The Hubble Law
Cosmology, in its broadest sense, encompasses the study of the universe's large-scale structure and aims to integrate our comprehensive understanding of it into a cohesive framework Our current perspective on the universe is grounded in observational evidence and key theoretical concepts, with Einstein's General Theory of Relativity playing a pivotal role in explaining the influence of gravity on the universe's evolution A significant milestone in modern cosmology is the discovery of the universe's expansion, which remains a fundamental aspect of our understanding.
In addition, the observation of theCosmic Microwave Background Radiation (CMB) provided a strong connection of the present cosmological picture to fundamental Particle Physics.
In 1929, Edwin Hubble revealed that the redshifts of galaxies increase with distance, a finding that aligns with the Doppler shift phenomenon This phenomenon indicates that the wavelength of light from a moving source expands, represented by the formula λ = λ(1 + V/c), which is adjusted for relativistic speeds The term z, defined as z ≡ ∆λ/λ, refers to the redshift, while the non-relativistic Doppler formula simplifies to z = V/c Hubble's relation shows that redshift is proportional to distance, expressed as z = ∆λ/λ ∝ L.
Subsequent measurements by him and others established beyond doubt the
Usually the name Hubble Law is reserved for the redshift-distance propor- tionality.
The Hubble parameter (H), currently valued at approximately 100 km/s/Mpc or 9.778 Gyr⁻¹, plays a crucial role in understanding the expanding Universe According to Hubble's Law, the redshift observed in the light from distant galaxies indicates that their separation is increasing due to the expansion of space While the Hubble parameter remains constant across space at any given moment, it is not constant over time, suggesting that the rate of expansion may have been more rapid in the past Observational data strongly support the notion of a homogeneous Universe.
(all places are alike) andisotropic (all directions are alike) The hypotheses of homogeneity and isotropy are referred to as the Cosmological Principle.
A uniform Universe maintains its uniformity through consistent motion, resulting in expansion that primarily involves dilation without significant shear or rotation This understanding leads to a straightforward derivation of Hubble's Law.
Comoving Coordinates and the Scale Factor
The homogeneity of the Universe ensures that all clocks synchronize their time intervals, leading to the concept of universal or cosmic time By focusing on uniform expansion, we can establish a comoving coordinate system where the distances between comoving points increase uniformly In this system, a universal scale factor R, which changes over time, characterizes the expansion or contraction of the Universe Specifically, R(t) is a function of cosmic time that maintains a consistent value across space Consequently, lengths, surfaces, and volumes expand in proportion to R, R², and R³, respectively, reflecting the dynamic nature of the Universe.
At the present time, the scale factor is represented by R₀, while L₀ denotes the distance between two comoving points The distance at any other time, t, can be expressed as L(t) = (L₀ / R₀) R(t) This relationship applies to an expanding volume V that encompasses these points.
N particles, we can write for the particle number density n =n 0 (R/R 0 ) 3
Using the current average density of matter in the Universe, which is approximately one hydrogen atom per cubic meter, we can estimate the average density of matter from an earlier epoch When the scale factor was just 1% of its present value, the average matter density reached one hydrogen atom per cubic centimeter.
In cosmology, the distance of a comoving body at a fixed coordinate distance is directly proportional to the scale factor, expressed as L = R × (coordinate distance) The recession velocity of this comoving body is also proportional to the rate of increase of the scale factor, represented by V = ˙R × (coordinate distance) By dividing these two relationships, we can derive important insights into the dynamics of the universe.
The age of the Universe is closely related to Hubble time, illustrating the Velocity-Distance Law in a different format These two expressions align when we equate the Hubble parameter with the scale factor's rate of change.
The Hubble parameter is a time-dependent measure that arises from the uniform expansion of the universe This expansion implies a consistent scale factor that applies universally across space and changes over time, directly resulting in the Velocity-Distance Law.
The Hubble time, denoted as t_H ≡ H_0^(-1), represents the time of expansion of the Universe, assuming a constant Hubble parameter or rate of expansion While most cosmological models indicate that the Hubble parameter varies, the Hubble time still provides an approximate measure of the Universe's age Numerically, the Hubble time is estimated to be t_H ∼ 10h^(-1) billion years, where the normalized Hubble parameter h ranges between 0.5 and 0.8.
Acceleration is by definition the rate of increase of the velocity, namely
V˙ = ¨R×(coordinate distance) As before, the coordinate distance of a comov- ing body is constant On the other hand, we know thatL=R×(coordinate distance) Thus,
We can define a deceleration parameter, independent of the particular body at comoving distanceL, as the dimensionless parameter q≡ − R¨
The kinematic classification of uniform Universes is based on the values of the Hubble parameter (H) and the deceleration parameter (q) When q is positive, it indicates deceleration, while a negative q signifies acceleration, which is why it is referred to as the acceleration parameter Uniform Universes can be categorized into several classes: (a) expanding and decelerating (H > 0, q > 0), (b) expanding and accelerating (H > 0, q < 0), (c) contracting and decelerating (H < 0, q > 0), (d) contracting and accelerating (H < 0, q < 0), (e) expanding with zero deceleration (H > 0, q = 0), (f) contracting with zero deceleration (H < 0, q = 0), and (g) static (H = 0, q = 0) This classification contrasts with geometric classifications based on curvature.
Current candidates for our Universe include (a), (b), and (e) By tracing the expansion of the Universe backwards, we reach a state of extremely high density as R approaches zero Evidence from Cosmic Microwave Background (CMB) radiation indicates that this high-density condition, often referred to as the Big Bang, likely existed in the Early Universe.
The Cosmic Microwave Background
The Hubble expansion arises naturally from the principles of homogeneity and isotropy, indicating that an expanding Universe must have had a significantly denser and hotter past In the Early Universe, prior to the formation of any structures, matter can be conceptualized as a gas of relativistic particles in thermodynamic equilibrium This equilibrium remains intact despite the expansion, as the rate of particle processes, which is proportional to the characteristic energy (T), is much greater than the rate of expansion, represented by the smaller scale H.
To understand the early universe, one must reference the Friedmann equation and the temperature-dependent energy density, ρ ∼ T^4, characteristic of radiation G Gamow and his collaborators, R Alpher and R Herman, initially proposed a model of the early universe as a gas of relativistic matter and electromagnetic radiation in equilibrium to explain nucleosynthesis This model also predicted the existence of relic black body radiation, with wavelengths in the microwave range corresponding to temperatures of just a few degrees Kelvin.
The term was first introduced by Fred Hoyle during a series of BBC radio talks, later published in his work "The Nature of the Universe" in 1950 Hoyle was a prominent advocate of the competing Steady State Theory regarding the Universe.
The Cosmic Microwave Background (CMB) radiation, discovered in 1965 by A Penzias and R Wilson, originated from the early Universe when it was extremely hot Over billions of years, this radiation has cooled and redshifted due to the expansion of the Universe, resulting in a current temperature of just a few degrees Kelvin According to black body radiation principles, the maximum wavelength of this radiation is approximately λ max ∼(1.26c/k B )T, where T represents the temperature.
Accurate measurements from the Cosmic Background Explorer (COBE) indicate that the intensity of the Cosmic Microwave Background (CMB) closely adheres to the blackbody radiation curve, with a deviation of merely one part in 10,000 Additionally, these observations account for a 24-hour anisotropy related to the Galaxy's motion at a speed of 600 km/sec.
