本文是数学专业的留学生作业范例，题目是“Mathematical Basis of Big Bang Cosmology(大爆炸宇宙学的数学基础)”宇宙学是对整个宇宙的研究，即宇宙的起源、演化和最终命运。在过去的100年里，我们取得了巨大的进步。宇宙最初被爱因斯坦认为是静止的，但后来被哈勃的观测证实，宇宙实际上是在膨胀的。这一发现导致了大爆炸理论的发展，该理论目前仍在完善中。本文旨在利用大爆炸模型来解释宇宙的起源。在第一节中，我们将简要回顾现代宇宙学的历史，然后总结形成大爆炸理论的数学基础，并提出支持大爆炸理论的三条主要证据。最后，关注科学理论/模型与现实世界的关系，指出模型只能部分地代表现实和现实。
Cosmology is the study of the entire universe i.e the origin, evolution and its ultimate fate. Within the past 100 years, tremendous advances have been made. The universe was first thought of as static by Einstein but was later confirmed by Hubble’s observation that the universe is in fact expanding. This discovery led to the development of the Big Bang theory which is still currently being refined. This paper aims to offer an explanation of the origins of the universe by using Big Bang model. In the first section, we would briefly look at the history of modern cosmology, followed by a summary of the mathematical basis that forms the Big Bang theory as well as present the three main lines of evidence that support the Big Bang theory. Lastly, attention is drawn to the relationship between the scientific theories/models and the real world, pointing out that models can only partially represent reality and
Rise of the Big Bang Model大爆炸模型的兴起
Soon after Einstein had formulated the general theory of relativity, in 1917 he postulated that the universe is homogenous, isotropic, static, with a constant mean mass density and curvature of space. However, this model was not permitted by his field equations as they suggested that the universe would collapse due to the gravitational force acting on matter. Therefore, he added a cosmological constant, Λ to the field equation to counteract this effect.
In 1922. Alexander Friedmann used Einstein’s field equations to show that a dynamical universe was also allowed. Five years later, George Lemaître, a physicist and Belgian Roman Catholic priest, unaware of Friedmann’s work, independently came up with a dynamical model that included mass and pressure. He managed to theoretically derive a linear relation between the recession velocity and distance of the nebulae using the latest observations provided by Slipher (whose data was published in Eddington’s book of 1923) and Hubble . This relationship is currently known as Hubble’s Law. Lemaître interpreted the observed redshifts as an effect of cosmic expansion. He published his findings that year in an obscure journal in French and sent a copy to his former mentor, Eddington. Unfortunately, it went unnoticed. During the 1927 Solvay congress, Lemaître took the opportunity to give a copy of the reprint of his findings to Einstein, who found it abominable as it made no sense to him from a physical point of view. According to Lemaître, it seemed like Einstein was unaware of the cosmological observations then .
In 1929. Hubble published a paper which led to the realisation of an expanding universe. Using data on Doppler redshifts mostly given by Slipher combined with the measured distances to the spiral nebulae, he too found a linear correlation between distances and velocities . Hubble’s work was very similar to Lemaître’s, however he merely interpreted them as Doppler effects and never considered them as a consequence of an expanding universe.
During the Royal Astronomical Society (RAS) meeting that took place in 1930. Eddington discussed the instability of Einstein’s model and called for the search of new explanations for the recession of velocities using dynamical models . Lemaître read the proceedings of the meeting and immediately wrote to Eddington to remind him of his paper published in 1927. Eddington was impressed and arranged for it to be translated into English. It was published in the monthly notices of the RAS in 1931. The paragraphs concerning Hubble’s law were deleted by Lemaître himself and thus he was never credited for Hubble’s law and the discovery of an expanding universe . In the same year, Lemaître hypothesised that the universe came into being, governed by the laws of quantum mechanics, from an initial point known as the “Primeval Atom” . However, the idea was poorly received by the scientific community due to Lemaître’s personal religious beliefs.
In 1949. Thomas Gold, Hermann Bondi together with Fred Hoyle came up with the “steady-state theory” which asserts that the universe was unchanging and eternal; matter is continuously created to maintain a constant density in the universe . Hoyle opposed the idea of an expanding universe from a singularity and made fun of it on a radio interview. He accidentally coined the term “Big Bang”, which was since then popularised even though it was meant in a pejorative sense.
