Astronomy & Celestial Mechanics
the sky, computed
Concept

Growth factor — where it appears

The factor by which a small density fluctuation in the matter has grown since some early time, in linear theory. It rises in proportion to the scale factor while matter dominates, and it stalls when radiation or dark energy sets the expansion rate. It links early fluctuations to later clustering.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Structure stops forming above about 14 times the observed Λ. The fraction of matter that ever collapses into a bound object, against the cosmological constant in units of the observed one, holding the primordial fluctuation amplitude fixed. Growth of structure stops once Λ dominates the expansion, so the linear growth factor approaches a finite limit rather than rising for ever, and a larger Λ freezes it earlier and smaller. The asymptotic growth factor at the observed Λ is 1.110; at a hundredth of it, 5.152; at a hundred times, 0.239. The collapsed fraction falls from 0.86 to 0.40 to 0.000 across the same range, and passes a tenth of its present value at 14 times the observed constant. The observed value sits a factor of 14 below the largest one that leaves anything at all — which is the whole of the anthropic argument, and that factor is what has to be compared against the 10¹²⁰ by which the naive theoretical estimate misses.

A coincidence that is a factor of fourteen

The cosmological constant is famously wrong by a hundred and twenty orders of magnitude, and famously coincidental in sitting near the matter density now. Computing how large it could be and still leave any structure at all turns the second complaint into a number — and the number is fourteen, not a hundred and twenty.

cosmology · Density parameters A clump that does not grow until the radiation is outweighed. The amplitude of a dark-matter overdensity inside the horizon against the scale factor in units of its value at matter–radiation equality, both logarithmic, from y = 10⁻⁴ to today near y = 3400. The solution of the growth equation, δ = 1 + 3y/2, stays flat until equality and then rises in proportion to the scale factor; matter domination alone would have grown it in proportion to the scale factor from the start. Over the whole range the true growth is 5.1e+3 against 3.4e+7 — radiation withheld a factor of 6666, which is the interval from y = 10⁻⁴ to about y = 0.67.

A clump that waits for the radiation to thin

Dark matter feels no pressure from light, so nothing should have stopped it clumping in the radiation era. Almost nothing did stop it, and yet a clump inside the horizon at that time barely grows. The radiation does not push on it. It only makes the universe expand too fast for the clump's own gravity to keep up — and the length of that wait is the bend in the spectrum of every structure in the sky.

cosmology · Large-scale structure

Named alongside it

The objects these essays reach for when they reach for this one.

Anthropic boundCoincidence problemCold dark matterCollapsed fractionCosmological constantCritical densityEquation of stateHorizon entryMatter radiation equalityMeszaros effectPower spectrumPress schechter

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