomogeneities of the matter,

which is obviously distributed in galaxies, clusters and voids. The resolution of the

naturalness of the initial conditions was solved by the postulate [32] of the existence

of a primordial accelerated expansion phase, called inflation.

The third and fourth periods were triggered by an analysis of the growth of the

density inhomogeneities by Lifshitz [33], opening the understanding of the evolution

of the large scale structure of the universe, that is of the distribution of the galax-

ies in cluster, filaments and voids. Technically, it opens the way to the theory of

cosmological perturbations [34, 35, 36] in which one considers the FL spacetime as

a background spacetime the geometry and matter content of which are perturbed.

The evolution of these perturbations can be derived from the Einstein equations. For

the mechanism studied by Evgeny Lifshitz to be e 10 J.-P. Uzan S´eminaire Poincar´e

results of the analysis of the Hubble diagram of type Ia supernovae in 1999 [40].

This ⇤CDM model is in agreement with all the existing observations of the large

survey (galaxy catalogs, CMB, weak lensing, Hubble diagram etc.) and its parame-

ters are measured with increasing accuracy. This has opened the era of observational

cosmology with the open question of the physical nature of the dark sector.

This is often advertised as precision cosmology, mostly because of the increase

of the quality of the observations, which allow one to derive sharp constraints on

the cosmological parameters. One has however to be aware, that these parameters

are defined within a very specific model and require many theoretical developments

(and approximations) to compare the predictions of the model to the data. Both

set a limit a on the accuracy of the interpretation of the data; see e.g. Refs. [41, 42]

for an example of the influence of the small scale structure of the universe on the

accuracy of the inference of the cosmological parameters. And indeed, measuring

these parameters with a higher accuracy often does not shed more light on their

physical nature.

The standard history of our universe, according to this model, is summarized in

Fig. 2. Interestingly, the evolution of the universe and of its structures spans a period

ranging from about 1 second after the big-bang to today. This is made possible by

the fact that (1) the relevant microphysics is well-known in the range of energies

reached by the thermal bath during that period (< 100 MeV typically) so that it

involves no speculative physics, and (2) most of the observables can be described by

a linear perturbation theory, which technically simplifies the analysis.

This description is in agreement with all observations performed so far (big-

bang nucleosynthesis abundances, cosmic microwave background temperature and

polarisation anisotropies, distribution of galaxies and galaxy clusters given by large

catalogs and weak lensing observations, supernovae data and their implication for

the Hubble diagram).

This short summary shows that today inflation is a cornerstone of the standard

cosmological model and emphasizes its roles in the development and architecture of

the model,

1. it was postulated in order to explain the required fine-tuning of the initial

conditions of the hot big-bang model,

2. it provides a mechanism for the origin of the large scale structure,

3. it gives a new and unexpected vision of the universe on large scale,

4. it connects, in principle [43], cosmology to high energy physics.

This construction is the endpoint of about one century of theoretical and observa-

tional developments, that we will now detail.

2.2 Relativistic cosmology

2.2.1 Einstein static universe (1917)

The Einstein static universe is a static homogeneous and isotropic universe with

compact spatial sections, hence enjoying the topology of a 3-sphere. It is thus char-

acterized by 3 quantities: the radius of the 3-sphere, the matter density and the L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 11

Figure 2: The standard history of our universe. The local universe provides observations on phenomena from big-

bang nucleosynthesis to today spanning a range between 10 12 J.-P. Uzan S´eminaire Poincar´e

Figure 3: Penrose diagrams of de Sitter space in the flat (left) and static (right) slicings that each cover only part

of the whole de Sitter space, and that are both geodesically incomplete. From Ref. [22].

sections and scale factor a / cosh Ht with H = p⇤/3 constant. It is the only

geodesically complete representation;

2. the flat slicing in which the metric has a FL form (5) with Euclidean spatial

section, in which case a / exp Ht;

3. the hyperbolic slicing in which the metric has a FL form (5) with K = L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 13

to be integrated. It is convenient to use the conformal time defined by dt = a(⌘)d⌘

and the normalized density parameters

⌦i = 8⇡G⇢i/3H2

0 , ⌦⇤ = ⇤/3H2

0 , ⌦K = 14 J.-P. Uzan S´eminaire Poincar´e

Hyperbolic

Spherical

3.0

2.0

1.0

0.0

– 1.0

Bouncing

N

M

F E

I

B

J

A C

H

G

D

Ωm0

ΩΛ0

0.0 1.0 2.0 3.0

– 3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

B I

J

Hyperbolic

A

F E

– 3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

Euclidean

–3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

B A

C

D

Λ = 0

H

D

G

–3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

Spherical

– 3 – 2 –1 0 1 2 3

3.0

2.0

1.0

0.0

K

L F

Hesitating

(Spherical

or

Euclidean)