The Cosmic Microwave Background (CMB) radiation exhibits surprising isotropy, with anisotropies measured at approximately 10^-5, and recent advancements have improved this accuracy to 10^-9 These anisotropies, remnants from the time of decoupling, reflect the density fluctuations that later formed galaxies and galaxy clusters Furthermore, the CMB's adherence to the Planck spectrum provides compelling evidence for an expanding Universe that underwent a hot phase, supported by the COBE's estimate of the CMB temperature.
The Cosmic Microwave Background (CMB) can be understood qualitatively by examining the early universe when temperatures exceeded 10^10 K, corresponding to an energy of about 1 MeV At these high energies, massless particles like photons, electrons, and neutrinos dominate the energy landscape In this state, reactions such as e+e- ↔ γ+γ maintain thermodynamic equilibrium, unaffected by the universe's expansion As the universe expands, the temperature decreases inversely with the scale factor, leading to significant changes only when the temperature falls below the threshold energy of k_B T ∼ m_e c^2, enabling photon-induced electron-positron pair creation Once the temperature drops below this threshold, electrons and positrons vanish from the plasma, resulting in the decoupling of photon radiation and a transparent universe.
It is exactly these photons which, redshifted, we observe as CMB.
The Hubble expansion alone is insufficient to support the Big Bang cosmology It was the discovery of the Cosmic Microwave Background and advancements in nucleosynthesis research that solidified the Big Bang Model as the leading framework for understanding the universe in standard cosmological terms.
The Friedmann Models
A Cosmological Model offers a simplified representation of the Universe, characterized by a geometric description of spacetime and a uniform distribution of matter and radiation Among these models, the homogeneous and isotropic Friedmann-Lemaître spacetimes (FL) stand out as fundamental solutions of General Relativity that embody the Cosmological Principle The mathematical formulation of a FL model is expressed by the line element ds² = dt² - R²(t)dσ².
The spatial line elementdσ 2 describes a three-dimensional space of constant curvature independent of time It is 2 dσ 2 =dχ 2 +f 2 (χ) dθ 2 + sin 2 θ dφ 2
These coordinates are comoving That means that the actual spatial distance of two points (χ, θ, φ) and (χ 0 , θ, φ) will be d =R(t)(χ−χ 0) There are three choices for f(χ), each corresponding to a different spatial curvaturek.
That is the value of the Ricci scalar (to be defined below) calculated from dσ 2 with the scale factor divided out They are f(χ)
In cosmology, the parameter k determines the geometry of spacetime, where k = +1 signifies a closed spacetime with spherical geometry, k = 0 indicates an infinite flat spacetime with Euclidean geometry, and k = -1 represents an open spacetime with hyperbolic geometry The Robertson-Walker metric can also be expressed using the variable r, defined as r ≡ f(χ), leading to the relationship dσ² = dr².
The above metric comes out as a solution of Einstein’s Equations
R àν is theRiemann Curvature Tensor andRis theRicci Scalar defined as
R=g àν R àν Gstands for Newton’s Constant of Gravitation The constantΛ is called theCosmological ConstantandT àν is theMatter Energy-Momentum Tensor A usual choice is that of a fluid
2 This is the so called Robertson-Walker metric A more complete name for these spacetime solutions is Friedmann-Lemaitre-Robertson-Walker or just FLRW models.
T à ν = (−ρ, p, p p), (1.11) withρthe energy density andpthe momentum density, related through some Equation of State.
In the context of the Robertson-Walker metric, light emitted from a source located at the point χ S at time t S travels along a null geodesic (dσ² = 0) and, assuming radial propagation (dΩ² = 0), arrives at us at χ 0 = 0 at the time t 0, which is determined by the equation t 0 = t S + dt R(t) = χ S.
A second signal emitted att S +δt S will satisfy t 0 +δt 0 t S +δt S dt R(t) =χ S ⇒ δt S
The ratio of the observed frequencies will be ω 0 ω S
This is theHubble Law The Velocity-Distance Law is a simple consequence of uniformity, namely
Inserting the Robertson-Walker metric into Einstein’s Equations, we ar- rive at the two equations
Multiplying the first of these equations by ˙Rand using the second, we arrive at the equivalent pair of two first order equations, namely ˙ ρ+ 3(ρ+p)
The Continuity Equation represents the conservation of energy within a comoving volume, denoted as R³ This concept becomes clearer when expressed in the form of a time derivative.
The other equation is purely dynamical and determines the evolution of the scale factor It is calledThe Friedmann Equation.
In the current epoch, the value of the present Hubble parameter \( H_0 \) and the deceleration parameter \( q_0 \) can be expressed with a critical density defined as \( \rho_c \equiv 3H^2 \) This relationship highlights the significance of understanding the dynamics of the universe in terms of these key cosmological parameters.
At the present time ρ c,0 = 1.05×10 − 5 h 2 GeV cm − 3 The name and the meaning ofρ c will become clear shortly We also introduce the dimensionless ratio
(1.19) in terms of which the Friedmann equations are written as k
In the case of vanishing cosmological constantΛ= 0, we have q 0=1
Thus, the measurable quantity Ω 0 = ρ 0 /ρ c,0 determines the sign of k, i.e. whether the present Universe is a hyperbolic or a spherical spacetime Note that for Λ = 0, H 0 and q 0 determine the spacetime and the present age completely.
In cosmology, it is essential to differentiate various contributions to the overall density, including the current density of pressureless matter (Ω m), the density of relativistic particles (Ω r), and the vacuum energy density represented as Ω Λ ≡ Λ/3H² Furthermore, in models where the vacuum's present-day contribution varies, an additional term can be incorporated to account for these changes.
Ω v Thus, in the general case, we have k
Simple Cosmological Solutions
Empty de Sitter Universe
In the case of the absence of matter (ρ=p= 0) and fork= 0, the Einstein- Friedmann equations take the very simple form
For positive Cosmological Constant Λ >0 we have a solution with an expo- nentially increasing scale factor
The de Sitter space describes an expanding Universe characterized by a constant Hubble parameter and a constant acceleration parameter This expansion is driven by a non-zero cosmological constant, resulting in a Universe with a constant positive curvature proportional to Λ.
Vacuum Energy Dominated Universe
When the primary source of the Energy-Momentum Tensor is Vacuum Energy, such as the vacuum expectation value of a Higgs field, the structure of the Energy-Momentum Tensor can be expressed in a specific mathematical form.
The relationship T à ν = −σδ à ν (1.27), where σ > 0 is a constant, indicates a significant aspect of the Equation of State, represented as p = −ρ = −σ (1.28) This formulation highlights the phenomenon of Negative Pressure, which is crucial in understanding the vacuum's influence on cosmic dynamics Notably, the presence of negative pressure can trigger an accelerated exponential expansion, similar to the behavior observed in empty de Sitter space.
ForΛ=k= 0, we obtain the Friedmann-Einstein equations
An Exponentially Expanding Vacuum Dominated Universe is essential for understanding Inflation Notably, the Vacuum Dominated Universe and the Empty de Sitter Universe exhibit physical indistinguishability This arises from the principle that a constant component of the Energy-Momentum Tensor, associated with matter, corresponds to a constant of opposite sign in Einstein’s Equations, functioning as a Cosmological Constant, which is typically linked to geometric properties.
In a more general case thatp =w ρ, the acceleration parameter isq (1 + 3w)Ω v /2 This shows that for an equation of state parameter w 1, the Universe is closed, and without a cosmological constant, the expansion would eventually reverse into contraction However, this reversal is not guaranteed if a non-zero cosmological constant is present Conversely, when Ω < 1, the Universe is open, leading to perpetual expansion, a scenario that also applies to the critical case where Ω = 1.
