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Showing 301–330 of 1709 reference entries

BiologyDevelopment and cell differentiation

Regulated cell states, division and spatial organization produce multicellular form.

ModelEarth & cosmosMinerals, structure and diagnostic properties

Mineral identification combines structural and physical evidence.

MethodEarth & cosmosRadiometric ages and sample history

Isotopic measurements estimate an age only with a suitable geological model.

MethodEarth & cosmosSeismic constraints on Earth's interior

Wave travel, refraction and transmission constrain an interior model.

MethodEarth & cosmosThe geodynamo model

Fluid motion in the outer core maintains much of Earth's magnetic field.

ModelEarth & cosmosWater budgets and storage change

A defined area's water storage changes with total inflow and outflow.

ModelEarth & cosmosDarcy flux and hydraulic conductivity

Groundwater flow is often approximately linear in hydraulic-head gradient.

ApproximationEarth & cosmosOcean circulation and overturning

Wind, buoyancy changes and mixing contribute to ocean transport.

ModelEarth & cosmosGlobal atmospheric circulation

Unequal heating and rotation organize broad circulation patterns.

ModelEarth & cosmosClimate forcing and energy-budget perturbations

External changes can disturb climate's radiation budget.

DefinitionEarth & cosmosClimate feedback mechanisms

Climate responses can amplify or damp a change.

ModelEarth & cosmosClimate detection and attribution methods

Attribution compares observed changes with plausible drivers and variability.

MethodEarth & cosmosStellar evolution and fuel stages

Stellar structure changes as core fuel and conditions change.

ModelEarth & cosmosWhite dwarfs, neutron stars and stellar black holes

Different evolutionary outcomes leave different compact remnants.

ModelEarth & cosmosAnnual stellar parallax and distance

Changing observing position permits geometric distance inference.

MethodEarth & cosmosCosmological redshift and distance inference

Spectral stretching carries expansion information but is not one universal distance ruler.

MethodEarth & cosmosGalactic rotation and inferred mass

Measured motions constrain the gravitational mass distribution.

MethodEarth & cosmosPlanetary formation in young-star disks

Disk observations and competing growth mechanisms inform planet-formation models.

ModelEarth & cosmosCosmological parameters inferred from the CMB

CMB statistics constrain parameters within a chosen cosmological model.

MethodEarth & cosmosGravitational evidence for dark matter

Motions and lensing constrain mass beyond that accounted for by luminous material.

Empirical findingEarth & cosmosWhat constitutes dark matter?

Gravitational evidence does not yet settle the unseen component's physical identity.

Open questionEarth & cosmosWhat explains accelerated cosmic expansion?

Expansion evidence motivates competing physical descriptions with unresolved origin.

Open questionMathematics & logicIntermediate values and compact-domain extrema

A continuous real function on a closed bounded interval attains a maximum and minimum and takes every value between its endpoint values.

Mathematical statementMathematics & logicProduct, quotient and chain rules

Differentiable real functions satisfy (fg)′=f′g+fg′ and (g∘f)′=(g′∘f)f′.

Mathematical statementMathematics & logicMean value theorem

If f is continuous on [a,b] and differentiable on (a,b), a<b, some c∈(a,b) satisfies f′(c)=(f(b)−f(a))/(b−a).

Mathematical statementMathematics & logicDarboux intermediate-value property of derivatives

For f differentiable throughout an open real interval, its derivative takes every value strictly between f′(a) and f′(b) at some c∈(a,b), whenever a<b are points of that interval.

Mathematical statementMathematics & logicSubstitution in a definite integral

For g of class C¹ on [a,b] and f continuous on an interval containing g([a,b]), ∫ₐᵇ f(g(x))g′(x)dx=∫g(a)ᵍ(b)f(u)du.

MethodMathematics & logicIntegration by parts

For u,v of class C¹ on [a,b], ∫ₐᵇuv′dx=[uv]ₐᵇ−∫ₐᵇu′v dx.

MethodMathematics & logicDarboux sums and Riemann integrability

For bounded real f on [a,b], lower and upper Darboux sums use each subinterval's infimum and supremum.

Mathematical statementMathematics & logicContinuity and the Riemann integrability criterion

A bounded real function on [a,b] is Riemann integrable exactly when its discontinuity set has Lebesgue measure zero.

Mathematical statement
MethodSource checked

Cosmological parameters inferred from the CMB

Earth & cosmos

CMB statistics constrain parameters within a chosen cosmological model.