M

N

Bouncing

(Spherical)

– 3 – 2 –1 0 1 2 3

3.0

2.0

1.0

0.0

a

a0

a

a0

a

a0

a

a0

a

a0

a

a0

H0t H0t H0t

H0t H0t H0t

LK

Figure 5: Depending on the values of the cosmological parameters (upper panel), the scale factor can have very

di↵erent evolutions. It includes bouncing solutions with no big-bang (i.e. no initial singularity), hesitating universes

(with a limiting case in which the universe is asymptotically initially Einstein static), collapsing universes. The

expansion can be accelerating or decelerating. From Ref. [19].

as first established by Lemaˆıtre [27]. This gives the first observational pillar

of the model and the order of magnitudes of the Hubble time and radius are

obtained by expressing the current value of the Hubble parameter as H0 =

100 h km · s L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 15

with E ⌘ H/H0. The use of standard candels, such as type Ia supernovae allow

to constrain the parameter of the models and measured the Hubble constant

(see Fig. 6).

10 16 J.-P. Uzan S´eminaire Poincar´e

2.3 The hot big-bang model

As challenged by Georges Lemaˆıtre [48] in 1931, "une cosmogonie vraiment compl`ete

devrait expliquer les atomes comme les soleils." This is mostly what the hot big-bang

model will achieved.

This next evolution takes into account a better description of the matter content

of the universe. Since for radiation ⇢ / a4 while for pressureless matter ⇢ / a3, it

can be concluded that the universe was dominated by radiation. The density of

radiation today is mostly determined by the temperature of the cosmic microwave

background (see below) so that equality takes place at a redshift

zeq ' 3612 ⇥ L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 17

the given species and, by symmetry, it is a function of t alone, Ti(t). Interacting

species have the same temperature. Among these particles, the universe contains

an electrodynamic radiation with black body spectrum (see below). Any species

interacting with photons will hence have the same temperature as these photons as

long as 18 J.-P. Uzan S´eminaire Poincar´e

Numerically, this amounts to

H(T) ⇠= 1.66g1/2 ⇤

T2

Mp

, t(T) ⇠= 0.3g L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 19

The evolution of the distribution function is obtained from the Boltzmann equa-

tion L[f] = C[f], where C describes the collisions and L = d/ds is the Liouville

operator, with s the length along a worldline. The operator L is a function of eight

variables taking the explicit form

L[f] = p↵ @

@x↵ 20 J.-P. Uzan S´eminaire Poincar´e

n

1

2

5

6

7

8

9

11

12

10

3 4

7Be

3He

1H 2H 3H

4He

7Li

1. p ´ n

2. p (n, g )d

3. d (p, g )3He

4. d (d,

The Big-Bang Theory: Construction, Evolution and Status

Jean-Philippe Uzan

Institut d'Astrophysique de Paris

UMR 7095 du CNRS,

98 bis, bd Arago

75014 Paris.

Abstract. Over the past century, rooted in the theory of general relativity, cos-

mology has developed a very successful physical model of the universe: the

big-bang model. Its construction followed di↵erent stages to incorporate nuclear

processes, the understanding of the matter present in the universe, a description

of the early universe and of the large scale structure. This model has been con-

fronted to a variety of observations that allow one to reconstruct its expansion

history, its thermal history and the structuration of matter. Hence, what we re-

fer to as the big-bang model today is radically di↵erent from what one may have

had in mind a century ago. This construction changed our vision of the universe,

both on observable scales and for the universe as a whole. It o↵ers in particular

physical models for the origins of the atomic nuclei, of matter and of the large

scale structure. This text summarizes the main steps of the construction of the

model, linking its main predictions to the observations that back them up. It

also discusses its weaknesses, the open questions and problems, among which

the need for a dark sector including dark matter and dark energy.