A lower bound forΩis supplied by the observed Visible Matter
Primordial Nucleosynthesis provides compelling evidence supporting the estimation of Ω vm ∼0.03, indicating that a significant portion of the Universe's mass exists as an unknown non-baryonic form known as Dark Matter This elusive matter can primarily be detected through its gravitational effects, leading to an estimated value of Ω dm ∼0.3.
The origin of the remaining contribution to the density parameter Ω is attributed to Dark Energy, which cannot be linked to either visible or dark matter and is represented by an effective vacuum term The value Ω = 1 is particularly appealing for theoretical reasons, suggesting that the Dark Energy contribution is approximately Ω_de ∼ 0.7, a figure supported by current data Specifically, data indicate that Ω_Λ = Λ/3H_0² is around 0.7, leading to an estimated cosmological constant Λ ∼ O(10⁻⁵⁶) cm⁻² This small cosmological constant is sometimes expressed as a scale Λ⁴ = Λ/M_P² ∼ (10⁻³ eV)⁴.
Thus, in the case of critical density, the various contributions are
Although it seems unavoidable, it is surprising that at least 90% of the matter in the Universe is of unknown form.
The Standard Cosmological Model
Thermal History
During the Radiation Dominated epoch the Friedmann equation is H 2 ∼
8πGρ/3, since the curvature term is irrelevant at small values of the scale factor Thus, the energy density has the critical value ρ∼3H 2
The solution (1.38), for each interval of constant effective number of degrees of freedomQ(T), gives
The value of Q(T) at specific temperatures is determined by the applicable Particle Physics model within that temperature or energy range The following table presents the values of Q(T) for temperatures up to O(100 GeV) based on the SU(3) C × SU(2) L × U(1) Y Standard Model.
In a relativistic gas assumed to be in thermodynamic equilibrium, the interaction rate is significantly higher than the expansion rate, ensuring stability The interaction rate, influenced by the cross section σ, is proportional to T^(-2) and particle number density n, which scales as T^3 Consequently, the reaction rate behaves like σn ∼ t^(-1/2), while the expansion rate follows H = 1/2t This relationship guarantees that σn remains greater than H as the Universe expands and cools, maintaining equilibrium.
Starting from approximately 1 second after the Big Bang, corresponding to a temperature of around 1 MeV (10 billion K), we can trace the universe's expansion backward in time At temperatures below 1 MeV, the plasma is primarily composed of photons and neutrinos However, as temperatures exceed 1 MeV, electron-positron pairs begin to form through the interaction of high-energy photons, specifically the process γ + γ → e⁻ + e⁺.
Protons and neutrons do not contribute to energy density, as their quantity is approximately 10⁻⁹ times that of light particles such as gamma rays, neutrinos, and electrons At shorter time scales, around 10⁻³ to 10⁻⁴ seconds, muons and pi mesons become involved in the plasma dynamics.
The deconfinement temperature \( T_c \) marks the phase transition from the hadron phase to the quark-gluon plasma Above \( T_c \), within the first \( 10^{-4} \) seconds, gluons and free quarks, including up, down, and later strange quarks, form the plasma As temperatures rise, charmed quarks, tau leptons, and bottom quarks also emerge, occurring around \( 10^{-10} \) seconds after the transition.
T ∼100GeV, theW ± andZbosons of Weak Interactions become abundant.
At even higher temperatures, the Higgs boson and top-quark appear At these temperatures, the full Electroweak symmetry SU(3) C ×SU(2) L ×U(1) Y is restored.
Nucleosynthesis
The period from 1 second to 200 seconds after the Big Bang is crucial in the history of the Universe, as it marks the formation of light nuclei and the emergence of ordinary matter During this time, the abundances of light nuclei were established, including Helium-4 at approximately 25%, Deuterium at around 3×10^-5, and Helium-3 at about 2×10^-5.
Heavier nuclei formed later in stars, while the primordial helium abundance, initially at 25%, has changed only slightly over billions of years of hydrogen conversion Remarkably, just 200 seconds during the early radiation era were enough to create nearly all of the helium present today The production of helium can be estimated for times less than one second.
T > 1M eV, protons and neutrons move freely in the primordial plasma. Their relative number can be expressed through the Boltzmann formula
At a temperature of approximately 0.7 MeV, the equilibrium in the reactions ν + pe + n and n + ν p + e slows significantly, causing the neutron-to-proton ratio to stabilize at about 0.16, indicating there is one neutron for every 5 to 6 protons The decay of free neutrons, which occurs over approximately 15 minutes, is insufficient to alter this ratio Additionally, protons and neutrons interact to form deuterium nuclei, or deuterons, through the reaction p + n → H₂ + γ.
Deuterons disintegrate through an inverse process, restoring protons and neutrons to the plasma After approximately 100 seconds, the temperature falls to a level where deuterons can stabilize By this stage, protons and neutrons have combined, resulting in a ratio of about two neutrons for every 14 protons This leads to the formation of two deuterons and 12 protons from a total of 16 nucleons The now stable deuterons can further combine to create a helium-4 nucleus.
Actually, one has to consider all the two-body processes, like p+H 2 ↔
He 3 +γ,n+He 3 ↔He 4 +γ, etc The whole process is over in roughly 200sec, and in that time 25 % of matter is converted into helium (four out of sixteen nucleons form a heliun nucleus) and the remainder consists predominantly of protons Slight amounts of deuterium,He 3 andLi are also produced.
Problems of Standard Cosmology
The Horizon Problem
The maximum size of a region in which causal relations can be established is given by the horizon r H (t) =R(t)∆χ=R(t) t
During the Radiation Dominated phase, R(t) ∼ t 1/2 and r H (t) = 2t For t → 0, r H shrinks much faster than R(t) Thus, at every epoch, most of
In the Standard Cosmological Model, regions within a typical dimension R(t) are causally unrelated, despite the extreme isotropy indicated by the Cosmic Microwave Background (CMB) data Prior to the time of hydrogen recombination (t R), radiation and matter existed in thermal equilibrium, leading to a transparent Universe post t R The current isotropy of the CMB suggests a similar isotropy existed at t R However, the radiation-temperature we observe today originates from regions that had not yet established causal contact during the epoch of t R The coordinate distance from our present epoch (t 0) to t R, with our position set at r = 0, highlights this phenomenon.
Since the horizon att R was
R(t ) (1.75) the number of horizon lengths contained in the distance 2∆χ(t 0 , t R ) between two opposite directions in the sky will be
In the context of Radiation Dominated expansion, applying this formula yields a significantly large result; however, an altered expansion law for the scale factor, such as that observed during inflation, has the potential to dramatically change this outcome.
R∼e Ht , we obtain forN a very small number.
The horizon problem in the Standard Cosmological Model is addressed through the concept of Inflation, which proposes a phase of exponential expansion in the Universe This framework accounts for the observed homogeneity and isotropy of the cosmos.
The Coincidence Puzzle and the Flatness Problem
The Friedmann equation for the present epoch has the form
Observations indicate that all three terms of the right hand side can be roughly of the same order of magnitude
In the early universe, the magnitudes of various terms differ significantly, with ρ ∝ R − 4 dominating initially This delicate balance among the three terms is crucial for the existence of our universe For example, in a k = +1 model, a balance between the first two terms would lead to a catastrophic collapse within a few Planck times Conversely, in a k = −1 universe, such a balance would result in an excessively rapid expansion, leaving the current density parameter Ω dangerously small This intriguing alignment of the magnitudes of these terms is commonly referred to as the Coincidence Puzzle.
