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Growth of cosmic structure

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We review one of the most fruitful areas in cosmology today that bridge theory and data—the temporal growth of large-scale structure. We go over the growth’s physical foundations, and derive its behavior in simple cosmological models. While doing so, we explain how measurements of growth can be used to understand theory. We then review how some of the most mature cosmological probes—galaxy clustering, gravitational lensing, the abundance of clusters of galaxies, cosmic velocities, and cosmic microwave background—can be used to probe the growth of structure. We report the current constraints on growth, which are summarized as measurements of the parameter combination \(f\sigma _8\) as a function of redshift, or else as the mass fluctuation amplitude parameter \(S_8\). We finally illustrate several statistical approaches, ranging from the “growth index” parameterization to more general comparisons of growth and geometry, that can sharply test the standard cosmological model and indicate the presence of modifications to general relativity.

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  1. The same happens during inflation, when the universe is completely vacuum-energy dominated.

  2. Typically tajen to be \(k_\text{piv}=0.05\,\text{Mpc}^{-1}\), which is close to the wavenumber at which the primordial power is best constrained.

  3. There is no way to break this degeneracy between the time dependence of bias and growth with galaxy clustering alone. However, as we discuss in Sect. 3.2, one can use weak gravitational lensing (which altogether avoids galaxy bias), as well as galaxy–galaxy lensing (proportional to bias, rather than bias squared), to break this degeneracy and isolate the growth of structure. Alternatively, simultaneous measurements of the power spectrum and other statistics of galaxy clustering (e.g., the three-point correlation function, or the bispectrum) can help separately constrain galaxy bias and the growth of structure.

  4. Note that \(S_8\) depends on redshift because \(S_8(z)\propto \sigma _8(z)\propto D(z)\); recall Eq. (16). Conventionally, the redshift-dependent part is taken out to quote constraints on \(S_8\) at \(z=0\).

  5. A slightly more accurate formulation replaces the density power spectrum \(P(k, z)\equiv P_{\delta \delta }\) on the right-hand side of Eq. (29) with a combination of \(P_{\delta \delta }\), the velocity power spectrum \(P_{vv}\), and the cross-power \(P_{\delta v}\), as (de la Torre and Guzzo 2012)

    $$\begin{aligned} P(\textbf{k}, z)^{(s)} = \left[ b^2P_{\delta \delta }+ 2bf\mu ^2P_{\delta v} + f^2\mu ^4P_{vv}\right] F(k^2\sigma _v^2\mu ^2). \end{aligned}$$
  6. In the concordance \(\Lambda \)CDM model with 30% matter and 70% dark energy, the energy densities in these two components are equal at \(z\simeq 0.33\).

  7. Also referred to as the “\(\sigma _8\) tension”, as a similar trend is seen in constraints on \(\sigma _8\).

  8. Not to be confused with shear which we discussed in Sect. 3.2.

  9. Of course, it is a good idea to implement the separation between geometry and growth terms that is physically sensible.

  10. We do not allow scale dependence of \(\mu \) and \(\Sigma \) in the following discussion, although such scale dependence may be observable with future data; see, e.g., Hojjati et al. (2014).


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I would like to thank Eric Linder and GongBo Zhao for helpful comments on the manuscript. Over the couple of years over which this manuscript evolved, my work has been supported by NASA under contract 19-ATP19-0058, DOE under Contract No. DE-FG02-95ER40899, NSF under contract AST-1812961, and Leinweber Center for Theoretical Physics at the University of Michigan. I would also like to thank Aspen Center for Physics and Max Planck Institute for Astrophysics for their hospitality.

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Correspondence to Dragan Huterer.

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Huterer, D. Growth of cosmic structure. Astron Astrophys Rev 31, 2 (2023).

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