1 Introduction

1.1 From General Relativity to cosmology

A cosmological model is a mathematical representation of our universe that is based

on the laws of nature that have been validated locally in our Solar system and on

their extrapolations (see Refs. [1, 2, 3] for a detailed discussion). It thus seats at the

crossroad between theoretical physics and astronomy. Its basic enterprise is thus to

use tested physical laws to understand the properties and evolution of our universe

and of the matter and the astrophysical objects it contains.

Cosmology is however peculiar among sciences at least on two foundational

aspects. The uniqueness of the universe limits the standard scientific method of

comparing similar objects in order to find regularities and to test for reproductibility;

indeed this limitation depends on the question that is asked. In particular, this will

tend to blur many discussions on chance and necessity. Its historical dimension

forces us to use abduction1 together with deduction (and sometime induction) to

reconstruct the most probable cosmological scenario2. One thus needs to reconstruct

1Abduction is a form of inference which goes from an observation to a theory, ideally looking for the simplest

and most likely explanation. In this reasoning, unlike with deduction, the premises do not guarantee the conclusion, 2 J.-P. Uzan S´eminaire Poincar´e

the conditions in the primordial universe to fit best what is observed at di↵erent

epochs, given a set of physical laws. Again the distinction between laws and initial

conditions may also be subtle.This means that cosmology also involves, whether we

like it or not, some philosophical issues [4].

In particular, one carefully needs to distinguish physical cosmology from the

Cosmology that aims to propose a global picture of the universe [1]. The former

has tremendously progressed during the past decades, both from a theory and an

observation point of view. Its goal is to relate the predictions of a physical theory

of the universe to actual observations. It is thus mostly limited to our observable

universe. The latter is aiming at answering broader questions on the universe as a

whole, such as questions on origins or its finiteness but also on the apparent fine-

tuning of the laws of nature for complexity to emerge or the universe to host a viable

form of life. The boundary between these two approaches is ill-defined and moving,

particularly when it comes to recent developments such as inflation or the multiverse

debate. They are related to the two notions, the universe, i.e. the ensemble of all

what exist, and our observable universe. Both have grown due to the progresses of

our theories, that allow us to conceptualize new continents, and of the technologies,

that have extended the domain of what we can observe and test.

Indeed the physical cosmology sets very strong passive constraints on Cosmol-

ogy. It is then important to evaluate to which extent our observable universe is

representative of the universe as a whole, a question whose answer depends dras-

tically of what is meant by "universe as a whole". Both approaches are legitimate

and the general public is mostly interested by the second. This is why we have the

moral duty to state to which of those approaches we are referring to when we talk

about cosmology.

While a topic of interest for many centuries – since any civilization needs to

be structured by an anthropology and a cosmology, through mythology or science –

we can safely declare [5] that scientific cosmology was born with Albert Einstein's

general relativity a century ago. His theory of gravitation made the geometry of

spacetime dynamical physical fields, gµ⌫, that need to be determined by solving

equations known as Einstein field equations,

Gµ⌫[g↵ L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 3

(c) L. Haddad & G. Duprat

Geological data

O

Figure 1: Astrophysical data are mostly located on our past lightcone and fade away with distance from us so

that we have access to a portion of a 3-dimensional null hypersurface – an object can be observed only when its

worldline (dashed lines) intersects our past lightcone. Some geological data can be extracted on our Solar system

neighborhood. It is important to keep in mind that the interpretation of the observations is not independent of the

spacetime structure, e.g. assigning distances. We are thus looking for compatibility between a universe model and

these observations.

in the universe. Even with perfect data, the fact that (almost)3 all the information

we can extract from the universe is under the form of electromagnetic signal implies

that observations are located on our past lightcone, that is on a 3-dimensional null

hypersurface (see Fig. 1). It can be demonstrated (see e.g. Ref. [6] and Ref. [7] for

a concrete of 2 di↵erent cosmological spacetimes which enjoy the same lightcone

observations) that the 4-dimensional metric cannot be reconstructed from this in-

formation alone. Indeed a further limitation arises from the fact that there is no such

thing as perfect observations. Galaxy catalogs are limited in magnitude or redshift,

evolution e↵ects have to be taken into account, some components of matter (such a

cold di↵use gas or dark matter) cannot be observed electromagnetically. We do not

observe the whole matter distribution but rather classes of particular objects (stars,

galaxies,...) and we need to deal with the variations in the properties of these indi-

vidual objects and evolution e↵ects. A di 4 J.-P. Uzan S´eminaire Poincar´e

them to describe our spacetime from stellar scales to the Hubble scale.