The balance of various terms can be effectively expressed through the Entropy of the Universe In the Radiation Dominated epoch, the entropy density \( s \) and the entropy \( S \) of a comoving volume \( R^3 \) are defined by the equation \( s = 2\pi^2 Q \).
Estimating the present time entropy density from the background of photons and neutrinos as s 0 ∼ n γ ∼ 10 3 cm − 3 , we obtain for the entropy the huge number
This number is an initial condition of the Standard Cosmological Model. The fact that there are so much more photons than baryons is something determined at the beginning.
4 The characteristic scale of gravitation, Newton’s gravitational constantGdefines a characteristic mass, thePlanck massM P ∼10 18 GeV, a characteristic length,thePlanck length, and a characteristic time, thePlanck time.
Rewriting the Friedman equation in terms of temperature and entropy, we obtain
It is clear that the curvature term at high temperature is negligible since
S is a large number The Friedmann equation can also be written as (Λ is negligible at early times)
This shows in a dramatic way thatΩ must have been terribly close to 1 at early epochs For instance
The Flatness Problem, also known as the Entropy Problem, highlights the Standard Cosmological Model's characteristic of maintaining a density parameter (Ω) close to 1 throughout cosmic history This phenomenon is not a conventional problem; instead, it connects the unique properties of our current Universe to specific initial conditions, such as the notably high value of entropy.
A theory of the Early Universe that begins with a value of S around 1 and evolves to the current number through natural physical processes would represent a significant advancement This approach would eliminate the need for highly specific initial conditions, making it a more robust framework for understanding the universe's development.
Phase Transitions in the Early Universe
The Standard Model of particle physics, which encompasses the SU(3) C × SU(2) L × U(1) Y framework for strong and electroweak interactions, introduces the concept of Spontaneous Symmetry Breaking This phenomenon indicates that while the fundamental laws of nature exhibit symmetry under a specific local gauge symmetry, the vacuum state does not maintain this symmetry Consequently, certain operators within the theory possess vacuum expectation values that violate this symmetry This violation is accomplished through the vacuum expectation value of a scalar Higgs field, which is an SU(2) L doublet and carries weak hypercharge In the resulting broken SU(3) C × U(1) electromagnetic vacuum, three out of the
Fig 1.4.Finite Temperature Effective Potential. four gauge bosons (W ± , Z 0 ) of SU(2) L ×U(1) Y obtain a mass, while the fourth (photon) remains massless, corresponding to the intact electromag- neticU(1) em gauge interaction.
In the Early Universe, matter exists in a thermodynamic equilibrium with a heat bath, described by the Hamiltonian of the SU(3) C ×SU(2) L ×U(1) Y gauge field theory The vacuum energy is determined by minimizing the Free Energy, which correlates with the Effective Potential that varies with temperature At high temperatures, the Universe's global vacuum state is symmetric, while at lower temperatures, it transitions to a broken state As the Universe cools during the Radiation-Dominated epoch, it undergoes a phase transition from the high-temperature symmetric phase to the low-temperature broken phase, as illustrated in the effective potential diagram.
The behavior of finite temperature in relation to the Higgs field vacuum expectation value (vev) mirrors phenomena observed in certain condensed matter systems, such as ferromagnets, which lose their magnetism when heated but exhibit non-vanishing magnetization at absolute zero, breaking rotational symmetry A more fitting analogy is the phase transition from water to ice, which typically occurs at 0°C; however, pure water can supercool below this freezing point before crystallizing into ice This transition releases latent heat from the false vacuum, causing the Universe to reheat Depending on the underlying theory, symmetry breaking may manifest as a first-order phase transition involving field tunneling through a potential barrier or as a second-order phase transition where the field transitions smoothly between states.
Inflation
In a phase transition where the Universe remains in a false vacuum state with an approximate value of < φ > ≈ 0, the energy density is primarily influenced by the vacuum contribution, leading to the condition ρ ≈ V(0).
A scalar potential illustrated in Fig 1.5 may lead to specific behaviors during the evolution of a false vacuum state In this scenario, the equation of state is characterized by negative pressure, expressed as p = -ρ = V(0), which results in a constant Hubble parameter and an exponentially increasing scale factor, as predicted by the Friedmann equation.
Once the transition to the true vacuum state, where < φ >= 0, is achieved, the latent energy from the false vacuum will be unleashed, causing the Universe to heat up to temperatures similar to those at its inception During this phase, the product of the scale factor and temperature, R T, will increase proportionally.
As a result, the entropy is expected to increase by a factor of 3η For a parameter value of η around 60 to 70, the current enormous entropy magnitude of 10^87 could have originated from an initial entropy level of O(1).
The slow evolution in the "false vacuum" provides a potential solution to the entropy puzzle and addresses the flatness problem through the Friedmann equation, expressed as k = R²H²(Ω - 1) In this context, the cosmological constant Λ does not influence the Early Universe but can be incorporated into the density parameter Ω, represented as ΩΛ = Λ/3H² Consequently, this leads to the relationship k = R²fH²(Ωf - 1) = R²iH²(Ωi - 1), resulting in Ωf = 1 + e^(-2η)(Ωi - 1).
The equation for η that accounts for entropy is Ω f = 1 + 10 − 58 (Ω i − 1), which remains remarkably close to 1 regardless of the initial value Ω i Additionally, as previously mentioned in relation to the horizon problem, exponential expansion effectively addresses this issue as well.
The Inflationary Scenario proposed by A Guth in 1982 introduces the concept of Old Inflation, which ultimately proves to be an unviable model This scenario describes the formation of bubbles representing a true (broken) vacuum within a false (unbroken) vacuum However, the rate at which these bubbles merge cannot match the rapid expansion of the Universe, leading to the formation of bubble concentrations that become dominated by a single bubble, resulting in a highly inhomogeneous cosmic landscape.
The New Inflation scenario, proposed by A Linde and P Steinhardt, suggests that the observable Universe originates from a single fluctuation region, diverging from earlier models by discarding the prolonged supercooled phase and bubble nucleation Instead, inflation occurs as the inflaton gradually grows from its initial to equilibrium value, requiring a duration significantly longer than H − 1, achievable with a suitably flat potential In this model, the Universe is heated post-inflation through particle creation from inflaton oscillations rather than bubble wall collisions, with any bubbles formed being too distant to allow causal interaction, thus preventing inhomogeneities from affecting the observable Universe However, implementing this scenario within realistic Particle Physics has presented challenges, leading cosmologists to adopt Chaotic Inflation, which assumes a chaotically distributed inflaton and necessitates specific conditions for the classical theory of the dilaton.
This is theslow-roll approximation quantified in terms of the parameters
Note that the last two conditions (slow-roll conditions) are necessary but not sufficient The slow-roll approximation requires ˙φto satisfy 3Hφ˙ ∼ −V The amount of inflation is N= ln R(t R(t) end ) ∼8πG φ φ end dφ V V
Inflation's historical significance primarily stems from its role in addressing the initial conditions necessary for the Hot Big Bang phase In contemporary understanding, the most crucial aspect of inflation is its ability to create irregularities in the Universe, which can ultimately facilitate the formation of cosmic structures.
