A Toy Model for the Large-Scale Structure of the Universe

We consider a toy model for the large-scale matter distribution in a static Universe. The model assumes a mass spectrum dN_{ m i}/dm_{ m i} = eta*m_{ m i}^{-alpha} (where alpha and eta are both positive constants) for low-mass particles with m_{ m i} ll M_{ m P}, where M_{ m P} is the Planck mass, and a particle mass-wavelength relation of the form lambda_{ m i} = hbar/(delta_{ m i}*m_{ m i}*c), where delta_{ m i} = eta*m_{ m i}^{gamma} and eta and gamma are both constants. Our model mainly concerns particles with masses far below those in the Standard Model of Particle Physics. We assume that, for such low-mass particles, locality can only be defined on large spatial scales, comparable to or exceeding the particle wavelengths. We use our model to derive the cosmological redshift characteristic of the Standard Model of Cosmology, which becomes a gravitational redshift in our model. We compare the results of our model to empirical data and show that, in order to reproduce the sub-linear form of the observed distance-redshift relation, our model requires alpha < 1+gamma. Taken at face value, the data also suggest that the particle mass function is relatively continuous (i.e., m_{ m i+1}/m_{ m i} < 10^2 for all i and assuming gamma = 0). We further place our toy model in the context of the Friedmann Universe, in order to better understand how a more dynamic version of our model would behave. Finally, we attempt to reconcile the static nature of our toy model with LambdaCDM, and discuss potentially observable distinctions.

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