The first relativistic cosmological model [9] was constructed by Einstein in 1917

and it can be considered as the birthdate of modern cosmology. As we shall see,

most models rely on some hypotheses that are di L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 5

it does not prove that it is the "correct" model of the universe, in the sense that it

is the correct cosmological extrapolation and solution of the local physical laws.

When confronted with an inconsistency in the model, one can either invoke the

need for new physics, i.e. a modification of the laws of physics we have extrapolated

in a regime outside of the domain of validity that has been established so far (e.g.

large cosmological distance, low curvature regimes etc.), or have a more conservative

attitude concerning fundamental physics and modify the cosmological hypotheses.

Let us start by reminding that the construction of any cosmological model relies

on 4 main hypotheses (see Ref. [3] for a detailed description),

(H1) a theory of gravity,

(H2) a description of the matter contained in the universe and their non-gravitational

interactions,

(H3) symmetry hypothesis,

(H4) a hypothesis on the global structure, i.e. the topology, of the universe.

These hypotheses are indeed not on the same footing since H1 and H2 refer to

the local (fundamental) physical theories. These two hypotheses are however not

su 6 J.-P. Uzan S´eminaire Poincar´e

1.2.2 Non-gravitational sector

Einstein equivalence principle, as the heart of general relativity, also implies that the

laws of non-gravitational physics validated locally can be extrapolated. In particular

the constants of nature shall remain constant, a prediction that can also be tested

on astrophysical scales [17, 18]. Our cosmological model assumes (H2) that the

matter and non-gravitational interactions are described by the standard model of

particle physics. As will be discussed later, but this is no breaking news, modern

cosmology requires the universe to contain some dark matter (DM) and a non-

vanishing cosmological constant (⇤). Their existence is inferred from cosmological

observations assuming the validity of general relativity (e.g. flat rotation curves,

large scale structure, dynamics of galaxy clusters for dark matter, accelerated cosmic

expansion for the cosmological constant; see chapters 7 and 12 of Ref. [19]). Dark

matter sets many questions on the standard model of particle physics and its possible

extensions since the physical nature of this new field has to be determined and

integrated consistently in the model. The cosmological constant problem is argued

to be a sign of a multiverse, indeed a very controversial statement. If solved then

one needs to infer some dark energy to be consistently included.

We thus assume that the action of the non-gravitational sector is of the form

S =

Z

L( , gµ⌫)

p L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 7

point of view, it can be shown that it implies that the metric of the universe reduces

to the Friedmann-Lemaˆıtre form (see e.g. chapter 3 of Ref. [19])

ds2 = 8 J.-P. Uzan S´eminaire Poincar´e

since then, for all g, L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 9

This allowed Ralph Alpher, Hans Bethe and George Gamow [31] to predict the ex-

istence and estimate the temperature of a cosmic microwave background (CMB)

radiation and understand the synthesis of light nuclei, the big-bang nucleosynthesis

(BBN), in the early universe. Both have led to theoretical developments compared

successfully to observation. It was understood that the universe is filled with a ther-

mal bath with a black-body spectrum, the temperature of which decreases with the

expansion of the universe. The universe cools down and has a thermal history, and

more important it was concluded that it emerges from a hot and dense phase at ther-

mal equilibrium (see e.g. Ref [19] for the details). This model has however several

problems, such as the fact that the universe is spatially extremely close to Euclidean

(flatness problem), the fact that it has an initial spacelike singularity (known as big-

bang) and the fact that thermal equilibrium, homogeneity and isotropy are set as

initial conditions and not explained (horizon problem). It is also too idealized since

it describes no structure, i.e. does not account for the inhomogeneities of the matter,

which is obviously distributed in galaxies, clusters and voids. The resolution of the

naturalness of the initial conditions was solved by the postulate [32] of the existence

of a primordial accelerated expansion phase, called inflation.

The third and fourth periods were triggered by an analysis of the growth of the

density inhomogeneities by Lifshitz [33], opening the understanding of the evolution

of the large scale structure of the universe, that is of the distribution of the galax-

ies in cluster, filaments and voids. Technically, it opens the way to the theory of

cosmological perturbations [34, 35, 36] in which one considers the FL spacetime as

a background spacetime the geometry and matter content of which are perturbed.