The Baryon Asymmetry in the Universe
Relativistic Quantum Theory posits that each elementary particle has a corresponding antiparticle, which generally differs in state Antiparticles share the same spacetime properties, such as mass and spin, but possess opposite electric charges and global quantum numbers, like Baryon and Lepton Number For instance, the antiparticle of the electron (e⁻) is the positron (e⁺), which carries a positive charge, while the antiparticle of the neutrino is the distinct antineutrino The relationship between the wave functions of these particles is expressed through a symmetry operator, denoted as ψ_e(x) ⇒ C{ψ_e(x)} = ψ_e⁺(x).
Antimatter consists of antiparticles, contrasting with the predominantly matter-filled Universe Existing antimatter primarily originates from secondary processes, such as relativistic collisions of matter, seen in both particle accelerators and Cosmic Rays Strong evidence suggests that primary forms of antimatter are rare in the Universe, as indicated by the observed ratio of baryons to photons, represented by η = N_B.
The Baryon Asymmetry, initially highlighted by A D Sakharov, necessitates interactions that violate Baryon Number (B), Charge Conjugation (C), and Parity (P), along with a deviation from thermodynamic equilibrium The Standard Model confirms the non-conservation of P and CP, while baryon violation is explored in Grand Unified Theories (GUTs) and the non-perturbative aspects of the Standard Model This departure from equilibrium can occur in the expanding Universe, where various interactions fluctuate in and out of equilibrium A prominent scenario involves the out-of-equilibrium decays of superheavy Higgs bosons predicted by GUTs.
The Standard Model's non-perturbative effects, specifically sphalerons, play a crucial role in generating Baryon Asymmetry at the Electroweak scale These processes conserve B+L, allowing for the initial creation of Lepton Asymmetry through the out-of-equilibrium decay of superheavy right-handed neutrinos, a phenomenon known as Leptogenesis.
The author expresses gratitude to the organizing committee of the 2nd Aegean Summer School on the Early Universe for the opportunity to present an introduction to Early Universe Cosmology Additionally, the author acknowledges the financial support received from the EU RTN "Supersymmetry and the Early Universe," contract No HPRN-CT-.
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Universit´e de Gen`eve, D´epartement de Physique Th´eorique, 24 Quai E Ansermet,
This review focuses on cosmological perturbation theory, beginning with an introduction to the challenges posed by gauge transformation It introduces gauge-invariant variables and reformulates the Einstein and conservation equations accordingly Detailed examples, including perfect fluids and scalar fields, are provided, alongside an examination of perturbation generation during inflation The significance of lightlike geodesics for cosmic microwave background (CMB) anisotropies is also briefly addressed, and the concept of perturbation theory in braneworlds is introduced.
Introduction
The idea that the large scale structure of our Universe might have grown out of small initial fluctuations via gravitational instability goes back to Newton (letter to Bentley, 1692 [1]).
In 1946, Lifshitz conducted the first relativistic analysis of linear perturbations in a Friedmann-Lemaître universe, revealing that, within the framework of linear perturbation theory, the gravitational potential cannot increase This led him to conclude that galaxies did not form through gravitational instability.
To explain the formation of cosmic structures, it is essential to recognize that significant initial matter density fluctuations, approximately 10^-5, are necessary These fluctuations exceed typical statistical variations found in galaxy scales, prompting the need for a mechanism to generate them Additionally, measurements of anisotropies in the cosmic microwave background reveal that the amplitude of these fluctuations remains constant across a broad range of scales, indicating a scale-independent spectrum Standard inflation theory typically yields a spectrum of nearly scale-invariant fluctuations, supporting these observations.
In this review I present gauge invariant cosmological perturbation theory.
This article begins with a definition of gauge invariant perturbation variables, followed by an introduction to the fundamental perturbation equations It explores examples of matter equations, focusing on perfect fluids and scalar fields Additionally, the discussion touches on lightlike geodesics and cosmic microwave background (CMB) anisotropies, noting that this section will be concise as it is supplemented by A Challinor's review on CMB anisotropies.
I shall make some brief comments on perturbation theory for braneworlds, a topic which is still wide open in my opinion.
The Background
I shall not come back to the homogeneous universe which has been discussed in depth by K Tamvakis I just specify our notation which is as follows:
A Friedmann-Lemaître universe represents a homogeneous and isotropic solution to Einstein's equations, characterized by hyper-surfaces of constant time that maintain this homogeneity and isotropy These surfaces are defined by a metric of constant curvature, expressed as a²(η)γij dxi dxj, where γij denotes the metric of a space with constant curvature κ This metric can further be detailed in the form γij dxi dxj = dr² + χ²(r) dϑ² + sin²ϑ dϕ².
The four-dimensional metric is expressed as \( g_{\mu\nu} dx^{\mu} dx^{\nu} = -a^2(\eta) d\eta^2 + a^2(\eta) \gamma_{ij} dx^i dx^j \), where \( \chi(r) \) is defined in equation (1.9) and the scale factor \( a(\eta) \) has been rescaled to ensure \( \kappa = \pm 1 \) or \( 0 \) This normalization gives the scale factor \( a \) dimensions of length, while \( \eta \) and \( r \) remain dimensionless when \( \kappa = 0 \).
Hereηis called theconformal time Thephysicalorcosmological timeis given bydt=adη.
Einstein’s equations reduce to ordinary differential equations for the func- tiona(η) (with ˙≡d/dη): ˙ a a
In the context of cosmology, the energy momentum tensor is defined with ρ = -T^0_0 and p = T^i_i (with no summation), ensuring that all other components vanish to maintain isotropy and homogeneity The cosmological constant, denoted as Λ, plays a crucial role in this framework Additionally, we introduce the Hubble parameter, H, defined as the rate of expansion ȧ/a.
Hubble parameter is defined by
Energy momentum “conservation” (which is also a consequence of (2.3) and (2.4) due to the contracted Bianchi identity) reads ˙ ρ=−3 a˙ a
(ρ+p) =−3(1 +w)H, (2.5) wherew≡p/ρ Later we will also usec 2 s ≡p/˙ ρ From the definition of˙ wand ρtogether with (2.5) one finds ˙ w= 3(1 +w)(w−c 2 s )H (2.6)
From the Friedmann equations one easily concludes that forκ=Λ= 0 and w=const the scale factor behaves like a power law, a∝η 1+3w 2 ∝t 3(1+w) 2 (2.7)
q= 2 for dust w= 0 q= 1 for radiation w= 1/3 q=−1 for inflation (or a cosm const.) w=−1
, where the index0 indicates the value of a given variable today Friedmann’s equation (2.3) then requires
Gauge Invariant Perturbation Variables
Gauge Transformation, Gauge Invariance
The first fundamental problem we want to discuss is the choice of gauge in cosmological perturbation theory:
For linear perturbation theory to be applicable, the spacetime manifold \( M \) with metric \( g \) and the energy momentum tensor \( T \) of the observable universe must closely resemble a Friedmann universe characterized by a Robertson–Walker metric \( \bar{g} \) and a homogeneous, isotropic energy momentum tensor \( T \) A significant challenge lies in accurately constructing the optimal \( \bar{g} \) and \( T \) from the actual physical fields \( g \) and \( T \) This task is complicated by two main issues: first, the spatial averaging methods are influenced by the selection of a constant time hypersurface and do not commute with derivatives, leading to averaged fields \( \bar{g} \) and \( T \) that typically fail to satisfy Einstein’s equations Second, practical averaging over super-horizon scales poses a significant hurdle.