Key knowledge

Temperature and polarization statistics of the cosmic microwave background can be compared with predicted spectra to constrain cosmological parameters. Planck releases provide likelihoods, parameter chains and best fits for stated models and data combinations; the inferred parameters are not direct readings from a sky-map pixel.

Conditions and scope

  • Specify the cosmological model, datasets, foreground treatment and instrumental assumptions.
  • Changing model freedom or data combinations can change parameter constraints and degeneracies.
Sources · 2

References

Sources retain their recorded scope. A source link alone does not establish verification of every statement.

  1. European Space Agency (ESA), Planck2 sources
    1. PR3 Cosmology Products — model grids, likelihoods and angular power spectraEuropean Space Agency (ESA), PlanckReference guideesdcdoi.esac.esa.intRead source
    2. Planck and the cosmic microwave background — model parametersEuropean Space Agency (ESA), PlanckTeaching resourceesa.intRead source

Evidence and review

Inspected ESA PR3 model grids, likelihood products, uncertainty descriptions and parameter explanation.

2026-10-10 · Checked conditional inference scope; no likelihood rerun or universally model-independent parameter claim.

Related reference entries

Explore relationships within this area of study and its connections to other areas.

Showing 1–7 of 7 relationships

Scientific relationships · 7 / 7

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

  1. Related to
    Relationship sources · 1

    Linked statements

    • Temperature and polarization statistics of the cosmic microwave background can be compared with predicted spectra to constrain cosmological parameters. Planck releases provide likelihoods, parameter chains and best fits for stated models and data combinations; the inferred parameters are not direct readings from a sky-map pixel.

    CMB statistics constrain parameters within an explicitly stated cosmological model.

  2. Related to
    Relationship sources · 1

    Linked statements

    • Observations interpreted within cosmological models indicate accelerated expansion. A cosmological constant, dynamical fields and modified-gravity descriptions are among approaches investigated to explain it. The physical origin of the acceleration remains unresolved, even when a particular model fits the measurements well.

    CMB and expansion observations inform competing models; parameter fits do not settle physical origin.

  3. Related to
    Relationship sources · 2

    Linked statements

    • The hot expanding universe passed through changing particle and radiation conditions. Light nuclei formed before neutral atoms; later declining free-electron abundance allowed photons to propagate much more freely. The cosmic microwave background preserves radiation from this last-scattering era, rather than light emitted at the first instant.

    Detailed anisotropies constrain a model, not just a thermal-history label.

  4. Related to
    Relationship sources · 1

    Linked statements

    • In the standard cosmological model, small density variations grow gravitationally and dark-matter structures help concentrate baryonic matter after photon coupling becomes less effective. Gas cooling, star formation, feedback and mergers then affect the luminous galaxies. Observed clustering tests this account alongside the CMB and expansion evidence.

    Early perturbation evidence constrains structure models.

  5. Related to
    Relationship sources · 2

    Linked statements

    • Atomic recombination binds many previously free electrons and lowers photon-scattering opacity. Photon last scattering describes the probabilistic final scattering along an observed path; it is related to recombination but is not an instantaneous atomic event. Later ultraviolet sources reionize intergalactic gas, creating free electrons that can scatter some CMB photons again. For homogeneous nonrelativistic Thomson scattering, optical depth from time t to now is τ(t)=∫ₜᵗ⁰ cσ_T nₑ(t′)dt′, with no-further-scattering probability e^(−τ).

    CMB parameter inference depends on scattering history.

  6. Related to
    Relationship sources · 1

    Linked statements

    • Before recombination, scattering tightly coupled photons and baryons. Gravitational perturbations drove small-amplitude acoustic oscillations, with photon pressure providing restoring force and baryons contributing inertia. Density/temperature and velocity contributions had different phases. Their projection at last scattering produces peak structure in the CMB angular spectrum; imperfect coupling, photon diffusion and finite last-scattering width suppress small-scale anisotropy.

    Parameter inference uses a spectrum model and uncertainty evidence, beyond this mechanism.

  7. Related to
    Relationship sources · 1

    Linked statements

    • Cosmological parameter inference combines data likelihoods with model priors and nuisance parameters for foregrounds, calibration and selection. Shared data/covariance prevent arbitrary multiplication of apparently independent probes. Constraints and discrepancies change when model freedom and systematics change.

    Conceptual association supported by the recorded research scope; this relation does not assert a proven derivation or mandatory prerequisite.

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University reference