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We exploit the recent determination of the cosmic star formation rate (SFR) density at high redshifts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>≳</mo><mn>4</mn></mrow></semantics></math></inline-formula> to derive astroparticle constraints on three common dark matter (DM) scenarios alternative to standard cold dark matter (CDM): warm dark matter (WDM), fuzzy dark matter (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>DM) and self-interacting dark matter (SIDM). Our analysis relies on the ultraviolet (UV) luminosity functions measured from blank field surveys by the Hubble Space Telescope out to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>≲</mo><mn>10</mn></mrow></semantics></math></inline-formula> and down to UV magnitudes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mi>UV</mi></msub><mo>≲</mo><mo>−</mo><mn>17</mn></mrow></semantics></math></inline-formula>. We extrapolate these to fainter yet unexplored magnitude ranges and perform abundance matching with the halo mass functions in a given DM scenario, thus, obtaining a redshift-dependent relationship between the UV magnitude and the halo mass. We then computed the cosmic SFR density by integrating the extrapolated UV luminosity functions down to a faint magnitude limit <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>M</mi><mrow><mi>UV</mi></mrow><mi>lim</mi></msubsup></semantics></math></inline-formula>, which is determined via the above abundance matching relationship by two free parameters: the minimum threshold halo mass <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>M</mi><mrow><mi mathvariant="normal">H</mi></mrow><mi>GF</mi></msubsup></semantics></math></inline-formula> for galaxy formation, and the astroparticle quantity <i>X</i> characterizing each DM scenario (namely, particle mass for WDM and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>DM, and kinetic temperature at decoupling <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>T</mi><mi>X</mi></msub></semantics></math></inline-formula> for SIDM). We perform Bayesian inference on such parameters using a Monte Carlo Markov Chain (MCMC) technique by comparing the cosmic SFR density from our approach to the current observational estimates at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>≳</mo><mn>4</mn></mrow></semantics></math></inline-formula>, constraining the WDM particle mass to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>m</mi><mi>X</mi></msub><mo>≈</mo><mn>1</mn><mo>.</mo><msubsup><mn>2</mn><mrow><mo>−</mo><mn>0.4</mn><mspace width="0.166667em"></mspace><mo>(</mo><mo>−</mo><mn>0.5</mn><mo>)</mo></mrow><mrow><mo>+</mo><mn>0.3</mn><mspace width="0.166667em"></mspace><mo>(</mo><mn>11.3</mn><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula> keV, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>DM particle mass to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>m</mi><mi>X</mi></msub><mo>≈</mo><mn>3</mn><mo>.</mo><msubsup><mn>7</mn><mrow><mo>−</mo><mn>0.4</mn><mspace width="0.166667em"></mspace><mo>(</mo><mo>−</mo><mn>0.5</mn><mo>)</mo></mrow><mrow><mo>+</mo><mn>1.8</mn><mspace width="0.166667em"></mspace><mo>(</mo><mo>+</mo><mn>12.9</mn><mo>.</mo><mn>3</mn><mo>)</mo></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>22</mn></mrow></msup></mrow></semantics></math></inline-formula> eV, and the SIDM temperature to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mi>X</mi></msub><mo>≈</mo><mn>0</mn><mo>.</mo><msubsup><mn>21</mn><mrow><mo>−</mo><mn>0.06</mn><mspace width="0.166667em"></mspace><mo>(</mo><mo>−</mo><mn>0.07</mn><mo>)</mo></mrow><mrow><mo>+</mo><mn>0.04</mn><mspace width="0.166667em"></mspace><mo>(</mo><mo>+</mo><mn>1.8</mn><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula> keV at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>68</mn><mo>%</mo></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>95</mn><mo>%</mo></mrow></semantics></math></inline-formula>) confidence level. Finally, we forecast how such constraints will be strengthened by upcoming refined estimates of the cosmic SFR density if the early data on the UV luminosity function at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>≳</mo><mn>10</mn></mrow></semantics></math></inline-formula> from the James Webb Space Telescope (JWST) will be confirmed down to ultra-faint magnitudes.