Mathematical Basis of Big Bang cosmology:
In General Relativity, the metric used to describe the geometry of spacetime is a fundamental quantity. In order to allow for a dynamic universe, that is also homogeneous and isotropic, Robertson and Walker independently came up with the most generic metric that allowed the radius of curvature to be a function of time . In spherical polar coordinates, this can be written as:
ds2= –c2dt2– dr̅1–Kr22– r̅ dθ2– r̅ sin θ dϕ2
By introducing a cosmic scale factor that is dependent on time t
, the radial coordinate is transformed to a comoving coordinate:
Furthermore, the radius of curvature is also affected by cosmic expansion so it can be expressed in terms of the scale factor and a constant k
that is independent of time:
After making substitutions for r̅t
into equation (1), the following is obtained:
ds2= –c2dt2– a2tdr21–kr2+r2dθ2+sin2θ dϕ2
With a final change of variables r→ Skr
, the metric can also be expressed as
ds2= –c2dt2+ a2tdr2+Skr2dθ2+sin2θ dϕ2
Depending on the curvature of space, Skr
takes on different values depending on k
The evolution of spacetime is determined by the application of Einstein’s field equations based on his General Theory of Relativity. This is assumed to be valid everywhere on a large-scale basis. For a given distribution of mass and energy:
is the stress-energy tensor. The evaluation of the effect of a given mass distribution and energy on the curvature of spacetime is described by the Einstein’s tensor, Gαβ
Gαβ= Rαβ – 12gαβR
is the Ricci tensor, R
is the Ricci scalar and gαβ
is the metric tensor. Note that α
run over the time and spatial coordinates. The FRW metric can be expressed in terms of a diagonal matrix. The only non-vanishing components of Rαβ
is when α= β=0
. When we equate equation (7) and (8) specifically for G00
(the time-time component), the Friedmann equation which describes the expansion of the universe is obtained:
H2= ȧa2= 8π3Gρ–kc2a2
is the Hubble parameter that evolves with time and ȧ
is the rate at which the universe expands. From the first law of thermodynamics, combined with the knowledge that the expansion of the universe is an adiabatic process, the fluid equation is derived:
This describes the evolution of energy density as the universe expands. Using the (9) and (10), we obtain the acceleration equation:
is the acceleration of expansion, P
is the pressure of matter and ρ
is its density. In addition, the critical density ρcrit= 3H2/8πG
. It is often convenient to express the densities in units of the critical density; thus, the density parameter, Ω
is defined as Ω= ρ/ρcrit
Hence, equation (9) can be rewritten as:
Ωt–1= kc2at2Ht2= kc2ȧt2
The Friedmann-Lemaître-Robertson-Walker (FLRW) universe is one that has a geometry with RW metric and dynamics governed by equations (9), (10), (11). The FLWR models represent a whole class of standard cosmological models that assume perfect isotropy and homogeneity in the universe. Equation (11) shows that if ρ+3P/c2
is positive, it follows that ä/a
is negative, which means that gravitational force is attractive.
In equation (9), we see that ȧ/a
is positive which means that if a graph of a(t)
is plotted, a(t)
reaches zero at t=0
. Furthermore, as ä
increases and a
decreases, the right-hand side of equation (11) tends to infinity. The vanishing of the scale factor implies that that RW metric becomes singular, i.e the shrinking of the 3D space until it reaches a single point. This means that the universe must have started from a point of infinite density, curvature and temperature, and have expanded until a(t)
reaches its present value.
3 Pillars of Big Bang:
If the universe was static, galaxies should move about randomly. However, observations showed that most galaxies were redshifted and receeding from us. Hubble’s law implies that the universe is expanding and that it had not always existed in its present state.
When the wavelength of light or sound emitted by objects that are receeding from us, it appears stretched out and this expansion factor, a
, can be defined as the redshift z:
λobsλem= 1+z= a(t0)a(t)=1a(t)
It is often convenient to use a cosmic scale factor to express distances at different epochs, i.e at time t
the scale was a(t)
and the present value of at0
is set to 1.
If the speed of recession of an object, v is small compared to c, the standard Doppler effect applies and z ≃ v/c
.Therefore, the measurement of the amount by which the absorption or emission lines have redshifted then gives us a direct measure of the recession velocity of the object.