While we cannot provide a definitive prescription, we propose that an averaging procedure exists that results in a Friedmann universe characterized by spatially averaged tensor fields Q, which account for the deviations.
(T àν −T àν )/max { αβ } {|T αβ |}and (g àν −g àν )/max { αβ } {g αβ }are small, and ¯g andT satisfy Friedmann’s equations Let us call such an averaging procedure
Various admissible averaging procedures, such as those applied over different hyper-surfaces, can yield slightly different Friedmann backgrounds However, the difference between these backgrounds, represented as |g−g¯|, remains minimal, indicating that the variations are of a small order.
Friedmann backgrounds must also be small of order and we can regard it as part of the perturbation.
In this discussion, we focus on a fixed admissible Friedmann background metric (¯g,T¯) and acknowledge that due to the invariance of the theory under diffeomorphisms, the resulting perturbations are not unique Specifically, for any arbitrary diffeomorphism φ and its push forward φ ∗, the metrics g and φ ∗(g) represent the same geometric structure By selecting the background metric ¯g, we restrict ourselves to diffeomorphisms that preserve ¯g, allowing only first-order deviations from the identity This infinitesimal diffeomorphism can be expressed as the infinitesimal flow of a vector field X, denoted as φ=φ X The flow is defined through the integral curve γ x (s) of X, starting from point x, where φ X s (x) equals γ x (s) Consequently, in terms of the vector field X and to first order, its pullback is represented as φ ∗ =id+L X.
(L X denotes the Lie derivative in direction X 1 The transformation g → φ ∗ (g) is equivalent to ¯g+a 2 h → ¯g+ (a 2 h+L X g) +¯ O( 2 ), i.e under
The Lie derivative represents the directional derivative along a vector field, providing a coordinate-independent definition It involves a vector field \(X\) and its flux \(\phi_s(x)\), which is the solution to the differential equation \(\frac{dx}{ds} = X(x)\) with an initial point \(x\) For sufficiently small values of \(s\), this concept is well-defined The Lie derivative of a tensor field \(T\) is introduced through an infinitesimal coordinate transformation, where the metric perturbation \(h\) transforms as \(h \to h + a - 2 L_X \bar{g}\).
In the context of cosmological perturbation theory, infinitesimal coordinate transformations are called ‘gauge transformations’ The perturbation of a arbitrary tensor fieldQ= ¯Q+Q (1) obeys the gauge transformation law
Every vector field \( X \) produces a gauge transformation \( \phi = \phi_X \), leading to the conclusion that only perturbations of tensor fields with \( L_X Q = 0 \) for all vector fields \( X \)—meaning those with a vanishing or constant 'background contribution'—are gauge invariant This fundamental result is often referred to as the gauge invariance principle.
The gauge dependence of perturbations has sparked significant debate in the literature, particularly regarding the interpretation of these perturbations on super-horizon scales To address this issue, we will employ gauge invariant perturbation theory throughout this review, which offers the advantage of using variables with clear geometric and physical meanings, free from gauge modes Although deriving these equations may require more effort, the resulting system is typically straightforward and conducive to numerical analysis Furthermore, we will demonstrate that on sub-horizon scales, gauge invariant matter perturbation variables converge with their gauge dependent counterparts Notably, one of the gauge invariant geometrical perturbation variables aligns with the Newtonian potential, facilitating an easy transition to the Newtonian limit.
All relativistic equations are covariant, meaning they can be represented as Q = 0 for a specific tensor field Q This allows the corresponding perturbation equations to be expressed using gauge invariant variables.
Harmonic Decomposition of Perturbation Variables
The homogeneous and isotropic nature of the {η = const} hyper-surfaces makes harmonic analysis a logical approach A spatial tensor field Q can be decomposed into components that transform irreducibly under translations and rotations, with each component evolving independently In the case of a scalar quantity f when κ= 0, this decomposition is represented by its Fourier series: f(x, η) = ∫ d³k f̂(k) e^(i kx).
Hereφ ∗ s denotes the pullback with the (local) diffeomorphismφ s [3].
The functions \( Y_k(x) = e^{ikx} \) represent the unitary irreducible representations of the Euclidean translation group For \( \kappa = 1 \), a decomposition is possible with discrete values \( k^2 = n(n + 2) \), while for \( \kappa = -1 \), the values are bounded from below as \( k^2 > 1 \) Notably, the functions \( Y_k \) differ when \( \kappa = 0 \) These functions constitute a complete orthogonal set of eigenfunctions of the Laplacian.
In addition, a tensorial variable (at fixed positionx) can be decomposed into irreducible components under the rotation groupSO(3).
For a vector field, this is its decomposition into a gradient and a rotation,
B | i i = 0, (2.18) where we used X | i to denote the three–dimensional covariant derivative of
X Hereϕis the spin 0 andB is the spin 1 component of the vector field V. For a symmetric tensor field we have
Here H L and H T are spin 0 components,H i (V ) is a spin 1 component and
We shall not need higher tensors (or spinors) As a basis for vector and tensor modes we use the vector and tensor type eigenfunctions of the La- placian,
∆Y ji (T ) =−k 2 Y ji (T ) , (2.22) whereY j (V ) is a transverse vector,Y j (V ) | j = 0 andY ji (T ) is a symmetric trans- verse traceless tensor,Y j (T )j =Y ji (T) | i = 0.
According to (2.17) and (2.19) we can construct scalar type vectors and tensors and vector type tensors To this goal we define
In the following we shall extensively use this decomposition and write down the perturbation equations for a given modek The decomposition of a vector
field is then of the form
B i =BY i (S) +B (V ) Y i (V ) (2.26) The decomposition of a tensor field is given by (compare (2.19))
H ij =H L Y (S) γ ij +H T Y ij (S) +H (V ) Y ij (V ) +H (T) Y ij (T ) (2.27)HereB,B (V ) , H L ,H T ,H (V ) andH (T) are functions ofη andk.
Metric Perturbations
Perturbations of the metric are of the form g àν = ¯g àν +a 2 h àν (2.28)
We parameterize them as h àν dx à dx ν =−2Adη 2 −2B i dηdx i + 2H ij dx i dx j , (2.29) and we decompose the perturbation variablesB i andH ij according to (2.26) and (2.27).
Let us consider the behavior ofh àν under gauge transformations We set the vector field defining the gauge transformation to
Using simple identities from differential geometry likeL X (df) =d(L X f) and (L X γ) ij =X i | j +X j | i , we obtain
Comparing this with (2.29) and using (2.13) we obtain the following be- haviour of our perturbation variables under gauge transformations (we de- compose the vectorL i =LY i (S) +L (V ) Y i (V ) ):
Two scalar and one vector variable can be set to zero by a cleverly chosen gauge transformations.
In cosmological studies, it is common to select the longitudinal gauge by setting the variables \( H_T \) and \( B \) to zero, resulting in scalar perturbations of the metric expressed as \( h^{(S)}_{\nu} = -2\Psi d\eta^2 - 2\Phi \gamma_{ij} dx^i dx^j \) Here, \( \Psi \) and \( \Phi \) are known as Bardeen potentials In a more general gauge, these potentials are defined by the equations \( \Psi = A - \frac{\dot{a}}{a} k - \frac{1}{\sigma} \dot{\sigma} \) and \( \Phi = -\frac{H}{L} \).
3H T +a˙ ak − 1 σ, (2.41) with σ=k − 1 H˙ T −B A short calculation using (2.32) to (2.36) shows that Ψ andΦare gauge invariant.
