Preparation: algebra, introductory mechanics, and the previous module's stated prerequisites. This is a 60-minute instructional module; use the slide timings as a guide and pause at the active check.
Terms Developed in Context
These terms organize the chain from recorded signal to calibrated observable, likelihood, and bounded cosmological claim.
- Density contrast
The fractional matter-density perturbation relative to the cosmic mean. In this module it is used to answer
How does gravity build structure after the CMB?
Observable/model connection: The term links
Growth curves in this package are model calculations, not measured galaxy power spectra; published DES S8 is used only as a separate summary constraint
to Growth factor and the quantitative relation \(delta'' + (2+H'/H)delta' - 3 Omega_m(a) delta/2 = 0\). State which part is measured, calibrated, derived, or model-dependent.Limit: The scale-independent growth equation assumes linear perturbations and general relativity with smooth dark energy; massive neutrinos and modified gravity alter the picture. That boundary must remain attached when the term is used in a cosmological claim.
- Growth factor
The time-dependent amplitude of the linear growing mode, usually normalized at a chosen epoch. In this module it is used to answer
How does gravity build structure after the CMB?
Observable/model connection: The term links
Growth curves in this package are model calculations, not measured galaxy power spectra; published DES S8 is used only as a separate summary constraint
to Growth rate and the quantitative relation \(delta'' + (2+H'/H)delta' - 3 Omega_m(a) delta/2 = 0\). State which part is measured, calibrated, derived, or model-dependent.Limit: The scale-independent growth equation assumes linear perturbations and general relativity with smooth dark energy; massive neutrinos and modified gravity alter the picture. That boundary must remain attached when the term is used in a cosmological claim.
- Growth rate
The logarithmic derivative f=d ln D/d ln a. In this module it is used to answer
How does gravity build structure after the CMB?
Observable/model connection: The term links
Growth curves in this package are model calculations, not measured galaxy power spectra; published DES S8 is used only as a separate summary constraint
to Redshift-space distortion and the quantitative relation \(delta'' + (2+H'/H)delta' - 3 Omega_m(a) delta/2 = 0\). State which part is measured, calibrated, derived, or model-dependent.Limit: The scale-independent growth equation assumes linear perturbations and general relativity with smooth dark energy; massive neutrinos and modified gravity alter the picture. That boundary must remain attached when the term is used in a cosmological claim.
- Redshift-space distortion
Anisotropy in observed clustering caused by line-of-sight peculiar velocities. In this module it is used to answer
How does gravity build structure after the CMB?
Observable/model connection: The term links
Growth curves in this package are model calculations, not measured galaxy power spectra; published DES S8 is used only as a separate summary constraint
to f sigma8 and the quantitative relation \(delta'' + (2+H'/H)delta' - 3 Omega_m(a) delta/2 = 0\). State which part is measured, calibrated, derived, or model-dependent.Limit: The scale-independent growth equation assumes linear perturbations and general relativity with smooth dark energy; massive neutrinos and modified gravity alter the picture. That boundary must remain attached when the term is used in a cosmological claim.
- f sigma8
A growth observable combining the logarithmic growth rate with fluctuation amplitude. In this module it is used to answer
How does gravity build structure after the CMB?
Observable/model connection: The term links
Growth curves in this package are model calculations, not measured galaxy power spectra; published DES S8 is used only as a separate summary constraint
to Density contrast and the quantitative relation \(delta'' + (2+H'/H)delta' - 3 Omega_m(a) delta/2 = 0\). State which part is measured, calibrated, derived, or model-dependent.Limit: The scale-independent growth equation assumes linear perturbations and general relativity with smooth dark energy; massive neutrinos and modified gravity alter the picture. That boundary must remain attached when the term is used in a cosmological claim.
Physical and observational argument
In linear theory, density perturbations grow under self-gravity while cosmic expansion damps their motion. The growth equation's friction term depends on H(a), and its source depends on the clustering matter fraction. The logarithmic rate f=d ln D/d ln a links density growth to coherent peculiar velocities; redshift-space anisotropy therefore adds a growth observable to geometric distance measures. The supplied integral solves a scale-independent smooth-dark-energy approximation and normalizes D(0)=1. It excludes radiation, massive-neutrino scale dependence and nonlinear evolution, so its values are model calculations rather than survey measurements.
Linear density growth responds to the expansion history and clustering matter. Redshift-space distortions probe coherent peculiar velocities and constrain f sigma8, complementing geometric distance probes.
Growth curves in this package are model calculations, not measured galaxy power spectra; published DES S8 is used only as a separate summary constraint.
Derivation and interpretation
\[delta'' + (2+H'/H)delta' - 3 Omega_m(a) delta/2 = 0\]
Begin by naming observables, coordinate convention, and model assumptions; each algebraic transformation must preserve units and the distinction between measured and inferred quantities. Test a limiting case, then perturb one input and state which observable changes.
The scale-independent growth equation assumes linear perturbations and general relativity with smooth dark energy; massive neutrinos and modified gravity alter the picture.
Worked numerical anchor
The normalized model growth factor is D(0.5)=0.76892 and D(1)=0.60675; these are model outputs.
Reproduce the value with the course computation module, retain full precision in code, and round only for communication. Report units and all conditional assumptions with the result.
Evidence, systematics, and limits
The scale-independent growth equation assumes linear perturbations and general relativity with smooth dark energy; massive neutrinos and modified gravity alter the picture.
Growth curves in this package are model calculations, not measured galaxy power spectra; published DES S8 is used only as a separate summary constraint.
Separate the directly recorded observable, its calibration, the derived intermediate quantity, and the model-dependent parameter. An uncertainty bar is not a complete likelihood when shared covariance or selection is omitted.
Study and retrieval practice
- Derive the linear growth equation and define the growth rate f.
- Explain how peculiar velocities distort clustering in redshift space.
- Distinguish model growth curves from observed f sigma8 constraints.
Checkpoint: Name scales and epochs where linear, scale-independent growth fails.
Answer in words before calculating; then identify one observation or null test that could falsify your interpretation.