In 1917. Slipher had already noted that most of the galaxies measured showed a redshift. Combining this available data with the distances to the 24 galaxies that Hubble measured by studying the Cepheid variables in them, in 1929. he proposed a relationship known as the Hubble’s law:
is the Hubble’s parameter with units km s-1 Mpc -1 and r is the distance. Hubble showed that the further away the galaxies, the faster they are receeding from us.
and it represents the constant rate of cosmic expansion due to the stretching of spacetime itself andchanges with time throughout the life of universe. Determining the Hubble’s parameter involves measuring distances and redshifts of distant objects. The main challenge is finding reliable indicators of distance.
Since the Cosmological Principle states that the universe is isotropic and homogenous, the observed recession of distant objects can be interpreted as a general expansion of the universe. When the expansion is traced backwards in time towards the Big Bang, we find that the universe must be denser in the distant past until a single mathematical point of infinite density is reached.
Cosmic Microwave Background Radiation(CMB) 宇宙微波背景辐射(CMB)
The first prediction for the residual radiation from the Big Bang was made by George Gamow and the calculations for the temperature of this CMB was done by Alpher and Herman [9.13]. The prediction of the blackbody radiation at around 5 K did not cause much excitement in the scientific community. Even though calculations were refined several times, no attempts were made to detect this CMB. It was only in 1964 that Arno Penzias and Robert Wilson accidentally discovered this CMB in attempt to calibrate the antenna of the radio telescope in a radiation free environment. They found a constant background noise corresponding to an antenna temperature of around 3.5K and was interpreted by Dicke et al as the left-over CMB from the hot past .
According to the Big Bang model, CMB would saturate space after the recombination epoch due to recombination and photon decoupling. This happened when the universe was around 380.000 years old. Before recombination, the universe was opaque and the plasma was in thermodynamic equilibrium. This is because the photons were coupled to the electrons through Compton scattering. Therefore, photons could not travel freely and no light escaped. When the universe cooled, through expansion, to around 3000 K, it became thermodynamically favourable for the electrons to combine with protons to form neutral hydrogen atoms. Hence, the photons decouple from the plasma and the universe became transparent, allowing photons to propagate freely in all directions and fill space around us. The collision of photons with electrons before last scattering ensures that the photons were in equilibrium which entails a blackbody spectrum.
In order to verify the prediction of this CMB, observations were made by the FIRAS (Far Infrared Absolute Spectrophotometer) instrument aboard the COBE (Cosmic Background Explorer) spacecraft. FIRAS measures the difference between the CMB and a precise blackbody spectrum. The data taken showed that the CMB agrees to a high accuracy with the blackbody radiation curve, with a temperature of 2.728±0.004 K
. This provides very strong evidence for the isotropy of the universe.
CMB anisotropies? Inflation?
Big Bang Nucleosynthesis(BBN) 大爆炸核合成(BBN)
In the 1950s, the most popular theory regarding the formation of chemical elements in the universe postulated that all elements are either formed in stars or during supernova explosions. However, the observed helium abundances are too high to be explained by stellar nucleosynthesis. As such, Gamow came up with another theory – the initial state of the universe was so hot that it provided the necessary temperatures for nuclear fusion to take place . BBN predicts that during the first three minutes since the universe was created, light elements such as hydrogen(H), deuterium(D), helium(4He) and lithium(Li) were formed. The temperature of the universe decreased as it expanded and thus heavy elements were only formed in stellar interiors many millions of years later.
Once the reaction rates are known, all the light element abundances can be determined in terms of the ratio of baryon to photon density, η.
According to the standard BBN, the value of η
at CMB decoupling should be the same as that obtained from BBN. Hence, a key test of the Big Bang theory is the comparison between the two values. CMB data provided by the Wilkinson Microwave Anisotropy Probe (WMAP) measures the baryonic density with a high precision. Thus, the CMB baryon density can be used as an input for BBN calculation to give the predictions for light element abundances, which ultimately is compared to the observed abundances .
