8.3 Cosmological parameters

Cosmological parameters are the building blocks of our understanding of the universe. They describe its fundamental properties, from expansion rate to composition, and help us piece together its history and future.

Determining these parameters is a complex task, involving various observational techniques and theoretical models. The tension between different measurements of the Hubble constant highlights the ongoing challenges in cosmology, pushing scientists to refine their methods and explore new ideas.

Key cosmological parameters

• Cosmological parameters describe the fundamental properties and evolution of the universe
• Determining accurate values for these parameters is crucial for understanding the universe's composition, structure, and ultimate fate
• Key parameters include the Hubble constant, density parameters, cosmological constant, curvature parameter, and age of the universe

Hubble constant

• The Hubble constant ($H_0$) represents the current expansion rate of the universe
• Its value determines the size and age of the universe, as well as the distance scale to astronomical objects

Value of Hubble constant

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• The Hubble constant is expressed in units of km/s/Mpc (kilometers per second per megaparsec)
• Current measurements suggest a value around 70 km/s/Mpc, with some tension between different measurement methods
  • Cosmic microwave background observations (Planck) indicate $H_0 \approx 67.4 \pm 0.5$ km/s/Mpc
  • Local measurements (Cepheid variables, supernovae) suggest $H_0 \approx 74.0 \pm 1.4$ km/s/Mpc
• Resolving this tension is an active area of research in cosmology

Implications for universe's age

• The Hubble constant is inversely related to the age of the universe
• A higher value of $H_0$ implies a younger universe, while a lower value suggests an older universe
• Current estimates based on the Hubble constant place the age of the universe around 13.8 billion years

Density parameters

• Density parameters ($\Omega$) describe the relative contributions of different components to the universe's total energy density
• The three main components are matter ($\Omega_m$), radiation ($\Omega_r$), and dark energy ($\Omega_\Lambda$)
• The sum of these density parameters determines the geometry and ultimate fate of the universe

Matter density parameter

• The matter density parameter ($\Omega_m$) represents the contribution of both ordinary (baryonic) and dark matter to the universe's energy density
• Current observations suggest $\Omega_m \approx 0.3$, indicating that matter makes up about 30% of the universe's total energy density
  • Baryonic matter (protons, neutrons, electrons) accounts for ~5% of the total energy density
  • Dark matter, which does not interact electromagnetically, makes up the remaining ~25%

Dark energy density parameter

• The dark energy density parameter ($\Omega_\Lambda$) represents the contribution of dark energy to the universe's total energy density
• Dark energy is a hypothetical form of energy that permeates all of space and drives the accelerated expansion of the universe
• Current observations suggest $\Omega_\Lambda \approx 0.7$, indicating that dark energy makes up about 70% of the universe's total energy density

Critical density

• The critical density ($\rho_c$) is the total energy density required for the universe to be spatially flat ($\Omega_{total} = 1$)
• It is defined as $\rho_c = \frac{3H_0^2}{8\pi G}$, where $G$ is the gravitational constant
• The ratio of the actual density to the critical density determines the universe's geometry (flat, open, or closed)

Cosmological constant

• The cosmological constant ($\Lambda$) is a term in Einstein's field equations of general relativity that represents a constant energy density of the vacuum
• It was originally introduced by Einstein to achieve a static universe, but later abandoned after the discovery of the universe's expansion

Einstein's original concept

• Einstein added the cosmological constant to his field equations to counteract the attractive force of gravity and achieve a static universe
• He later referred to this as his "greatest blunder" after the discovery of the expanding universe by Edwin Hubble

Modern interpretation as dark energy

• In the context of modern cosmology, the cosmological constant is interpreted as a form of dark energy
• It provides a simple explanation for the observed accelerated expansion of the universe
• The cosmological constant is the simplest form of dark energy, with a constant energy density throughout space and time

Curvature parameter

• The curvature parameter ($\Omega_k$) describes the geometry of the universe
• It is defined as $\Omega_k = 1 - \Omega_{total}$, where $\Omega_{total}$ is the sum of the matter, radiation, and dark energy density parameters

Flat vs curved universe

• A flat universe has $\Omega_k = 0$, meaning that the total energy density is equal to the critical density
  • Parallel lines remain parallel, and the interior angles of a triangle sum to 180 degrees
• A positively curved (closed) universe has $\Omega_k < 0$, with a total energy density greater than the critical density
  • Parallel lines converge, and the interior angles of a triangle sum to more than 180 degrees
• A negatively curved (open) universe has $\Omega_k > 0$, with a total energy density less than the critical density
  • Parallel lines diverge, and the interior angles of a triangle sum to less than 180 degrees

