Secondary organic aerosols (SOA) affect visibility, health and global climate. Current chemical transport models cannot represent SOA in the free troposphere. Fog/cloud processing, which is the dominant source of atmospheric sulfate, has been recognized as a missing source of SOA globally. Aqueous photooxidation of water-soluble products (e.g., glyoxal and methylglyoxal) of gas-phase photochemistry yields low-volatility compounds including oxalic acid. When this chemistry takes place in clouds and fogs followed by droplet evaporation (or if this chemistry occurs in aerosol water) then products remain in part in the particle phase, forming SOA. However, current aqueous SOA formation mechanism has not shown how the starting concentrations of precursors and presence of acidic sulfate affect product formation. Aqueous phase photochemical batch reactions were conducted with glyoxal and methylglyoxal at cloud relevant concentrations, using hydrogen peroxide photolysis as the hydroxyl radical (OH) source. Experiments were repeated at higher concentrations and with/without sulfuric acid. Precursors and products were investigated using ion chromatography (IC), electrospray ionization mass spectrometry (ESI-MS), and IC-ESI-MS. Products included carboxylic acids and higher molecular weight compounds, which are major constituents of aerosols. Sulfuric acid shows little effect on product formation. Dilute aqueous chemistry models successfully reproduced product formation for glyoxal and methylglyoxal at cloud relevant conditions, but measurements deviated from predictions from predictions at elevated concentrations. Higher molecular weight products become increasingly important as precursor concentration increases. Aqueous radical-radical reactions provide explanations for observed higher molecular weight products. Additionally, acetic acid is identified as an SOA precursor for the first time. This work provides an improved understanding of aqueous phase dicarbonyl oxidation mechanism and the overall significance of aqueous SOA formation. Kinetic data are made available to regional and global atmospheric models, and the mechanism described in this work will help people to mitigate adverse aerosol effects.