In a Friedmann universe the Weyl tensor vanishes It therefore is a gauge invariant perturbation For scalar perturbations one finds
For vector perturbations it is convenient to set kL (V ) = H (V ) so that
H (V ) vanishes and we have h (V àν ) dx à dx ν = 2σ (V ) Y i (V ) dηdx i (2.43)
We shall call this gauge the “vector gauge” In generalσ (V ) =k − 1 H˙ (V ) −B (V ) is gauge invariant 2 The Weyl tensor from vector perturbation is given by
The absence of tensorial (spin 2) gauge transformations indicates that H ij (T) is gauge invariant Notably, the expression for the Weyl tensor derived from tensor perturbations mirrors that of vector perturbations when substituting σ ij (V) with ˙H ij (T).
2 Y ij (V ) σ (V ) is the shear of the hyper-surfaces of constant time.
Perturbations of the Energy Momentum Tensor
Let T ν à =T à ν +θ à ν be the full energy momentum tensor We define its en- ergy densityρand its energy flow 4-vectoruas the time-like eigenvalue and eigenvector ofT ν à :
We then parameterize their perturbations by ρ= ¯ρ(1 +δ), u=u 0 ∂ t +u i ∂ i (2.47) u 0 is fixed by the normalization condition, u 0 =1 a(1−A) (2.48)
We further set u i = 1 av i =vY (S)i +v (V ) Y (V )i (2.49)
We defineP ν à ≡u à u ν +δ à ν , the projection tensor onto the part of tangent space normal touand set the stress tensor τ àν =P α à P β ν T αβ (2.50)
In the unperturbed case we haveτ 0 0 = 0, τ j i = ¯pδ j i Including perturba- tions, to first order we still obtain τ 0 0 =τ i 0 =τ 0 i = 0 (2.51) Butτ j i contains in general perturbations We set τ j i = ¯p
We shall not derive the gauge transformation properties of these pertur- bation variables in detail, but just state some results which can be obtained as an exercise (see also [6]):
The gauge-invariant variables defined above, specifically Π (S,V,T), characterize the anisotropic stress tensor, expressed as Π ν à = τ ν à − 1 / 3 τ α α δ à ν Their gauge invariance is established through the Stewart–Walker lemma, which indicates that ¯Π = 0 In the case of perfect fluids, the condition simplifies to Π ν à = 0.
A second gauge invariant variable is defined as Γ = πL - c²s wδ, where c²s represents the adiabatic sound speed and w denotes the enthalpy It can be demonstrated that Γ is proportional to the divergence of the entropy flux of the perturbations, with adiabatic perturbations being characterized by Γ = 0.
– Gauge invariant density and velocity perturbations can be found by com- biningδ, vandv (V i ) with metric perturbations.
In the context of cosmological perturbations, the variables v (long), δ (long), and v i (vec) represent the velocity and density fluctuations in the longitudinal and vector gauges, while σ (V) denotes the metric perturbation in the vector gauge These variables can be effectively understood through their relationship with the gradients of energy density, as well as the shear and vorticity of the velocity field.
Here I just want to show that on scales much smaller than the Hubble scale,kη 1, the metric perturbations are much smaller thanδ andv and we can thus “forget them” (which will be important when comparing experi- mental results with calculations in this formalism): The perturbations of the Einstein tensor are given by second derivatives of the metric perturbations. Einstein’s equations yield the following order of magnitude estimate:
O(h) On sub-horizon scales the difference betweenδ,δ (long) ,D g andD is negligible as well as the difference betweenv andV orv (V ) , V (V ) andΩ (V )
Einstein’s Equations
Constraint Equations
Dynamical Equations
A secondary dynamical scalar equation exists but is largely unnecessary, as one of the conservation equations can be utilized instead In the case of perfect fluids, where Π j i is equal to zero, the equations simplify to Φ=Ψ, with σ (V) inversely proportional to the square of the scale factor (a²), and H (T) following a damped wave equation The damping term can be disregarded on small scales and short time intervals when η − 2 is approximately less than or equal to zero.
2κ+k 2 , and H ij represents propagating gravitational waves For vanishing curvature, these are just the sub-horizon scales, kη ∼ > 1 For κ < 0, waves oscillate with a somewhat smaller frequency,ω =√
2κ+k 2 , while for κ >0 the frequency is somewhat larger thank.
Energy Momentum Conservation
The conservation equations,T ;ν àν = 0 lead to the following perturbation equa- tions
The Einstein equations, which are equivalent to the contracted Bianchi identities, provide essential insights into perturbations in spacetime For scalar perturbations, there are four independent equations governing six variables, while vector perturbations consist of two equations with three variables In the case of tensor perturbations, the relationships are similarly structured, highlighting the complexity of these gravitational phenomena.
In a system of equations with two variables, it is essential to incorporate matter equations to achieve closure The most straightforward approach is to set Γ and Π ij to zero This matter equation, which addresses adiabatic perturbations in a perfect fluid, provides two additional equations for scalar perturbations, along with one equation each for vector and tensor perturbations.
In the following section, we will explore a universe characterized by a scalar field as its matter content Additionally, we will examine more complex scenarios involving multiple interacting particle species, some of which require a Boltzmann equation for accurate description This reflects the state of our universe at late times, specifically when the redshift is approximately z < 10^7.
A Special Case
Here we want to rewrite the scalar perturbation equations for a simple but important special case We consider adiabatic perturbations of a perfect fluid.
In this case Π = 0 since there are no anisotropic stresses and Γ = 0. Equation (2.66) then implies Φ=Ψ Using the first equation of (2.64) and (2.58,2.57) to replaceD g in the second of (2.69) byΨ andV, finally replacing
V by (2.64) one can derive a second order equation forΨ, which is, in this case the only dynamical degree of freedom Ψ¨+ 3H(1 +c 2 s ) ˙Ψ+ [(1 + 3c 2 s )(H 2 −κ)−(1 + 3w)H 2 +c 2 s k 2 ]Ψ = 0 (2.71)
In the scalar field case, which is particularly relevant in discussions of inflation, we find that while Π = 0, the condition Γ = 0 generally holds true due to the relationship δp/δρ = ˙p/ρ This scenario features a single dynamical degree of freedom, allowing us to simplify the perturbation equations into a single second-order equation for Ψ In Section 2.6, we derive the following equation for perturbed scalar field cosmology: Ψ¨ + 3H(1 + c²s) ˙Ψ + [(1 + 3c²s)(H² - κ) - (1 + 3w)H² + k²]Ψ = 0.
The distinction between the perfect fluid and scalar field perturbation equations lies in the absence of the factor \( c_s^2 \) in the latter's oscillatory \( k^2 \) term Additionally, when \( \kappa = 0 \) and \( w = c_s^2 \) remains constant, the time-dependent mass term \( m^2(\eta) \) simplifies to zero, expressed as \( - (1 + 3c_s^2)(H^2 - \kappa) + (1 + 3w)H^2 \) It is also beneficial to define the variable \( u = a \).
$ − 1/2 Ψ, (2.73) which satisfies the equation ¨ u+ (Υ k 2 −θ/θ)u¨ = 0, (2.74) where Υ = c 2 s or Υ = 1 for a perfect fluid or a scalar field background respectively, and θ= 3H
Using (2.71) and (2.72) respectively one obtains ζ˙=−k 2 ΥH
H 2 −H˙Ψ , (2.77) hence on super horizon scales,k/H 1, this variable is conserved.