When the temperature of the universe, T≥1
MeV, the universe is dominated by radiation. The ratio of neutron to proton number density is fixed to be n/p =exp–Q/kBT
, where Q=1.3 MeV
is the mass difference between a neutron and a proton. Thermodynamic equilibrium is maintained via the following weak interactions :
; n+e+ ↔p+v¯e
When T ~
0.7 MeV, the expansion rate of the universe over takes the reaction rates. At this point, neutrinos decouple and the neutrons “freeze-out”. Free neutrons are unstable with a half-life of around 886s and will further decay into protons . This causes the neutron fraction to drop to approximately n/p ⋍1/7
. Deuterium formation then takes place:
n+p ↔D+ γ
However, the annihilation of electrons and positrons reheats the photon bath, resulting in photodissociation of the deuterium formed. This happens as long as the mean energy of particles and photons is higher than the binding energy (B.E) of deuterium. When the universe finally cools to T≤0.1
MeV, two deuterium will combine to make 4He which is stable due to its high B.E:
2 D↔ He23+n↔ H13+p;
H13+D → He24+n
; He13+D → p+He24
Formation of elements heavier than 4He are restricted due to the lack of stable nuclei of mass number 5 or 8. Hence, BBN does not produce nuclei beyond He in significant amounts. Almost all the neutrons are now 4He bound. The primordial 4He mass fraction is given by:
In order to confirm this predicted mass fraction based on the BBN theory, 4He and H emission lines from blue compact dwarf galaxies with low metallicity are observed . This is because they are pristine and have undergone little chemical evolution. By plotting a graph of 4He abundance as a function of oxygen abundance, one finds a positive correlation between the two. Therefore, when one extrapolates back to zero metallicity, the primordial abundance for 4He was found to be YP=0.2449 ±0.0040
. This agrees with the predicted helium abundance in equation (14).
During the synthesis of 4He, D is produced as one of the intermediate products. If η
is high and the nuclear reaction is fast, most D would be converted to 4He, resulting in less left-over D. It is important to note that BBN is the main source of D because it is destroyed in stars during stellar nucleosynthesis. Due to such astration, the ratio of D/HP
has decreased from its primordial value. Observed abundances therefore only provide a lower bound to its primordial abundance and an upper bound on η
Damped Lyman alpha systems with low metallicity display the Lyman series absorption lines of D. Hence, by looking at such systems, its primordial abundance can be measured. The weighted mean value of the six measurements that were carried out is D/HP=2.547 ±0.033 × 10–5
, corresponding to a baryon density of 100 Ωb,0h2=2.156 ±0.020
. This result is comparable to CMB observations recorded by the Planck satellite which gives a baryon density of 100 Ωb,0h2=2.226 ±0.023 
. Although there is a 2.3σ
discrepancy between the two results, it is impressive that they somewhat agree with each other given that the measurements are based on two completely different physical processes, at two widely separated epochs. The former is determined by the theory of BBN and the latter comes from taking the ratio of the amplitudes of the first and second acoustic peaks in the angular power spectrum of the CMB anisotropies [book: the physics of early universe- E. Papantonopoulos pg 114-116].
However, an issue that remains is the inconsistency between the calculated and observed Li/HP 
. It is not yet known whether this disagreement is due to systematic errors, and/or uncertainties or that new physics beyond the standard model is required in the early universe.
Nonetheless, there is a broad agreement between the predictions and measured primordial abundances. This concordance shows that the Big Bang theory although might not be perfect, but is still a success and makes BBN a reliable probe into the early universe. This whole process has been a difficult because the primordial abundances have been altered by various nuclear processes during the course evolution of the universe. Furthermore, each nucleus is modified differently hence various methods as well as environments are used in order to measure these abundances accurately.
Classical Problems of the Big Bang Model
Horizon Problem :
A key assumption made in the Big Bang model is the cosmological principle. In order to show that the universe is indeed isotropic, one may be tempted to use the CMB radiation as a justification of this principle.
During the recombination epoch at around 400.000 years after the big bang, the casual region at decoupling corresponds to an angle of about 1 °
on the sky. This means the Big Bang model predicts that regions in the sky separated by more than 1 °
should not have been in causal contact at the time of decoupling. We expect that two regions that were not causally connected should display different CMB properties. Contrary to expectations, the observations of the CMB in opposite directions of the sky that had not established casual contact have the same CMB temperature up to 0.001%. This is known as the horizon problem, where it is difficult to understand how causally independent regions can appear to be nearly isotropic.
Observations suggest that the universe has a flat spatial geometry, i.e k=1
. This means that the energy density ρ
is equal to the critical density ρcrit
. In particular,the angular power spectrum of the CMB shows a dominating peak at a spherical harmonic multipole, lpeak≈200
. This result was interpreted as evidence for a flat universe.
