What Is the Price of Abandoning Dark Matter? Cosmological Constraints on Alternative Gravity Theories

Any successful alternative gravity theory that obviates the need for dark matter must fit our cosmological observations. Measurements of microwave background polarization trace the large-scale baryon velocity field at recombination and show very strong $O(1)$ baryon acoustic oscillations. Measurements of the large-scale structure of galaxies at low redshift show much weaker features in the spectrum. If the alternative gravity theory’s dynamical equations for the growth rate of structure are linear, then the density field growth can be described by a Green’s function: $\delta (\overrightarrow{x},t)=\delta (\overrightarrow{x},t^{\prime })G(x,t,t^{\prime })$. We show that the Green’s function $G(x,t,t^{\prime })$ must have dramatic features that erase the initial baryon oscillations. This implies an acceleration law that changes sign on the $\sim 150\mathrm{Mpc}$ scale. On the other hand, if the alternative gravity theory has a large nonlinear term that couples modes on different scales, then the theory would predict large-scale non-Gaussian features in large-scale structure. These are not seen in the distribution of galaxies nor in the distribution of quasars. No proposed alternative gravity theory for dark matter seems to satisfy these constraints.

Figure 1

Baryon power spectra at $z=0.38$ (solid circles) and $z=1100$ (open circles). The $z=0.38$ data is taken from Ref. [29]. The $z=1100$ points use the polarization data from Refs. [35, 36] and our analytical model, described in the text. The difference between the circles and the spectrum produced by a full treatment by the camb code for each redshift (blue dotted line), assuming the Ref. [10] values, provides an estimate of the error in the approximation we use. The majority of this error is due to the lack of a DM potential driving term in Eq. (6). This would shift the peaks to align with the camb case. The black, dashed line gives the acoustic scale, as given by Ref. [10]. All high-redshift curves are arbitrarily normalized.

Figure 2

Baryon transfer function from $z=1100$ to $z=0.38$ using the data (black line) and camb, assuming $\mathrm{\Lambda }\mathrm{CDM}$ and the values from Ref. [10] (blue, dotted line). The difference between the two shows the limitations of the analytical approximation used to derive the transfer function. The gray region shows the $1\sigma$ error from the data.

Figure 3

Green’s function for the baryon-only transfer function in Fig. 2. The shape depends on the value of the transfer function at all $k$. We extrapolate the transfer function from Fig. 2 using Gaussian process regression and the $\mathrm{\Lambda }\mathrm{CDM}$ baryon transfer function as a prior. The black curve gives the mean result and the blue shows the standard deviation. The errors are dominated by the extrapolation choice, thus we exclude the statistical error bars.