The evolution of these perturbations can be derived from the Einstein equations. For

the mechanism studied by Evgeny Lifshitz to be e 10 J.-P. Uzan S´eminaire Poincar´e

results of the analysis of the Hubble diagram of type Ia supernovae in 1999 [40].

This ⇤CDM model is in agreement with all the existing observations of the large

survey (galaxy catalogs, CMB, weak lensing, Hubble diagram etc.) and its parame-

ters are measured with increasing accuracy. This has opened the era of observational

cosmology with the open question of the physical nature of the dark sector.

This is often advertised as precision cosmology, mostly because of the increase

of the quality of the observations, which allow one to derive sharp constraints on

the cosmological parameters. One has however to be aware, that these parameters

are defined within a very specific model and require many theoretical developments

(and approximations) to compare the predictions of the model to the data. Both

set a limit a on the accuracy of the interpretation of the data; see e.g. Refs. [41, 42]

for an example of the influence of the small scale structure of the universe on the

accuracy of the inference of the cosmological parameters. And indeed, measuring

these parameters with a higher accuracy often does not shed more light on their

physical nature.

The standard history of our universe, according to this model, is summarized in

Fig. 2. Interestingly, the evolution of the universe and of its structures spans a period

ranging from about 1 second after the big-bang to today. This is made possible by

the fact that (1) the relevant microphysics is well-known in the range of energies

reached by the thermal bath during that period (< 100 MeV typically) so that it

involves no speculative physics, and (2) most of the observables can be described by

a linear perturbation theory, which technically simplifies the analysis.

This description is in agreement with all observations performed so far (big-

bang nucleosynthesis abundances, cosmic microwave background temperature and

polarisation anisotropies, distribution of galaxies and galaxy clusters given by large

catalogs and weak lensing observations, supernovae data and their implication for

the Hubble diagram).

This short summary shows that today inflation is a cornerstone of the standard

cosmological model and emphasizes its roles in the development and architecture of

the model,

1. it was postulated in order to explain the required fine-tuning of the initial

conditions of the hot big-bang model,

2. it provides a mechanism for the origin of the large scale structure,

3. it gives a new and unexpected vision of the universe on large scale,

4. it connects, in principle [43], cosmology to high energy physics.

This construction is the endpoint of about one century of theoretical and observa-

tional developments, that we will now detail.

2.2 Relativistic cosmology

2.2.1 Einstein static universe (1917)

The Einstein static universe is a static homogeneous and isotropic universe with

compact spatial sections, hence enjoying the topology of a 3-sphere. It is thus char-

acterized by 3 quantities: the radius of the 3-sphere, the matter density and the L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 11

Figure 2: The standard history of our universe. The local universe provides observations on phenomena from big-

bang nucleosynthesis to today spanning a range between 10 12 J.-P. Uzan S´eminaire Poincar´e

Figure 3: Penrose diagrams of de Sitter space in the flat (left) and static (right) slicings that each cover only part

of the whole de Sitter space, and that are both geodesically incomplete. From Ref. [22].

sections and scale factor a / cosh Ht with H = p⇤/3 constant. It is the only

geodesically complete representation;

2. the flat slicing in which the metric has a FL form (5) with Euclidean spatial

section, in which case a / exp Ht;

3. the hyperbolic slicing in which the metric has a FL form (5) with K = L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 13

to be integrated. It is convenient to use the conformal time defined by dt = a(⌘)d⌘

and the normalized density parameters

⌦i = 8⇡G⇢i/3H2

0 , ⌦⇤ = ⇤/3H2

0 , ⌦K = 14 J.-P. Uzan S´eminaire Poincar´e

Hyperbolic

Spherical

3.0

2.0

1.0

0.0

– 1.0

Bouncing

N

M

F E

I

B

J

A C

H

G

D

Ωm0

ΩΛ0

0.0 1.0 2.0 3.0

– 3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

B I

J

Hyperbolic

A

F E

– 3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

Euclidean

–3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

B A

C

D

Λ = 0

H

D

G

–3 – 2 – 1 0 1 2 3

3.0

2.0

1.0

0.0

Spherical

– 3 – 2 –1 0 1 2 3

3.0

2.0

1.0

0.0

K

L F

Hesitating

(Spherical

or

Euclidean)