Observational evidence

• Observations of the cosmic microwave background (CMB) and large-scale structure suggest that the universe is very close to being spatially flat
• Current measurements indicate $|\Omega_k| < 0.005$, consistent with a flat universe within the uncertainties
• A flat universe is also predicted by the theory of cosmic inflation, which proposes a period of rapid expansion in the early universe

Age of the universe

• The age of the universe is the time elapsed since the Big Bang, the event that marked the beginning of the universe as we know it
• Determining the age of the universe is crucial for understanding its evolution and constraining cosmological models

Estimation methods

• The age of the universe can be estimated using several methods, including:
  • Hubble constant measurements: The age is approximately the inverse of the Hubble constant, $t_0 \approx H_0^{-1}$
  • Cosmic microwave background observations: The CMB provides a snapshot of the universe at an age of ~380,000 years, allowing the total age to be inferred
  • Stellar and galactic evolution models: The ages of the oldest stars and galaxies provide a lower limit on the age of the universe

Current best estimates

• The most precise estimates of the universe's age come from measurements of the cosmic microwave background
• The Planck satellite's observations of the CMB, combined with other cosmological data, yield an age of $13.797 \pm 0.023$ billion years
• This estimate assumes the standard $\Lambda$CDM cosmological model, which includes dark energy and cold dark matter

Cosmic microwave background

• The cosmic microwave background (CMB) is the oldest observable light in the universe, originating from the epoch of recombination ~380,000 years after the Big Bang
• It provides a wealth of information about the early universe and constrains key cosmological parameters

Discovery and significance

• The CMB was discovered accidentally in 1965 by Arno Penzias and Robert Wilson, who detected a uniform background noise in their radio antenna
• The existence of the CMB had been predicted by George Gamow and others as a consequence of the Big Bang theory
• The discovery of the CMB provided strong evidence for the Big Bang model and ruled out steady-state models of the universe

Temperature and fluctuations

• The CMB has a nearly perfect blackbody spectrum with a temperature of $2.7255 \pm 0.0006$ K
• Small temperature fluctuations (~1 part in 100,000) are observed across the sky, representing density variations in the early universe
• These fluctuations are believed to be the seeds of cosmic structure, giving rise to galaxies and galaxy clusters through gravitational instability

Constraints on cosmological parameters

• Measurements of the CMB temperature and polarization anisotropies provide tight constraints on cosmological parameters
• Key parameters constrained by the CMB include:
  • Hubble constant ($H_0$)
  • Matter density ($\Omega_m$)
  • Baryon density ($\Omega_b$)
  • Dark energy density ($\Omega_\Lambda$)
  • Curvature parameter ($\Omega_k$)
  • Scalar spectral index ($n_s$)
• The CMB constraints are often combined with other cosmological probes (baryon acoustic oscillations, supernovae, galaxy clusters) to break degeneracies and improve parameter estimates

Concordance model

• The concordance model, also known as the $\Lambda$CDM model, is the standard cosmological model that best describes the universe on large scales
• It incorporates dark energy ($\Lambda$) and cold dark matter (CDM) to explain the observed accelerated expansion and structure formation

Key components and parameters

• The $\Lambda$CDM model includes the following key components:
  • Dark energy (cosmological constant, $\Lambda$): ~70% of the universe's energy density
  • Cold dark matter (CDM): ~25% of the energy density, non-relativistic and non-baryonic matter
  • Ordinary (baryonic) matter: ~5% of the energy density
  • Radiation (photons and neutrinos): <0.1% of the energy density at present
• The model is characterized by six main parameters: $H_0$, $\Omega_m$, $\Omega_b$, $\Omega_\Lambda$, $n_s$, and $\sigma_8$ (amplitude of matter fluctuations)

Observational support

• The $\Lambda$CDM model is supported by a wide range of observational evidence, including:
  • Cosmic microwave background temperature and polarization anisotropies
  • Baryon acoustic oscillations in the distribution of galaxies
  • Type Ia supernova distance measurements, indicating accelerated expansion
  • Abundance of light elements (hydrogen, helium, lithium) from Big Bang nucleosynthesis
  • Large-scale structure formation and galaxy clustering
• The model provides a consistent description of the universe's evolution and structure across multiple observational probes

Remaining uncertainties and challenges

• Despite its success, the $\Lambda$CDM model faces some remaining uncertainties and challenges:
  • The nature of dark energy and dark matter remains unknown
  • Tensions exist between different measurements of the Hubble constant ($H_0$)
  • The model struggles to explain some small-scale issues, such as the "cuspy halo" and "missing satellites" problems in galaxy formation
  • The origin of the universe's initial conditions and the cause of cosmic inflation are not fully understood
• Addressing these challenges and refining the concordance model is an ongoing effort in cosmology, driving future observations and theoretical work