The evolution ofζis closely related to the canonical variablevdefined by v=−a
√4πGΥH ζ (2.78) which satisfies the equation ¨ v+ (Υ k 2 −z/z)v¨ = 0, (2.79) for z= a
More details on the significance of the canonical variablev will be found inSects 2.6 and 2.7.
Simple Examples
The Pure Dust Fluid for κ = 0, Λ = 0
In the context of cosmological perturbations, we assume that the dust has parameters w = c²s = p = 0 and Π = Γ = 0 This leads to the simplified equation Ψ¨ + 6ηΨ˙ = 0, with the general solution Ψ = Ψ₀ + Ψ₁/η⁵, where Ψ₀ and Ψ₁ are arbitrary constants Given that the initial perturbations are small, we must choose the growing mode Ψ₁ = 0 to prevent divergence as η approaches 0 Lifshitz, who first analyzed these perturbations, concluded that linear perturbations do not grow in a Friedman universe, implying that cosmic structure cannot evolve through gravitational instability However, it is crucial to note that even with a constant gravitational potential, matter density fluctuations can grow on sub-horizon scales, allowing for the evolution of inhomogeneities and structure on scales smaller than the Hubble scale This can be understood through the conservation equations and the Poisson equation applicable to the pure dust case.
(Poisson),(2.85) where we have used the relation
The Friedmann equation for dust gives 4πGρa 2 = 3 / 2( ˙a/a) 2 = 6/η 2 Setting kη=xand =d/dx, the system (2.83-2.85) becomes
We use (2.89) to eliminateΨ and (2.87) to eliminateD g , leading to
V = 0 (2.90) The general solution of (2.90) is
TheV 1 mode is the decaying mode (corresponding toΨ 1 ) which we neglect. The perturbation variables are then given by
We distinguish two regimes: i ) super-horizon,x1 where we have
Despite the growth of V, it remains significantly smaller than Ψ or Dg on super-horizon scales, indicating that the largest fluctuations are primarily determined by the constant Ψ In the sub-horizon region, the solution is predominantly influenced by these terms.
In the variableD the constant term has disappeared and we haveDΨ on super-horizon scales, x1.
On sub-horizon scales, density fluctuations increase in proportion to the scale factor, following the relationship ∝ x² ∝ a However, Lifshitz's assertion that pure gravitational instability is insufficient for structure formation holds some validity Starting from minuscule thermal fluctuations around 10⁻³⁵, these can only grow to approximately 10⁻³⁰ during the matter-dominated era due to this gradual power law instability Therefore, for gravitational instability to effectively contribute to structure formation, initial fluctuations must be at least of the order of 10⁻³⁵.
During cosmic inflation, quantum particle production in a classical gravitational field can generate fluctuations significantly larger than thermal fluctuations The universe's rapid expansion during this period stretches microscopic scales, causing quantum fluctuations to become relevant at cosmological scales These fluctuations are subsequently "frozen in" as classical perturbations in energy density and geometry Further details on the induced spectrum of these fluctuations will be discussed in Section 2.7.
The Pure Radiation Fluid, κ = 0, Λ = 0
In this limit we set w = c 2 s = 1 / 3 and Π = Γ = 0 so that Φ = −Ψ We conclude from ρ ∝ a − 4 that a ∝ η For radiation, the u–equation (2.74) becomes ¨ u+ (1
3k 2 − 2 η 2 )u= 0, (2.101) with general solution u(x) =A sin(x) x −cos(x)
3 =c s kη For the Bardeen potential we obtain with (2.73), up to constant factors, Ψ(x) =u(x) x 2 (2.103)
On super-horizon scales,x1, we have Ψ(x) A
Assuming that the perturbations were initialized at an early time x in 1, and that both modes were comparable at that time, we can conclude that BA is significant while the B-mode can be disregarded in later instances.
To analyze density and velocity perturbations, we solve the radiation equations alongside the conservation and Poisson equations, similar to the dust scenario In the context of radiation, the perturbation equations are expressed using the notation x = c_s kη.
The general solution of this system is
Again, regularity at x = 0 requires D 1 = 0 Comparing with (2.102,2.103) givesD 2 = 2A In thesuper-horizon regime,x1, we obtain Ψ = A
Onsub-horizon scales, x1, we find oscillating solutions with constant amplitude and with frequency of k/√
D g = 2Acos(x), Ψ =−Acos(x)/x 2 (2.113) Note that also for radiation perturbations
On super horizon scales, the perturbation amplitude is characterized by the largest gauge invariant variable, indicating that perturbations outside the Hubble horizon remain frozen to first order As these perturbations cross into the horizon, they begin to collapse; however, pressure counteracts the gravitational force, causing radiation fluid fluctuations to oscillate at a constant amplitude Inside the horizon, the gravitational potential perturbations oscillate and decay at a rate of 1/a².
Adiabatic Initial Conditions
Adiabaticity necessitates that all contributions to energy density perturbations start in thermal equilibrium, establishing a fixed ratio of density perturbations among different components In this scenario, there is no entropy flux, resulting in Γ = 0 For illustration, we examine non-relativistic matter and radiation perturbations, which exhibit similar behavior on super-horizon scales.
D g (r) =A+Bx 2 , D g (m) =A +B x 2 , V (r) ∝V (m) ∝x, (2.114) we may require a constant ratio between matter and radiation perturbations.
In the sub-horizon region (x > 1), radiation perturbations begin to oscillate, while matter perturbations adhere to a power law, leading to a breakdown of the constant ratio between them This scenario presents two noteworthy types of perturbations: adiabatic and isocurvature Our focus here is on adiabatic perturbations, which appear to be the primary contributors to the observed anisotropies in the Cosmic Microwave Background (CMB).
From Γ = 0 one easily derives that two components with p i /ρ i w i =constant, i = 1,2, are adiabatically coupled if (1 +w 1)D g (2) = (1 + w 2)D (1) g Energy conservation then implies that also their velocity fields agree,
V (1) =V (2) This result is also a consequence of the Boltzmann equation in the strong coupling regime We therefore require
V (r) =V (m) , (2.115) so that the energy flux in the two fluids is coupled initially.
In a matter-dominated universe, we focus on radiation perturbations relevant to this era The gravitational potential, denoted as Ψ, is primarily influenced by matter, remaining constant at Ψ₀ due to its proportional relationship with the background density Consequently, we disregard the minor contributions from radiation to Ψ Applying energy-momentum conservation for radiation, we express the relationship in terms of x = kη, where k represents the wave number and η is the conformal time.
NowΨ is just a constant given by the matter perturbations, and it acts like a constant source term The general solution of this system is then
3 is the sound speed of radiation Our adiabatic initial con- ditions require x lim → 0
ThereforeB= 0 andV 0=A/4−2Ψ Using in additionΨ = 3V 0(see (2.100)) we obtain
On super-horizon scales,x1 we have
In the context of adiabatic initial conditions, the relationship between the density perturbations D(r) and D(m) is given by D(r) = (4/3)D(m), with the velocity perturbations V(r) and V(m) being equal Alternatively, isocurvature initial conditions can result in non-vanishing perturbations D(r), D(m), V(r), and V(m) that balance each other, leading to Ψ = 0 on super-horizon scales However, the simplest inflationary models do not generate such perturbations, and current observations suggest they do not dominate the anisotropies observed in the Cosmic Microwave Background (CMB), although their presence could complicate the accurate determination of cosmological parameters from CMB anisotropies.