Current observations show that Ω0t–1<0.01
Evaluation of Big Bang cosmology 对大爆炸宇宙学的评价
The standard Big Bang model is adequate in describing the universe on large scales where it is assumed to have an isotropic and homogenous space-time. The Big Bang model relies on an extrapolation of laws of physics to regimes where they have not been tested yet. As mentioned in section xxxx, the expansion of the universe implies that when traced backwards in time, the diameter decreases and approaches a limit of zero. At this limit, laws of physics, specifically general relativity, breaks down at least at the Planck time (10-45 seconds) which is when quantum effects start becoming significant. Precisely because of this reason, Hawking believes that Einstein’s theory would only suffice in predicting how the universe will evolve once it has begun and not how the universe will begin . Thus, in the standard Big Bang model problems arise due to the lack of understanding of these initial conditions that one can only speculate about . Nevertheless, it is important for primordial cosmology to have a standard model such as the Big Bang to build on.
Scientific models facilitate access to phenomena based on theories and observations known to be valid. It is a mere attempt to represent reality and must not be confused with reality itself. Furthermore, every model has a domain of validity which should be specified and not exceeded . Whilst theories aim to be general, models aim to match specific empirical evidences well . In order to formulate theories, hypotheses (mathematical relationships or causal mechanisms) are first proposed based on existing background knowledge and then verified through experimentation. This process is commonly known as the Scientific Method. A well tested hypothesis then becomes accepted as a theory i.e. an explanation of a phenomenon. According to Khun, a good scientific theory should fit the following criteria :
Observational and experimental support – Testability and confirmation
Consistency – Agreement with other currently accepted theories
Broad scope of theory – The unification of otherwise separate phenomena as well as providing as basis for further development
Satisfactoriness – Aesthetic qualities such as simplicity and elegance
However, a criterion is a standard of judgement. These criteria are philosophical in nature and cannot be subjected to experimental verification for proof. Hence, scientists following the same list of criteria can still reach different conclusions.
The uniqueness of the universe raises problems when one assesses the desirability of scientific theories following the aforementioned criteria. Most importantly, the universe itself cannot be subjected to physical experimentation. It is impossible perform any “repeats” for one cannot recreate or alter the initial conditions of the universe.
The Big Bang theory is developed using the laws of physics to explain how the universe came about. For many cases, an explanation of an event is an account of how it was caused . However, in the context of cosmology, the issue of what constitutes as an explanation becomes central. Often times, when asked deeper “Why” questions – why they are that way instead of another way, only partial and intermediate answers are offered. Even so, the explanation provided will be in terms of another structure or relationship. We can again pose the same question and end up with an infinite regress of “Why”s . The significance of cosmological philosophy depends on the extent one intends to pursue a theory, i.e. the scope of enquiry. Since the understanding of physical processes at the beginning of the universe is not well understood, the philosophical stance here becomes more important because of the limits of testability. A theory has very little explanatory power if such “Why” questions are answered in a strictly scientific manner, avoiding any philosophical issues that arise. Thus, the scientific method has its own limits and complete solutions cannot be provided from physics alone.
One needs to recognise that these so called “laws” are provisional, idealised and descriptive. They are contingent to the models and do not contain within themselves an explanation for why they and not other laws obtain instead. Although the Big Bang model has its strengths, it only depicts the universe aptly to an extent. Amongst the three cornerstones that supports the Big Bang theory, none of which we had access to and directly observe. They are a result of imaginative and conceptualised abstraction guided by careful experimentation through observations. This gives rise to qualitative and quantitative relationships that appear to exist amongst them. Nonetheless, it is through these relationships that we can perfect the models constructed and improve one’s ability to apply our understanding and knowledge of the physical reality.
The Big Bang model offers a convincing description of the evolution of the universe. It is successful in accounting for the expansion of the universe, the abundance of light elements as well as the accurate prediction of the CMB. However, at higher temperatures, the model becomes less reliable because it involves an extrapolation of well-tested local physics to regimes that are not well-understood and have yet to be tested. Laws of physics breakdown at energies close to the Planck scale and quantum effects must be considered. The flatness, horizon and monopole problem are some of the failures of the Big Bang model. However, in science, models can never completely represent reality. One should understand the model’s limits and not use it in attempt to answer questions that are beyond its capacity. The main challenge in cosmology is the testing and evaluation of cosmological models given our observational horizon in this unique universe. Due to the limitation of observability and testability, theoretical assumptions have to be made. Hence, there will always be philosophical considerations underlying the study of cosmology.