M

N

Bouncing

(Spherical)

– 3 – 2 –1 0 1 2 3

3.0

2.0

1.0

0.0

a

a0

a

a0

a

a0

a

a0

a

a0

a

a0

H0t H0t H0t

H0t H0t H0t

LK

Figure 5: Depending on the values of the cosmological parameters (upper panel), the scale factor can have very

di↵erent evolutions. It includes bouncing solutions with no big-bang (i.e. no initial singularity), hesitating universes

(with a limiting case in which the universe is asymptotically initially Einstein static), collapsing universes. The

expansion can be accelerating or decelerating. From Ref. [19].

as first established by Lemaˆıtre [27]. This gives the first observational pillar

of the model and the order of magnitudes of the Hubble time and radius are

obtained by expressing the current value of the Hubble parameter as H0 =

100 h km · s L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 15

with E ⌘ H/H0. The use of standard candels, such as type Ia supernovae allow

to constrain the parameter of the models and measured the Hubble constant

(see Fig. 6).

10 16 J.-P. Uzan S´eminaire Poincar´e

2.3 The hot big-bang model

As challenged by Georges Lemaˆıtre [48] in 1931, "une cosmogonie vraiment compl`ete

devrait expliquer les atomes comme les soleils." This is mostly what the hot big-bang

model will achieved.

This next evolution takes into account a better description of the matter content

of the universe. Since for radiation ⇢ / a4 while for pressureless matter ⇢ / a3, it

can be concluded that the universe was dominated by radiation. The density of

radiation today is mostly determined by the temperature of the cosmic microwave

background (see below) so that equality takes place at a redshift

zeq ' 3612 ⇥ L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 17

the given species and, by symmetry, it is a function of t alone, Ti(t). Interacting

species have the same temperature. Among these particles, the universe contains

an electrodynamic radiation with black body spectrum (see below). Any species

interacting with photons will hence have the same temperature as these photons as

long as 18 J.-P. Uzan S´eminaire Poincar´e

Numerically, this amounts to

H(T) ⇠= 1.66g1/2 ⇤

T2

Mp

, t(T) ⇠= 0.3g L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 19

The evolution of the distribution function is obtained from the Boltzmann equa-

tion L[f] = C[f], where C describes the collisions and L = d/ds is the Liouville

operator, with s the length along a worldline. The operator L is a function of eight

variables taking the explicit form

L[f] = p↵ @

@x↵ 20 J.-P. Uzan S´eminaire Poincar´e

n

1

2

5

6

7

8

9

11

12

10

3 4

7Be

3He

1H 2H 3H

4He

7Li

1. p ´ n

2. p (n, g )d

3. d (p, g )3He

4. d (d,

a

a0

H0t H0t H0t

H0t H0t H0t

LK

Figure 5: Depending on the values of the cosmological parameters (upper panel), the scale factor can have very

di↵erent evolutions. It includes bouncing solutions with no big-bang (i.e. no initial singularity), hesitating universes

(with a limiting case in which the universe is asymptotically initially Einstein static), collapsing universes. The

expansion can be accelerating or decelerating. From Ref. [19].

as first established by Lemaˆıtre [27]. This gives the first observational pillar

of the model and the order of magnitudes of the Hubble time and radius are

obtained by expressing the current value of the Hubble parameter as H0 =

100 h km · s L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 15

with E ⌘ H/H0. The use of standard candels, such as type Ia supernovae allow

to constrain the parameter of the models and measured the Hubble constant

(see Fig. 6).

10 16 J.-P. Uzan S´eminaire Poincar´e

2.3 The hot big-bang model

As challenged by Georges Lemaˆıtre [48] in 1931, "une cosmogonie vraiment compl`ete

devrait expliquer les atomes comme les soleils." This is mostly what the hot big-bang

model will achieved.

This next evolution takes into account a better description of the matter content

of the universe. Since for radiation ⇢ / a4 while for pressureless matter ⇢ / a3, it

can be concluded that the universe was dominated by radiation. The density of

radiation today is mostly determined by the temperature of the cosmic microwave

background (see below) so that equality takes place at a redshift

zeq ' 3612 ⇥ L'Univers, Vol. XX, 2015 The Big-Bang Theory: Construction, Evolution and Status 17

the given species and, by symmetry, it is a function of t alone, Ti(t). Interacting

species have the same temperature. Among these particles, the un